cs270: lecture 1. - people
TRANSCRIPT
![Page 1: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/1.jpg)
CS270: Lecture 1.
1. Overview2. Administration3. Dueling Subroutines: Congestion/Tolls.
![Page 2: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/2.jpg)
Algorithms.
Undergraduate.
This class.
1. Classical.
Flavor of the week?Modern.
2. Cleanly Stated Problems. Shortest Paths, max-flow, MST.
Vaguely stated problems!Address problems; messy or not.
3. Solutions: effective precise bounds!
Ineffective ..imprecise!Analysis sometimes based on modelling world.
4. Techniques: Greedy Dyn. Programming LinearProgramming.
Heuristic, in practice.
5. Techniques tend to be Combinatorial.
Probabilistic, linear algebra methods, continuous.
![Page 3: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/3.jpg)
Algorithms.
Undergraduate.
This class.
1. Classical.
Flavor of the week?Modern.2. Cleanly Stated Problems. Shortest Paths, max-flow, MST.
Vaguely stated problems!Address problems; messy or not.
3. Solutions: effective precise bounds!
Ineffective ..imprecise!Analysis sometimes based on modelling world.
4. Techniques: Greedy Dyn. Programming LinearProgramming.
Heuristic, in practice.
5. Techniques tend to be Combinatorial.
Probabilistic, linear algebra methods, continuous.
![Page 4: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/4.jpg)
Algorithms.
Undergraduate.
This class.
1. Classical.
Flavor of the week?Modern.
2. Cleanly Stated Problems. Shortest Paths, max-flow, MST.
Vaguely stated problems!Address problems; messy or not.3. Solutions: effective precise bounds!
Ineffective ..imprecise!Analysis sometimes based on modelling world.
4. Techniques: Greedy Dyn. Programming LinearProgramming.
Heuristic, in practice.
5. Techniques tend to be Combinatorial.
Probabilistic, linear algebra methods, continuous.
![Page 5: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/5.jpg)
Algorithms.
Undergraduate.
This class.
1. Classical.
Flavor of the week?Modern.
2. Cleanly Stated Problems. Shortest Paths, max-flow, MST.
Vaguely stated problems!Address problems; messy or not.
3. Solutions:
effective precise bounds!
Ineffective ..imprecise!Analysis sometimes based on modelling world.
4. Techniques: Greedy Dyn. Programming LinearProgramming.
Heuristic, in practice.
5. Techniques tend to be Combinatorial.
Probabilistic, linear algebra methods, continuous.
![Page 6: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/6.jpg)
Algorithms.
Undergraduate.
This class.
1. Classical.
Flavor of the week?Modern.
2. Cleanly Stated Problems. Shortest Paths, max-flow, MST.
Vaguely stated problems!Address problems; messy or not.
3. Solutions: effective
precise bounds!
Ineffective ..imprecise!Analysis sometimes based on modelling world.
4. Techniques: Greedy Dyn. Programming LinearProgramming.
Heuristic, in practice.
5. Techniques tend to be Combinatorial.
Probabilistic, linear algebra methods, continuous.
![Page 7: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/7.jpg)
Algorithms.
Undergraduate.
This class.
1. Classical.
Flavor of the week?Modern.
2. Cleanly Stated Problems. Shortest Paths, max-flow, MST.
Vaguely stated problems!Address problems; messy or not.
3. Solutions: effective precise bounds!
Ineffective ..imprecise!Analysis sometimes based on modelling world.4. Techniques: Greedy Dyn. Programming Linear
Programming.
Heuristic, in practice.
5. Techniques tend to be Combinatorial.
Probabilistic, linear algebra methods, continuous.
![Page 8: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/8.jpg)
Algorithms.
Undergraduate.
This class.
1. Classical.
Flavor of the week?Modern.
2. Cleanly Stated Problems. Shortest Paths, max-flow, MST.
Vaguely stated problems!Address problems; messy or not.
3. Solutions: effective precise bounds!
Ineffective ..imprecise!Analysis sometimes based on modelling world.
4. Techniques:
Greedy Dyn. Programming LinearProgramming.
Heuristic, in practice.
5. Techniques tend to be Combinatorial.
Probabilistic, linear algebra methods, continuous.
![Page 9: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/9.jpg)
Algorithms.
Undergraduate.
This class.
1. Classical.
Flavor of the week?Modern.
2. Cleanly Stated Problems. Shortest Paths, max-flow, MST.
Vaguely stated problems!Address problems; messy or not.
3. Solutions: effective precise bounds!
Ineffective ..imprecise!Analysis sometimes based on modelling world.
4. Techniques: Greedy
Dyn. Programming LinearProgramming.
Heuristic, in practice.
5. Techniques tend to be Combinatorial.
Probabilistic, linear algebra methods, continuous.
![Page 10: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/10.jpg)
Algorithms.
Undergraduate.
This class.
1. Classical.
Flavor of the week?Modern.
2. Cleanly Stated Problems. Shortest Paths, max-flow, MST.
Vaguely stated problems!Address problems; messy or not.
3. Solutions: effective precise bounds!
Ineffective ..imprecise!Analysis sometimes based on modelling world.
4. Techniques: Greedy Dyn. Programming
LinearProgramming.
Heuristic, in practice.
5. Techniques tend to be Combinatorial.
Probabilistic, linear algebra methods, continuous.
![Page 11: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/11.jpg)
Algorithms.
Undergraduate.
This class.
1. Classical.
Flavor of the week?Modern.
2. Cleanly Stated Problems. Shortest Paths, max-flow, MST.
Vaguely stated problems!Address problems; messy or not.
3. Solutions: effective precise bounds!
Ineffective ..imprecise!Analysis sometimes based on modelling world.
4. Techniques: Greedy Dyn. Programming LinearProgramming.
Heuristic, in practice.5. Techniques tend to be Combinatorial.
Probabilistic, linear algebra methods, continuous.
![Page 12: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/12.jpg)
Algorithms.
Undergraduate.
This class.
1. Classical.
Flavor of the week?Modern.
2. Cleanly Stated Problems. Shortest Paths, max-flow, MST.
Vaguely stated problems!Address problems; messy or not.
3. Solutions: effective precise bounds!
Ineffective ..imprecise!Analysis sometimes based on modelling world.
4. Techniques: Greedy Dyn. Programming LinearProgramming.
Heuristic, in practice.
5. Techniques tend to be Combinatorial.
Probabilistic, linear algebra methods, continuous.
![Page 13: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/13.jpg)
Algorithms.
Undergraduate.This class.
1. Classical.
Flavor of the week?Modern.
2. Cleanly Stated Problems. Shortest Paths, max-flow, MST.
Vaguely stated problems!Address problems; messy or not.
3. Solutions: effective precise bounds!
Ineffective ..imprecise!Analysis sometimes based on modelling world.
4. Techniques: Greedy Dyn. Programming LinearProgramming.
Heuristic, in practice.
5. Techniques tend to be Combinatorial.
Probabilistic, linear algebra methods, continuous.
![Page 14: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/14.jpg)
Algorithms.
Undergraduate.This class.
1. Classical.Flavor of the week?
Modern.
2. Cleanly Stated Problems. Shortest Paths, max-flow, MST.
Vaguely stated problems!Address problems; messy or not.
3. Solutions: effective precise bounds!
Ineffective ..imprecise!Analysis sometimes based on modelling world.
4. Techniques: Greedy Dyn. Programming LinearProgramming.
Heuristic, in practice.
5. Techniques tend to be Combinatorial.
Probabilistic, linear algebra methods, continuous.
![Page 15: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/15.jpg)
Algorithms.
Undergraduate.This class.
1. Classical.
Flavor of the week?
Modern.2. Cleanly Stated Problems. Shortest Paths, max-flow, MST.
Vaguely stated problems!Address problems; messy or not.
3. Solutions: effective precise bounds!
Ineffective ..imprecise!Analysis sometimes based on modelling world.
4. Techniques: Greedy Dyn. Programming LinearProgramming.
Heuristic, in practice.
5. Techniques tend to be Combinatorial.
Probabilistic, linear algebra methods, continuous.
![Page 16: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/16.jpg)
Algorithms.
Undergraduate.This class.
1. Classical.
Flavor of the week?
Modern.2. Cleanly Stated Problems. Shortest Paths, max-flow, MST.
Vaguely stated problems!
Address problems; messy or not.
3. Solutions: effective precise bounds!
Ineffective ..imprecise!Analysis sometimes based on modelling world.
4. Techniques: Greedy Dyn. Programming LinearProgramming.
Heuristic, in practice.
5. Techniques tend to be Combinatorial.
Probabilistic, linear algebra methods, continuous.
![Page 17: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/17.jpg)
Algorithms.
Undergraduate.This class.
1. Classical.
Flavor of the week?
Modern.2. Cleanly Stated Problems. Shortest Paths, max-flow, MST.
Vaguely stated problems!
Address problems; messy or not.3. Solutions: effective precise bounds!
Ineffective ..imprecise!Analysis sometimes based on modelling world.
4. Techniques: Greedy Dyn. Programming LinearProgramming.
Heuristic, in practice.
5. Techniques tend to be Combinatorial.
Probabilistic, linear algebra methods, continuous.
![Page 18: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/18.jpg)
Algorithms.
Undergraduate.This class.
1. Classical.
Flavor of the week?
Modern.2. Cleanly Stated Problems. Shortest Paths, max-flow, MST.
Vaguely stated problems!
Address problems; messy or not.3. Solutions: effective precise bounds!
Ineffective ..imprecise!
Analysis sometimes based on modelling world.
4. Techniques: Greedy Dyn. Programming LinearProgramming.
Heuristic, in practice.
5. Techniques tend to be Combinatorial.
Probabilistic, linear algebra methods, continuous.
![Page 19: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/19.jpg)
Algorithms.
Undergraduate.This class.
1. Classical.
Flavor of the week?
Modern.2. Cleanly Stated Problems. Shortest Paths, max-flow, MST.
Vaguely stated problems!
Address problems; messy or not.3. Solutions: effective precise bounds!
Ineffective ..imprecise!
Analysis sometimes based on modelling world.4. Techniques: Greedy Dyn. Programming Linear
Programming.
Heuristic, in practice.
5. Techniques tend to be Combinatorial.
Probabilistic, linear algebra methods, continuous.
![Page 20: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/20.jpg)
Algorithms.
Undergraduate.This class.
1. Classical.
Flavor of the week?
Modern.2. Cleanly Stated Problems. Shortest Paths, max-flow, MST.
Vaguely stated problems!
Address problems; messy or not.3. Solutions: effective precise bounds!
Ineffective ..imprecise!
Analysis sometimes based on modelling world.4. Techniques: Greedy Dyn. Programming Linear
Programming.Heuristic, in practice.
5. Techniques tend to be Combinatorial.
Probabilistic, linear algebra methods, continuous.
![Page 21: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/21.jpg)
Algorithms.
Undergraduate.This class.
1. Classical.
Flavor of the week?
Modern.2. Cleanly Stated Problems. Shortest Paths, max-flow, MST.
Vaguely stated problems!
Address problems; messy or not.3. Solutions: effective precise bounds!
Ineffective ..imprecise!
Analysis sometimes based on modelling world.4. Techniques: Greedy Dyn. Programming Linear
Programming.Heuristic, in practice.
5. Techniques tend to be Combinatorial.Probabilistic, linear algebra methods, continuous.
![Page 22: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/22.jpg)
Example Problem: clustering.
I Points: documents, dna, preferences.I Graphs: applications to VLSI, parallel processing, image
segmentation.
![Page 23: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/23.jpg)
Image example.
![Page 24: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/24.jpg)
Image Segmentation
Which region?
Normalized Cut: Find S, which minimizes
w(S,S)
w(S)×w(S).
Ratio Cut: minimizew(S,S)
w(S),
w(S) no more than half the weight. (Minimize cost per unitweight that is removed.)
Either is generally useful!
![Page 25: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/25.jpg)
Image Segmentation
Which region?
Normalized Cut: Find S, which minimizes
w(S,S)
w(S)×w(S).
Ratio Cut: minimizew(S,S)
w(S),
w(S) no more than half the weight. (Minimize cost per unitweight that is removed.)
Either is generally useful!
![Page 26: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/26.jpg)
Image Segmentation
Which region?
Normalized Cut: Find S, which minimizes
w(S,S)
w(S)×w(S).
Ratio Cut: minimizew(S,S)
w(S),
w(S) no more than half the weight. (Minimize cost per unitweight that is removed.)
Either is generally useful!
![Page 27: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/27.jpg)
Image Segmentation
Which region? Normalized Cut: Find S, which minimizes
w(S,S)
w(S)×w(S).
Ratio Cut: minimizew(S,S)
w(S),
w(S) no more than half the weight. (Minimize cost per unitweight that is removed.)
Either is generally useful!
![Page 28: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/28.jpg)
Image Segmentation
Which region? Normalized Cut: Find S, which minimizes
w(S,S)
w(S)×w(S).
Ratio Cut: minimizew(S,S)
w(S),
w(S) no more than half the weight. (Minimize cost per unitweight that is removed.)
Either is generally useful!
![Page 29: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/29.jpg)
Image Segmentation
Which region? Normalized Cut: Find S, which minimizes
w(S,S)
w(S)×w(S).
Ratio Cut: minimizew(S,S)
w(S),
w(S) no more than half the weight. (Minimize cost per unitweight that is removed.)Either is generally useful!
![Page 30: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/30.jpg)
Example: recommendations.Sarah Palin likes True Grit (the old one.)
Sarah Palin doesn’t like The Social Network.Sarah Palin doesn’t like Black Swan.Sarah Palin likes Sarah Palin on Discovery channel.
Hillary Clinton doesn’t like True Grit (the old one.)Hillary Clinton likes The Social Network.Hillary Clinton likes Black Swan.
Should you recommend the discovery channel to Hillary?
What about you?
Are you Hillary? Are you Sarah? A bit of both?
High dimensional data: dimension for each movie.More than three dimensions!Nearest neighbors. Principal Components methods.Topic Models.Reasoning about these methods.
![Page 31: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/31.jpg)
Example: recommendations.Sarah Palin likes True Grit (the old one.)Sarah Palin doesn’t like The Social Network.
Sarah Palin doesn’t like Black Swan.Sarah Palin likes Sarah Palin on Discovery channel.
Hillary Clinton doesn’t like True Grit (the old one.)Hillary Clinton likes The Social Network.Hillary Clinton likes Black Swan.
Should you recommend the discovery channel to Hillary?
What about you?
Are you Hillary? Are you Sarah? A bit of both?
High dimensional data: dimension for each movie.More than three dimensions!Nearest neighbors. Principal Components methods.Topic Models.Reasoning about these methods.
![Page 32: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/32.jpg)
Example: recommendations.Sarah Palin likes True Grit (the old one.)Sarah Palin doesn’t like The Social Network.Sarah Palin doesn’t like Black Swan.
Sarah Palin likes Sarah Palin on Discovery channel.
Hillary Clinton doesn’t like True Grit (the old one.)Hillary Clinton likes The Social Network.Hillary Clinton likes Black Swan.
Should you recommend the discovery channel to Hillary?
What about you?
Are you Hillary? Are you Sarah? A bit of both?
High dimensional data: dimension for each movie.More than three dimensions!Nearest neighbors. Principal Components methods.Topic Models.Reasoning about these methods.
![Page 33: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/33.jpg)
Example: recommendations.Sarah Palin likes True Grit (the old one.)Sarah Palin doesn’t like The Social Network.Sarah Palin doesn’t like Black Swan.Sarah Palin likes Sarah Palin on Discovery channel.
Hillary Clinton doesn’t like True Grit (the old one.)Hillary Clinton likes The Social Network.Hillary Clinton likes Black Swan.
Should you recommend the discovery channel to Hillary?
What about you?
Are you Hillary? Are you Sarah? A bit of both?
High dimensional data: dimension for each movie.More than three dimensions!Nearest neighbors. Principal Components methods.Topic Models.Reasoning about these methods.
![Page 34: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/34.jpg)
Example: recommendations.Sarah Palin likes True Grit (the old one.)Sarah Palin doesn’t like The Social Network.Sarah Palin doesn’t like Black Swan.Sarah Palin likes Sarah Palin on Discovery channel.
Hillary Clinton doesn’t like True Grit (the old one.)
Hillary Clinton likes The Social Network.Hillary Clinton likes Black Swan.
Should you recommend the discovery channel to Hillary?
What about you?
Are you Hillary? Are you Sarah? A bit of both?
High dimensional data: dimension for each movie.More than three dimensions!Nearest neighbors. Principal Components methods.Topic Models.Reasoning about these methods.
![Page 35: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/35.jpg)
Example: recommendations.Sarah Palin likes True Grit (the old one.)Sarah Palin doesn’t like The Social Network.Sarah Palin doesn’t like Black Swan.Sarah Palin likes Sarah Palin on Discovery channel.
Hillary Clinton doesn’t like True Grit (the old one.)Hillary Clinton likes The Social Network.
Hillary Clinton likes Black Swan.
Should you recommend the discovery channel to Hillary?
What about you?
Are you Hillary? Are you Sarah? A bit of both?
High dimensional data: dimension for each movie.More than three dimensions!Nearest neighbors. Principal Components methods.Topic Models.Reasoning about these methods.
![Page 36: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/36.jpg)
Example: recommendations.Sarah Palin likes True Grit (the old one.)Sarah Palin doesn’t like The Social Network.Sarah Palin doesn’t like Black Swan.Sarah Palin likes Sarah Palin on Discovery channel.
Hillary Clinton doesn’t like True Grit (the old one.)Hillary Clinton likes The Social Network.Hillary Clinton likes Black Swan.
Should you recommend the discovery channel to Hillary?
What about you?
Are you Hillary? Are you Sarah? A bit of both?
High dimensional data: dimension for each movie.More than three dimensions!Nearest neighbors. Principal Components methods.Topic Models.Reasoning about these methods.
![Page 37: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/37.jpg)
Example: recommendations.Sarah Palin likes True Grit (the old one.)Sarah Palin doesn’t like The Social Network.Sarah Palin doesn’t like Black Swan.Sarah Palin likes Sarah Palin on Discovery channel.
Hillary Clinton doesn’t like True Grit (the old one.)Hillary Clinton likes The Social Network.Hillary Clinton likes Black Swan.
Should you recommend the discovery channel to Hillary?
What about you?
Are you Hillary? Are you Sarah? A bit of both?
High dimensional data: dimension for each movie.More than three dimensions!Nearest neighbors. Principal Components methods.Topic Models.Reasoning about these methods.
![Page 38: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/38.jpg)
Example: recommendations.Sarah Palin likes True Grit (the old one.)Sarah Palin doesn’t like The Social Network.Sarah Palin doesn’t like Black Swan.Sarah Palin likes Sarah Palin on Discovery channel.
Hillary Clinton doesn’t like True Grit (the old one.)Hillary Clinton likes The Social Network.Hillary Clinton likes Black Swan.
Should you recommend the discovery channel to Hillary?
What about you?
Are you Hillary? Are you Sarah? A bit of both?
High dimensional data: dimension for each movie.More than three dimensions!Nearest neighbors. Principal Components methods.Topic Models.Reasoning about these methods.
![Page 39: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/39.jpg)
Example: recommendations.Sarah Palin likes True Grit (the old one.)Sarah Palin doesn’t like The Social Network.Sarah Palin doesn’t like Black Swan.Sarah Palin likes Sarah Palin on Discovery channel.
Hillary Clinton doesn’t like True Grit (the old one.)Hillary Clinton likes The Social Network.Hillary Clinton likes Black Swan.
Should you recommend the discovery channel to Hillary?
What about you?
Are you Hillary?
Are you Sarah? A bit of both?
High dimensional data: dimension for each movie.More than three dimensions!Nearest neighbors. Principal Components methods.Topic Models.Reasoning about these methods.
![Page 40: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/40.jpg)
Example: recommendations.Sarah Palin likes True Grit (the old one.)Sarah Palin doesn’t like The Social Network.Sarah Palin doesn’t like Black Swan.Sarah Palin likes Sarah Palin on Discovery channel.
Hillary Clinton doesn’t like True Grit (the old one.)Hillary Clinton likes The Social Network.Hillary Clinton likes Black Swan.
Should you recommend the discovery channel to Hillary?
What about you?
Are you Hillary? Are you Sarah?
A bit of both?
High dimensional data: dimension for each movie.More than three dimensions!Nearest neighbors. Principal Components methods.Topic Models.Reasoning about these methods.
![Page 41: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/41.jpg)
Example: recommendations.Sarah Palin likes True Grit (the old one.)Sarah Palin doesn’t like The Social Network.Sarah Palin doesn’t like Black Swan.Sarah Palin likes Sarah Palin on Discovery channel.
Hillary Clinton doesn’t like True Grit (the old one.)Hillary Clinton likes The Social Network.Hillary Clinton likes Black Swan.
Should you recommend the discovery channel to Hillary?
What about you?
Are you Hillary? Are you Sarah? A bit of both?
High dimensional data: dimension for each movie.More than three dimensions!Nearest neighbors. Principal Components methods.Topic Models.Reasoning about these methods.
![Page 42: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/42.jpg)
Example: recommendations.Sarah Palin likes True Grit (the old one.)Sarah Palin doesn’t like The Social Network.Sarah Palin doesn’t like Black Swan.Sarah Palin likes Sarah Palin on Discovery channel.
Hillary Clinton doesn’t like True Grit (the old one.)Hillary Clinton likes The Social Network.Hillary Clinton likes Black Swan.
Should you recommend the discovery channel to Hillary?
What about you?
Are you Hillary? Are you Sarah? A bit of both?
High dimensional data: dimension for each movie.
More than three dimensions!Nearest neighbors. Principal Components methods.Topic Models.Reasoning about these methods.
![Page 43: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/43.jpg)
Example: recommendations.Sarah Palin likes True Grit (the old one.)Sarah Palin doesn’t like The Social Network.Sarah Palin doesn’t like Black Swan.Sarah Palin likes Sarah Palin on Discovery channel.
Hillary Clinton doesn’t like True Grit (the old one.)Hillary Clinton likes The Social Network.Hillary Clinton likes Black Swan.
Should you recommend the discovery channel to Hillary?
What about you?
Are you Hillary? Are you Sarah? A bit of both?
High dimensional data: dimension for each movie.More than three dimensions!
Nearest neighbors. Principal Components methods.Topic Models.Reasoning about these methods.
![Page 44: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/44.jpg)
Example: recommendations.Sarah Palin likes True Grit (the old one.)Sarah Palin doesn’t like The Social Network.Sarah Palin doesn’t like Black Swan.Sarah Palin likes Sarah Palin on Discovery channel.
Hillary Clinton doesn’t like True Grit (the old one.)Hillary Clinton likes The Social Network.Hillary Clinton likes Black Swan.
Should you recommend the discovery channel to Hillary?
What about you?
Are you Hillary? Are you Sarah? A bit of both?
High dimensional data: dimension for each movie.More than three dimensions!Nearest neighbors.
Principal Components methods.Topic Models.Reasoning about these methods.
![Page 45: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/45.jpg)
Example: recommendations.Sarah Palin likes True Grit (the old one.)Sarah Palin doesn’t like The Social Network.Sarah Palin doesn’t like Black Swan.Sarah Palin likes Sarah Palin on Discovery channel.
Hillary Clinton doesn’t like True Grit (the old one.)Hillary Clinton likes The Social Network.Hillary Clinton likes Black Swan.
Should you recommend the discovery channel to Hillary?
What about you?
Are you Hillary? Are you Sarah? A bit of both?
High dimensional data: dimension for each movie.More than three dimensions!Nearest neighbors. Principal Components methods.
Topic Models.Reasoning about these methods.
![Page 46: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/46.jpg)
Example: recommendations.Sarah Palin likes True Grit (the old one.)Sarah Palin doesn’t like The Social Network.Sarah Palin doesn’t like Black Swan.Sarah Palin likes Sarah Palin on Discovery channel.
Hillary Clinton doesn’t like True Grit (the old one.)Hillary Clinton likes The Social Network.Hillary Clinton likes Black Swan.
Should you recommend the discovery channel to Hillary?
What about you?
Are you Hillary? Are you Sarah? A bit of both?
High dimensional data: dimension for each movie.More than three dimensions!Nearest neighbors. Principal Components methods.Topic Models.
Reasoning about these methods.
![Page 47: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/47.jpg)
Example: recommendations.Sarah Palin likes True Grit (the old one.)Sarah Palin doesn’t like The Social Network.Sarah Palin doesn’t like Black Swan.Sarah Palin likes Sarah Palin on Discovery channel.
Hillary Clinton doesn’t like True Grit (the old one.)Hillary Clinton likes The Social Network.Hillary Clinton likes Black Swan.
Should you recommend the discovery channel to Hillary?
What about you?
Are you Hillary? Are you Sarah? A bit of both?
High dimensional data: dimension for each movie.More than three dimensions!Nearest neighbors. Principal Components methods.Topic Models.Reasoning about these methods.
![Page 48: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/48.jpg)
Linear Systems.Revolution!
Physical Simulation. Airflow.
Solve Ax = b.
How long?
n×n matrix A.
Middle School: substitution, adding equations ...Time: O(n3).
Now: O(m). Hmmm. What’s that tilde?
Techniques:Relate graph theory to matrix properties.Dense matrix (graph) to sparse matrix (graph).Approximating distances by trees.Electrical networks analysis.
Combinatorial Applications: Better Max Flow!
![Page 49: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/49.jpg)
Linear Systems.Revolution!
Physical Simulation. Airflow.
Solve Ax = b.
How long?
n×n matrix A.
Middle School: substitution, adding equations ...Time: O(n3).
Now: O(m). Hmmm. What’s that tilde?
Techniques:Relate graph theory to matrix properties.Dense matrix (graph) to sparse matrix (graph).Approximating distances by trees.Electrical networks analysis.
Combinatorial Applications: Better Max Flow!
![Page 50: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/50.jpg)
Linear Systems.Revolution!
Physical Simulation. Airflow.
Solve Ax = b.
How long?
n×n matrix A.
Middle School: substitution, adding equations ...Time: O(n3).
Now: O(m). Hmmm. What’s that tilde?
Techniques:Relate graph theory to matrix properties.Dense matrix (graph) to sparse matrix (graph).Approximating distances by trees.Electrical networks analysis.
Combinatorial Applications: Better Max Flow!
![Page 51: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/51.jpg)
Linear Systems.Revolution!
Physical Simulation. Airflow.
Solve Ax = b.
How long?
n×n matrix A.
Middle School: substitution, adding equations ...Time: O(n3).
Now: O(m). Hmmm. What’s that tilde?
Techniques:Relate graph theory to matrix properties.Dense matrix (graph) to sparse matrix (graph).Approximating distances by trees.Electrical networks analysis.
Combinatorial Applications: Better Max Flow!
![Page 52: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/52.jpg)
Linear Systems.Revolution!
Physical Simulation. Airflow.
Solve Ax = b.
How long?
n×n matrix A.
Middle School: substitution, adding equations ...Time: O(n3).
Now: O(m). Hmmm. What’s that tilde?
Techniques:Relate graph theory to matrix properties.Dense matrix (graph) to sparse matrix (graph).Approximating distances by trees.Electrical networks analysis.
Combinatorial Applications: Better Max Flow!
![Page 53: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/53.jpg)
Linear Systems.Revolution!
Physical Simulation.
Airflow.
Solve Ax = b.
How long?
n×n matrix A.
Middle School: substitution, adding equations ...Time: O(n3).
Now: O(m). Hmmm. What’s that tilde?
Techniques:Relate graph theory to matrix properties.Dense matrix (graph) to sparse matrix (graph).Approximating distances by trees.Electrical networks analysis.
Combinatorial Applications: Better Max Flow!
![Page 54: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/54.jpg)
Linear Systems.Revolution!
Physical Simulation. Airflow.
Solve Ax = b.
How long?
n×n matrix A.
Middle School: substitution, adding equations ...Time: O(n3).
Now: O(m). Hmmm. What’s that tilde?
Techniques:Relate graph theory to matrix properties.Dense matrix (graph) to sparse matrix (graph).Approximating distances by trees.Electrical networks analysis.
Combinatorial Applications: Better Max Flow!
![Page 55: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/55.jpg)
Linear Systems.Revolution!
Physical Simulation. Airflow.
Solve Ax = b.
How long?
n×n matrix A.
Middle School: substitution, adding equations ...Time: O(n3).
Now: O(m). Hmmm. What’s that tilde?
Techniques:Relate graph theory to matrix properties.Dense matrix (graph) to sparse matrix (graph).Approximating distances by trees.Electrical networks analysis.
Combinatorial Applications: Better Max Flow!
![Page 56: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/56.jpg)
Linear Systems.Revolution!
Physical Simulation. Airflow.
Solve Ax = b.
How long?
n×n matrix A.
Middle School: substitution, adding equations ...Time: O(n3).
Now: O(m). Hmmm. What’s that tilde?
Techniques:Relate graph theory to matrix properties.Dense matrix (graph) to sparse matrix (graph).Approximating distances by trees.Electrical networks analysis.
Combinatorial Applications: Better Max Flow!
![Page 57: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/57.jpg)
Linear Systems.Revolution!
Physical Simulation. Airflow.
Solve Ax = b.
How long?
n×n matrix A.
Middle School: substitution, adding equations ...Time: O(n3).
Now: O(m). Hmmm. What’s that tilde?
Techniques:Relate graph theory to matrix properties.Dense matrix (graph) to sparse matrix (graph).Approximating distances by trees.Electrical networks analysis.
Combinatorial Applications: Better Max Flow!
![Page 58: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/58.jpg)
Linear Systems.Revolution!
Physical Simulation. Airflow.
Solve Ax = b.
How long?
n×n matrix A.
Middle School:
substitution, adding equations ...Time: O(n3).
Now: O(m). Hmmm. What’s that tilde?
Techniques:Relate graph theory to matrix properties.Dense matrix (graph) to sparse matrix (graph).Approximating distances by trees.Electrical networks analysis.
Combinatorial Applications: Better Max Flow!
![Page 59: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/59.jpg)
Linear Systems.Revolution!
Physical Simulation. Airflow.
Solve Ax = b.
How long?
n×n matrix A.
Middle School: substitution,
adding equations ...Time: O(n3).
Now: O(m). Hmmm. What’s that tilde?
Techniques:Relate graph theory to matrix properties.Dense matrix (graph) to sparse matrix (graph).Approximating distances by trees.Electrical networks analysis.
Combinatorial Applications: Better Max Flow!
![Page 60: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/60.jpg)
Linear Systems.Revolution!
Physical Simulation. Airflow.
Solve Ax = b.
How long?
n×n matrix A.
Middle School: substitution, adding equations ...
Time: O(n3).
Now: O(m). Hmmm. What’s that tilde?
Techniques:Relate graph theory to matrix properties.Dense matrix (graph) to sparse matrix (graph).Approximating distances by trees.Electrical networks analysis.
Combinatorial Applications: Better Max Flow!
![Page 61: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/61.jpg)
Linear Systems.Revolution!
Physical Simulation. Airflow.
Solve Ax = b.
How long?
n×n matrix A.
Middle School: substitution, adding equations ...Time: O(n3).
Now: O(m). Hmmm. What’s that tilde?
Techniques:Relate graph theory to matrix properties.Dense matrix (graph) to sparse matrix (graph).Approximating distances by trees.Electrical networks analysis.
Combinatorial Applications: Better Max Flow!
![Page 62: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/62.jpg)
Linear Systems.Revolution!
Physical Simulation. Airflow.
Solve Ax = b.
How long?
n×n matrix A.
Middle School: substitution, adding equations ...Time: O(n3).
Now: O(m). Hmmm. What’s that tilde?
Techniques:Relate graph theory to matrix properties.Dense matrix (graph) to sparse matrix (graph).Approximating distances by trees.Electrical networks analysis.
Combinatorial Applications: Better Max Flow!
![Page 63: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/63.jpg)
Linear Systems.Revolution!
Physical Simulation. Airflow.
Solve Ax = b.
How long?
n×n matrix A.
Middle School: substitution, adding equations ...Time: O(n3).
Now: O(m).
Hmmm. What’s that tilde?
Techniques:Relate graph theory to matrix properties.Dense matrix (graph) to sparse matrix (graph).Approximating distances by trees.Electrical networks analysis.
Combinatorial Applications: Better Max Flow!
![Page 64: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/64.jpg)
Linear Systems.Revolution!
Physical Simulation. Airflow.
Solve Ax = b.
How long?
n×n matrix A.
Middle School: substitution, adding equations ...Time: O(n3).
Now: O(m). Hmmm.
What’s that tilde?
Techniques:Relate graph theory to matrix properties.Dense matrix (graph) to sparse matrix (graph).Approximating distances by trees.Electrical networks analysis.
Combinatorial Applications: Better Max Flow!
![Page 65: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/65.jpg)
Linear Systems.Revolution!
Physical Simulation. Airflow.
Solve Ax = b.
How long?
n×n matrix A.
Middle School: substitution, adding equations ...Time: O(n3).
Now: O(m). Hmmm. What’s that tilde?
Techniques:Relate graph theory to matrix properties.Dense matrix (graph) to sparse matrix (graph).Approximating distances by trees.Electrical networks analysis.
Combinatorial Applications: Better Max Flow!
![Page 66: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/66.jpg)
Linear Systems.Revolution!
Physical Simulation. Airflow.
Solve Ax = b.
How long?
n×n matrix A.
Middle School: substitution, adding equations ...Time: O(n3).
Now: O(m). Hmmm. What’s that tilde?
Techniques:
Relate graph theory to matrix properties.Dense matrix (graph) to sparse matrix (graph).Approximating distances by trees.Electrical networks analysis.
Combinatorial Applications: Better Max Flow!
![Page 67: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/67.jpg)
Linear Systems.Revolution!
Physical Simulation. Airflow.
Solve Ax = b.
How long?
n×n matrix A.
Middle School: substitution, adding equations ...Time: O(n3).
Now: O(m). Hmmm. What’s that tilde?
Techniques:Relate graph theory to matrix properties.
Dense matrix (graph) to sparse matrix (graph).Approximating distances by trees.Electrical networks analysis.
Combinatorial Applications: Better Max Flow!
![Page 68: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/68.jpg)
Linear Systems.Revolution!
Physical Simulation. Airflow.
Solve Ax = b.
How long?
n×n matrix A.
Middle School: substitution, adding equations ...Time: O(n3).
Now: O(m). Hmmm. What’s that tilde?
Techniques:Relate graph theory to matrix properties.Dense matrix (graph) to sparse matrix (graph).
Approximating distances by trees.Electrical networks analysis.
Combinatorial Applications: Better Max Flow!
![Page 69: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/69.jpg)
Linear Systems.Revolution!
Physical Simulation. Airflow.
Solve Ax = b.
How long?
n×n matrix A.
Middle School: substitution, adding equations ...Time: O(n3).
Now: O(m). Hmmm. What’s that tilde?
Techniques:Relate graph theory to matrix properties.Dense matrix (graph) to sparse matrix (graph).Approximating distances by trees.
Electrical networks analysis.
Combinatorial Applications: Better Max Flow!
![Page 70: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/70.jpg)
Linear Systems.Revolution!
Physical Simulation. Airflow.
Solve Ax = b.
How long?
n×n matrix A.
Middle School: substitution, adding equations ...Time: O(n3).
Now: O(m). Hmmm. What’s that tilde?
Techniques:Relate graph theory to matrix properties.Dense matrix (graph) to sparse matrix (graph).Approximating distances by trees.Electrical networks analysis.
Combinatorial Applications: Better Max Flow!
![Page 71: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/71.jpg)
Linear Systems.Revolution!
Physical Simulation. Airflow.
Solve Ax = b.
How long?
n×n matrix A.
Middle School: substitution, adding equations ...Time: O(n3).
Now: O(m). Hmmm. What’s that tilde?
Techniques:Relate graph theory to matrix properties.Dense matrix (graph) to sparse matrix (graph).Approximating distances by trees.Electrical networks analysis.
Combinatorial Applications:
Better Max Flow!
![Page 72: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/72.jpg)
Linear Systems.Revolution!
Physical Simulation. Airflow.
Solve Ax = b.
How long?
n×n matrix A.
Middle School: substitution, adding equations ...Time: O(n3).
Now: O(m). Hmmm. What’s that tilde?
Techniques:Relate graph theory to matrix properties.Dense matrix (graph) to sparse matrix (graph).Approximating distances by trees.Electrical networks analysis.
Combinatorial Applications: Better Max Flow!
![Page 73: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/73.jpg)
Other Algorithmic Techiniques
Sketching:Large stream of data: a1,a2, . . .
Find digest.Graphs: Sparse graph.Data: average, statistics.Points: center point, k -medians, .
High Dimensional optimization.Gradient Descent. Convexity.
Linear Algebra.Eigenvalues.Semidefinite Programming.
Dueling Subroutines. Duality.Lower bounder, upper bounder.Upper uses lower’s evidence to find better solutions.Lower uses upper’s evidence to prove better lower bounds.
![Page 74: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/74.jpg)
Other Algorithmic Techiniques
Sketching:Large stream of data: a1,a2, . . .
Find digest.
Graphs: Sparse graph.Data: average, statistics.Points: center point, k -medians, .
High Dimensional optimization.Gradient Descent. Convexity.
Linear Algebra.Eigenvalues.Semidefinite Programming.
Dueling Subroutines. Duality.Lower bounder, upper bounder.Upper uses lower’s evidence to find better solutions.Lower uses upper’s evidence to prove better lower bounds.
![Page 75: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/75.jpg)
Other Algorithmic Techiniques
Sketching:Large stream of data: a1,a2, . . .
Find digest.Graphs: Sparse graph.
Data: average, statistics.Points: center point, k -medians, .
High Dimensional optimization.Gradient Descent. Convexity.
Linear Algebra.Eigenvalues.Semidefinite Programming.
Dueling Subroutines. Duality.Lower bounder, upper bounder.Upper uses lower’s evidence to find better solutions.Lower uses upper’s evidence to prove better lower bounds.
![Page 76: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/76.jpg)
Other Algorithmic Techiniques
Sketching:Large stream of data: a1,a2, . . .
Find digest.Graphs: Sparse graph.Data: average, statistics.
Points: center point, k -medians, .
High Dimensional optimization.Gradient Descent. Convexity.
Linear Algebra.Eigenvalues.Semidefinite Programming.
Dueling Subroutines. Duality.Lower bounder, upper bounder.Upper uses lower’s evidence to find better solutions.Lower uses upper’s evidence to prove better lower bounds.
![Page 77: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/77.jpg)
Other Algorithmic Techiniques
Sketching:Large stream of data: a1,a2, . . .
Find digest.Graphs: Sparse graph.Data: average, statistics.Points: center point, k -medians, .
High Dimensional optimization.Gradient Descent. Convexity.
Linear Algebra.Eigenvalues.Semidefinite Programming.
Dueling Subroutines. Duality.Lower bounder, upper bounder.Upper uses lower’s evidence to find better solutions.Lower uses upper’s evidence to prove better lower bounds.
![Page 78: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/78.jpg)
Other Algorithmic Techiniques
Sketching:Large stream of data: a1,a2, . . .
Find digest.Graphs: Sparse graph.Data: average, statistics.Points: center point, k -medians, .
High Dimensional optimization.Gradient Descent. Convexity.
Linear Algebra.Eigenvalues.Semidefinite Programming.
Dueling Subroutines. Duality.Lower bounder, upper bounder.Upper uses lower’s evidence to find better solutions.Lower uses upper’s evidence to prove better lower bounds.
![Page 79: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/79.jpg)
Other Algorithmic Techiniques
Sketching:Large stream of data: a1,a2, . . .
Find digest.Graphs: Sparse graph.Data: average, statistics.Points: center point, k -medians, .
High Dimensional optimization.
Gradient Descent. Convexity.
Linear Algebra.Eigenvalues.Semidefinite Programming.
Dueling Subroutines. Duality.Lower bounder, upper bounder.Upper uses lower’s evidence to find better solutions.Lower uses upper’s evidence to prove better lower bounds.
![Page 80: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/80.jpg)
Other Algorithmic Techiniques
Sketching:Large stream of data: a1,a2, . . .
Find digest.Graphs: Sparse graph.Data: average, statistics.Points: center point, k -medians, .
High Dimensional optimization.Gradient Descent. Convexity.
Linear Algebra.Eigenvalues.Semidefinite Programming.
Dueling Subroutines. Duality.Lower bounder, upper bounder.Upper uses lower’s evidence to find better solutions.Lower uses upper’s evidence to prove better lower bounds.
![Page 81: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/81.jpg)
Other Algorithmic Techiniques
Sketching:Large stream of data: a1,a2, . . .
Find digest.Graphs: Sparse graph.Data: average, statistics.Points: center point, k -medians, .
High Dimensional optimization.Gradient Descent. Convexity.
Linear Algebra.
Eigenvalues.Semidefinite Programming.
Dueling Subroutines. Duality.Lower bounder, upper bounder.Upper uses lower’s evidence to find better solutions.Lower uses upper’s evidence to prove better lower bounds.
![Page 82: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/82.jpg)
Other Algorithmic Techiniques
Sketching:Large stream of data: a1,a2, . . .
Find digest.Graphs: Sparse graph.Data: average, statistics.Points: center point, k -medians, .
High Dimensional optimization.Gradient Descent. Convexity.
Linear Algebra.Eigenvalues.
Semidefinite Programming.
Dueling Subroutines. Duality.Lower bounder, upper bounder.Upper uses lower’s evidence to find better solutions.Lower uses upper’s evidence to prove better lower bounds.
![Page 83: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/83.jpg)
Other Algorithmic Techiniques
Sketching:Large stream of data: a1,a2, . . .
Find digest.Graphs: Sparse graph.Data: average, statistics.Points: center point, k -medians, .
High Dimensional optimization.Gradient Descent. Convexity.
Linear Algebra.Eigenvalues.Semidefinite Programming.
Dueling Subroutines. Duality.Lower bounder, upper bounder.Upper uses lower’s evidence to find better solutions.Lower uses upper’s evidence to prove better lower bounds.
![Page 84: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/84.jpg)
Other Algorithmic Techiniques
Sketching:Large stream of data: a1,a2, . . .
Find digest.Graphs: Sparse graph.Data: average, statistics.Points: center point, k -medians, .
High Dimensional optimization.Gradient Descent. Convexity.
Linear Algebra.Eigenvalues.Semidefinite Programming.
Dueling Subroutines. Duality.
Lower bounder, upper bounder.Upper uses lower’s evidence to find better solutions.Lower uses upper’s evidence to prove better lower bounds.
![Page 85: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/85.jpg)
Other Algorithmic Techiniques
Sketching:Large stream of data: a1,a2, . . .
Find digest.Graphs: Sparse graph.Data: average, statistics.Points: center point, k -medians, .
High Dimensional optimization.Gradient Descent. Convexity.
Linear Algebra.Eigenvalues.Semidefinite Programming.
Dueling Subroutines. Duality.Lower bounder, upper bounder.
Upper uses lower’s evidence to find better solutions.Lower uses upper’s evidence to prove better lower bounds.
![Page 86: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/86.jpg)
Other Algorithmic Techiniques
Sketching:Large stream of data: a1,a2, . . .
Find digest.Graphs: Sparse graph.Data: average, statistics.Points: center point, k -medians, .
High Dimensional optimization.Gradient Descent. Convexity.
Linear Algebra.Eigenvalues.Semidefinite Programming.
Dueling Subroutines. Duality.Lower bounder, upper bounder.Upper uses lower’s evidence to find better solutions.
Lower uses upper’s evidence to prove better lower bounds.
![Page 87: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/87.jpg)
Other Algorithmic Techiniques
Sketching:Large stream of data: a1,a2, . . .
Find digest.Graphs: Sparse graph.Data: average, statistics.Points: center point, k -medians, .
High Dimensional optimization.Gradient Descent. Convexity.
Linear Algebra.Eigenvalues.Semidefinite Programming.
Dueling Subroutines. Duality.Lower bounder, upper bounder.Upper uses lower’s evidence to find better solutions.Lower uses upper’s evidence to prove better lower bounds.
![Page 88: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/88.jpg)
Other Algorithmic Techiniques
Sketching:Large stream of data: a1,a2, . . .
Find digest.Graphs: Sparse graph.Data: average, statistics.Points: center point, k -medians, .
High Dimensional optimization.Gradient Descent. Convexity.
Linear Algebra.Eigenvalues.Semidefinite Programming.
Dueling Subroutines. Duality.Lower bounder, upper bounder.Upper uses lower’s evidence to find better solutions.Lower uses upper’s evidence to prove better lower bounds.
![Page 89: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/89.jpg)
CS270: Administration.
1. Staff:Satish Rao
Benjamin Weitz2. Piazza. Log in! Pay attention to “bypass email preferences”
especially.3. Assessment.
3.1 Homeworks (40%).Homework 1 out tonight/tomorrow.
3.2 1 Takehome Midterm (25 %)3.3 Project (35%)
Groups of 2 or 3.Connect research to class.Or explore/digest a topic from class.
3.4 No Discussion this week.
![Page 90: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/90.jpg)
CS270: Administration.
1. Staff:Satish RaoBenjamin Weitz
2. Piazza. Log in! Pay attention to “bypass email preferences”especially.
3. Assessment.3.1 Homeworks (40%).
Homework 1 out tonight/tomorrow.3.2 1 Takehome Midterm (25 %)3.3 Project (35%)
Groups of 2 or 3.Connect research to class.Or explore/digest a topic from class.
3.4 No Discussion this week.
![Page 91: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/91.jpg)
CS270: Administration.
1. Staff:Satish RaoBenjamin Weitz
2. Piazza. Log in! Pay attention to “bypass email preferences”especially.
3. Assessment.3.1 Homeworks (40%).
Homework 1 out tonight/tomorrow.
3.2 1 Takehome Midterm (25 %)3.3 Project (35%)
Groups of 2 or 3.Connect research to class.Or explore/digest a topic from class.
3.4 No Discussion this week.
![Page 92: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/92.jpg)
CS270: Administration.
1. Staff:Satish RaoBenjamin Weitz
2. Piazza. Log in! Pay attention to “bypass email preferences”especially.
3. Assessment.3.1 Homeworks (40%).
Homework 1 out tonight/tomorrow.3.2 1 Takehome Midterm (25 %)
3.3 Project (35%)Groups of 2 or 3.Connect research to class.Or explore/digest a topic from class.
3.4 No Discussion this week.
![Page 93: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/93.jpg)
CS270: Administration.
1. Staff:Satish RaoBenjamin Weitz
2. Piazza. Log in! Pay attention to “bypass email preferences”especially.
3. Assessment.3.1 Homeworks (40%).
Homework 1 out tonight/tomorrow.3.2 1 Takehome Midterm (25 %)3.3 Project (35%)
Groups of 2 or 3.Connect research to class.Or explore/digest a topic from class.
3.4 No Discussion this week.
![Page 94: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/94.jpg)
CS270: Administration.
1. Staff:Satish RaoBenjamin Weitz
2. Piazza. Log in! Pay attention to “bypass email preferences”especially.
3. Assessment.3.1 Homeworks (40%).
Homework 1 out tonight/tomorrow.3.2 1 Takehome Midterm (25 %)3.3 Project (35%)
Groups of 2 or 3.
Connect research to class.Or explore/digest a topic from class.
3.4 No Discussion this week.
![Page 95: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/95.jpg)
CS270: Administration.
1. Staff:Satish RaoBenjamin Weitz
2. Piazza. Log in! Pay attention to “bypass email preferences”especially.
3. Assessment.3.1 Homeworks (40%).
Homework 1 out tonight/tomorrow.3.2 1 Takehome Midterm (25 %)3.3 Project (35%)
Groups of 2 or 3.Connect research to class.
Or explore/digest a topic from class.3.4 No Discussion this week.
![Page 96: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/96.jpg)
CS270: Administration.
1. Staff:Satish RaoBenjamin Weitz
2. Piazza. Log in! Pay attention to “bypass email preferences”especially.
3. Assessment.3.1 Homeworks (40%).
Homework 1 out tonight/tomorrow.3.2 1 Takehome Midterm (25 %)3.3 Project (35%)
Groups of 2 or 3.Connect research to class.Or explore/digest a topic from class.
3.4 No Discussion this week.
![Page 97: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/97.jpg)
Path Routing.
Given G = (V ,E), (s1, t1), . . . ,(sk , tk ), find a set of k pathsconnecting si and ti and minimize max load on any edge.
s1 t1
s2
t2
s3
t3
Value: 3—————
Value: 2
![Page 98: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/98.jpg)
Path Routing.
Given G = (V ,E), (s1, t1), . . . ,(sk , tk ), find a set of k pathsconnecting si and ti and minimize max load on any edge.
s1 t1
s2
t2
s3
t3
Value: 3—————
Value: 2
![Page 99: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/99.jpg)
Path Routing.
Given G = (V ,E), (s1, t1), . . . ,(sk , tk ), find a set of k pathsconnecting si and ti and minimize max load on any edge.
s1 t1
s2
t2
s3
t3
Value: 3—————
Value: 2
![Page 100: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/100.jpg)
Path Routing.
Given G = (V ,E), (s1, t1), . . . ,(sk , tk ), find a set of k pathsconnecting si and ti and minimize max load on any edge.
s1 t1
s2
t2
s3
t3
Value: 3—————
Value: 2
![Page 101: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/101.jpg)
Path Routing.
Given G = (V ,E), (s1, t1), . . . ,(sk , tk ), find a set of k pathsconnecting si and ti and minimize max load on any edge.
s1 t1
s2
t2
s3
t3
Value: 3—————
Value: 2
![Page 102: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/102.jpg)
Path Routing.
Given G = (V ,E), (s1, t1), . . . ,(sk , tk ), find a set of k pathsconnecting si and ti and minimize max load on any edge.
s1 t1
s2
t2
s3
t3
Value: 3
—————
Value: 2
![Page 103: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/103.jpg)
Path Routing.
Given G = (V ,E), (s1, t1), . . . ,(sk , tk ), find a set of k pathsconnecting si and ti and minimize max load on any edge.
s1 t1
s2
t2
s3
t3
Value: 3
—————
Value: 2
![Page 104: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/104.jpg)
Path Routing.
Given G = (V ,E), (s1, t1), . . . ,(sk , tk ), find a set of k pathsconnecting si and ti and minimize max load on any edge.
s1 t1
s2
t2
s3
t3
Value: 3—————
Value: 2
![Page 105: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/105.jpg)
Path Routing.
Given G = (V ,E), (s1, t1), . . . ,(sk , tk ), find a set of k pathsconnecting si and ti and minimize max load on any edge.
s1 t1
s2
t2
s3
t3
Value: 3—————
Value: 2
![Page 106: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/106.jpg)
Terminology
Routing: Paths p1,p2, . . . ,pk , pi connects si and ti .
Congestion of edge, e: c(e)number of paths in routing that contain e.
Congestion of routing: maximum congestion of any edge.
Find routing that minimizes congestion (or maximumcongestion.)
![Page 107: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/107.jpg)
Terminology
Routing: Paths p1,p2, . . . ,pk , pi connects si and ti .
Congestion of edge, e: c(e)
number of paths in routing that contain e.
Congestion of routing: maximum congestion of any edge.
Find routing that minimizes congestion (or maximumcongestion.)
![Page 108: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/108.jpg)
Terminology
Routing: Paths p1,p2, . . . ,pk , pi connects si and ti .
Congestion of edge, e: c(e)number of paths in routing that contain e.
Congestion of routing: maximum congestion of any edge.
Find routing that minimizes congestion (or maximumcongestion.)
![Page 109: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/109.jpg)
Terminology
Routing: Paths p1,p2, . . . ,pk , pi connects si and ti .
Congestion of edge, e: c(e)number of paths in routing that contain e.
Congestion of routing:
maximum congestion of any edge.
Find routing that minimizes congestion (or maximumcongestion.)
![Page 110: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/110.jpg)
Terminology
Routing: Paths p1,p2, . . . ,pk , pi connects si and ti .
Congestion of edge, e: c(e)number of paths in routing that contain e.
Congestion of routing: maximum congestion of any edge.
Find routing that minimizes congestion (or maximumcongestion.)
![Page 111: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/111.jpg)
Terminology
Routing: Paths p1,p2, . . . ,pk , pi connects si and ti .
Congestion of edge, e: c(e)number of paths in routing that contain e.
Congestion of routing: maximum congestion of any edge.
Find routing that minimizes congestion (or maximumcongestion.)
![Page 112: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/112.jpg)
Algorithms?Route along any path.
Feasible...but is it as good as possible?How far from optimal could it be?(A) It is optimal!(B) A factor of two.(C) A factor of k , in general.
(C)
s1s2sk
t1t2tk
· · ·
· · · Value: k .
Opt: 1.
Stupid..but this could be depth first search lexicographically!Route along shortest path! Duh.
Optimal use of “resources” ..or edges.
![Page 113: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/113.jpg)
Algorithms?Route along any path.Feasible...
but is it as good as possible?How far from optimal could it be?(A) It is optimal!(B) A factor of two.(C) A factor of k , in general.
(C)
s1s2sk
t1t2tk
· · ·
· · · Value: k .
Opt: 1.
Stupid..but this could be depth first search lexicographically!Route along shortest path! Duh.
Optimal use of “resources” ..or edges.
![Page 114: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/114.jpg)
Algorithms?Route along any path.Feasible...but is it as good as possible?
How far from optimal could it be?(A) It is optimal!(B) A factor of two.(C) A factor of k , in general.
(C)
s1s2sk
t1t2tk
· · ·
· · · Value: k .
Opt: 1.
Stupid..but this could be depth first search lexicographically!Route along shortest path! Duh.
Optimal use of “resources” ..or edges.
![Page 115: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/115.jpg)
Algorithms?Route along any path.Feasible...but is it as good as possible?How far from optimal could it be?
(A) It is optimal!(B) A factor of two.(C) A factor of k , in general.
(C)
s1s2sk
t1t2tk
· · ·
· · · Value: k .
Opt: 1.
Stupid..but this could be depth first search lexicographically!Route along shortest path! Duh.
Optimal use of “resources” ..or edges.
![Page 116: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/116.jpg)
Algorithms?Route along any path.Feasible...but is it as good as possible?How far from optimal could it be?(A) It is optimal!
(B) A factor of two.(C) A factor of k , in general.
(C)
s1s2sk
t1t2tk
· · ·
· · · Value: k .
Opt: 1.
Stupid..but this could be depth first search lexicographically!Route along shortest path! Duh.
Optimal use of “resources” ..or edges.
![Page 117: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/117.jpg)
Algorithms?Route along any path.Feasible...but is it as good as possible?How far from optimal could it be?(A) It is optimal!(B) A factor of two.
(C) A factor of k , in general.
(C)
s1s2sk
t1t2tk
· · ·
· · · Value: k .
Opt: 1.
Stupid..but this could be depth first search lexicographically!Route along shortest path! Duh.
Optimal use of “resources” ..or edges.
![Page 118: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/118.jpg)
Algorithms?Route along any path.Feasible...but is it as good as possible?How far from optimal could it be?(A) It is optimal!(B) A factor of two.(C) A factor of k , in general.
(C)
s1s2sk
t1t2tk
· · ·
· · · Value: k .
Opt: 1.
Stupid..but this could be depth first search lexicographically!Route along shortest path! Duh.
Optimal use of “resources” ..or edges.
![Page 119: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/119.jpg)
Algorithms?Route along any path.Feasible...but is it as good as possible?How far from optimal could it be?(A) It is optimal!(B) A factor of two.(C) A factor of k , in general.
(C)
s1s2sk
t1t2tk
· · ·
· · · Value: k .
Opt: 1.
Stupid..but this could be depth first search lexicographically!Route along shortest path! Duh.
Optimal use of “resources” ..or edges.
![Page 120: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/120.jpg)
Algorithms?Route along any path.Feasible...but is it as good as possible?How far from optimal could it be?(A) It is optimal!(B) A factor of two.(C) A factor of k , in general.
(C)
s1s2sk
t1t2tk
· · ·
· · ·
Value: k .
Opt: 1.
Stupid..but this could be depth first search lexicographically!Route along shortest path! Duh.
Optimal use of “resources” ..or edges.
![Page 121: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/121.jpg)
Algorithms?Route along any path.Feasible...but is it as good as possible?How far from optimal could it be?(A) It is optimal!(B) A factor of two.(C) A factor of k , in general.
(C)
s1s2sk
t1t2tk
· · ·
· · ·
Value: k .
Opt: 1.
Stupid..but this could be depth first search lexicographically!Route along shortest path! Duh.
Optimal use of “resources” ..or edges.
![Page 122: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/122.jpg)
Algorithms?Route along any path.Feasible...but is it as good as possible?How far from optimal could it be?(A) It is optimal!(B) A factor of two.(C) A factor of k , in general.
(C)
s1s2sk
t1t2tk
· · ·
· · ·
Value: k .
Opt: 1.
Stupid..but this could be depth first search lexicographically!Route along shortest path! Duh.
Optimal use of “resources” ..or edges.
![Page 123: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/123.jpg)
Algorithms?Route along any path.Feasible...but is it as good as possible?How far from optimal could it be?(A) It is optimal!(B) A factor of two.(C) A factor of k , in general.
(C)
s1s2sk
t1t2tk
· · ·
· · ·
Value: k .
Opt: 1.
Stupid..but this could be depth first search lexicographically!Route along shortest path! Duh.
Optimal use of “resources” ..or edges.
![Page 124: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/124.jpg)
Algorithms?Route along any path.Feasible...but is it as good as possible?How far from optimal could it be?(A) It is optimal!(B) A factor of two.(C) A factor of k , in general.
(C)
s1s2sk
t1t2tk
· · ·
· · ·
Value: k .
Opt: 1.
Stupid..but this could be depth first search lexicographically!Route along shortest path! Duh.
Optimal use of “resources” ..or edges.
![Page 125: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/125.jpg)
Algorithms?Route along any path.Feasible...but is it as good as possible?How far from optimal could it be?(A) It is optimal!(B) A factor of two.(C) A factor of k , in general.
(C)
s1s2sk
t1t2tk
· · ·
· · · Value: k .
Opt: 1.
Stupid..but this could be depth first search lexicographically!Route along shortest path! Duh.
Optimal use of “resources” ..or edges.
![Page 126: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/126.jpg)
Algorithms?Route along any path.Feasible...but is it as good as possible?How far from optimal could it be?(A) It is optimal!(B) A factor of two.(C) A factor of k , in general.
(C)
s1s2sk
t1t2tk
· · ·
· · · Value: k .
Opt: 1.
Stupid..but this could be depth first search lexicographically!Route along shortest path! Duh.
Optimal use of “resources” ..or edges.
![Page 127: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/127.jpg)
Algorithms?Route along any path.Feasible...but is it as good as possible?How far from optimal could it be?(A) It is optimal!(B) A factor of two.(C) A factor of k , in general.
(C)
s1s2sk
t1t2tk
· · ·
· · · Value: k .
Opt: 1.
Stupid..
but this could be depth first search lexicographically!Route along shortest path! Duh.
Optimal use of “resources” ..or edges.
![Page 128: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/128.jpg)
Algorithms?Route along any path.Feasible...but is it as good as possible?How far from optimal could it be?(A) It is optimal!(B) A factor of two.(C) A factor of k , in general.
(C)
s1s2sk
t1t2tk
· · ·
· · · Value: k .
Opt: 1.
Stupid..but this could be depth first search lexicographically!
Route along shortest path! Duh.Optimal use of “resources” ..or edges.
![Page 129: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/129.jpg)
Algorithms?Route along any path.Feasible...but is it as good as possible?How far from optimal could it be?(A) It is optimal!(B) A factor of two.(C) A factor of k , in general.
(C)
s1s2sk
t1t2tk
· · ·
· · · Value: k .
Opt: 1.
Stupid..but this could be depth first search lexicographically!Route along shortest path!
Duh.Optimal use of “resources” ..or edges.
![Page 130: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/130.jpg)
Algorithms?Route along any path.Feasible...but is it as good as possible?How far from optimal could it be?(A) It is optimal!(B) A factor of two.(C) A factor of k , in general.
(C)
s1s2sk
t1t2tk
· · ·
· · · Value: k .
Opt: 1.
Stupid..but this could be depth first search lexicographically!Route along shortest path! Duh.
Optimal use of “resources” ..or edges.
![Page 131: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/131.jpg)
Algorithms?Route along any path.Feasible...but is it as good as possible?How far from optimal could it be?(A) It is optimal!(B) A factor of two.(C) A factor of k , in general.
(C)
s1s2sk
t1t2tk
· · ·
· · · Value: k .
Opt: 1.
Stupid..but this could be depth first search lexicographically!Route along shortest path! Duh.
Optimal use of “resources”
..or edges.
![Page 132: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/132.jpg)
Algorithms?Route along any path.Feasible...but is it as good as possible?How far from optimal could it be?(A) It is optimal!(B) A factor of two.(C) A factor of k , in general.
(C)
s1s2sk
t1t2tk
· · ·
· · · Value: k .
Opt: 1.
Stupid..but this could be depth first search lexicographically!Route along shortest path! Duh.
Optimal use of “resources” ..or edges.
![Page 133: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/133.jpg)
Shortest Path Routing.
Minimizes ∑i `(pi).si , ti routed along pi`(pi) is number of edges in pi .
Also minimizes total congestion: ∑e c(e)where c(e) congestion of edge.
Why? Let `(pi) be the length of path pi .
(A) ∑i `(pi) = ∑e c(e)?(B) ∑i `(pi)> ∑e c(e)?(C) ∑i `(pi)< ∑e c(e)?
(A). Proof?Path i uses `(pi) edges. Edge e used by c(e) paths.
Both count “uses”→ Sums are the same.
∑i `(pi) = ∑i ∑e∈pi1 = ∑e ∑pi3e 1 = ∑e c(e)
Shortest path routing minimizes total congestion ! !
![Page 134: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/134.jpg)
Shortest Path Routing.
Minimizes ∑i `(pi).
si , ti routed along pi`(pi) is number of edges in pi .
Also minimizes total congestion: ∑e c(e)where c(e) congestion of edge.
Why? Let `(pi) be the length of path pi .
(A) ∑i `(pi) = ∑e c(e)?(B) ∑i `(pi)> ∑e c(e)?(C) ∑i `(pi)< ∑e c(e)?
(A). Proof?Path i uses `(pi) edges. Edge e used by c(e) paths.
Both count “uses”→ Sums are the same.
∑i `(pi) = ∑i ∑e∈pi1 = ∑e ∑pi3e 1 = ∑e c(e)
Shortest path routing minimizes total congestion ! !
![Page 135: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/135.jpg)
Shortest Path Routing.
Minimizes ∑i `(pi).si , ti routed along pi
`(pi) is number of edges in pi .Also minimizes total congestion: ∑e c(e)
where c(e) congestion of edge.
Why? Let `(pi) be the length of path pi .
(A) ∑i `(pi) = ∑e c(e)?(B) ∑i `(pi)> ∑e c(e)?(C) ∑i `(pi)< ∑e c(e)?
(A). Proof?Path i uses `(pi) edges. Edge e used by c(e) paths.
Both count “uses”→ Sums are the same.
∑i `(pi) = ∑i ∑e∈pi1 = ∑e ∑pi3e 1 = ∑e c(e)
Shortest path routing minimizes total congestion ! !
![Page 136: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/136.jpg)
Shortest Path Routing.
Minimizes ∑i `(pi).si , ti routed along pi`(pi) is number of edges in pi .
Also minimizes total congestion: ∑e c(e)where c(e) congestion of edge.
Why? Let `(pi) be the length of path pi .
(A) ∑i `(pi) = ∑e c(e)?(B) ∑i `(pi)> ∑e c(e)?(C) ∑i `(pi)< ∑e c(e)?
(A). Proof?Path i uses `(pi) edges. Edge e used by c(e) paths.
Both count “uses”→ Sums are the same.
∑i `(pi) = ∑i ∑e∈pi1 = ∑e ∑pi3e 1 = ∑e c(e)
Shortest path routing minimizes total congestion ! !
![Page 137: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/137.jpg)
Shortest Path Routing.
Minimizes ∑i `(pi).si , ti routed along pi`(pi) is number of edges in pi .
Also minimizes total congestion: ∑e c(e)
where c(e) congestion of edge.
Why? Let `(pi) be the length of path pi .
(A) ∑i `(pi) = ∑e c(e)?(B) ∑i `(pi)> ∑e c(e)?(C) ∑i `(pi)< ∑e c(e)?
(A). Proof?Path i uses `(pi) edges. Edge e used by c(e) paths.
Both count “uses”→ Sums are the same.
∑i `(pi) = ∑i ∑e∈pi1 = ∑e ∑pi3e 1 = ∑e c(e)
Shortest path routing minimizes total congestion ! !
![Page 138: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/138.jpg)
Shortest Path Routing.
Minimizes ∑i `(pi).si , ti routed along pi`(pi) is number of edges in pi .
Also minimizes total congestion: ∑e c(e)where c(e) congestion of edge.
Why? Let `(pi) be the length of path pi .
(A) ∑i `(pi) = ∑e c(e)?(B) ∑i `(pi)> ∑e c(e)?(C) ∑i `(pi)< ∑e c(e)?
(A). Proof?Path i uses `(pi) edges. Edge e used by c(e) paths.
Both count “uses”→ Sums are the same.
∑i `(pi) = ∑i ∑e∈pi1 = ∑e ∑pi3e 1 = ∑e c(e)
Shortest path routing minimizes total congestion ! !
![Page 139: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/139.jpg)
Shortest Path Routing.
Minimizes ∑i `(pi).si , ti routed along pi`(pi) is number of edges in pi .
Also minimizes total congestion: ∑e c(e)where c(e) congestion of edge.
Why?
Let `(pi) be the length of path pi .
(A) ∑i `(pi) = ∑e c(e)?(B) ∑i `(pi)> ∑e c(e)?(C) ∑i `(pi)< ∑e c(e)?
(A). Proof?Path i uses `(pi) edges. Edge e used by c(e) paths.
Both count “uses”→ Sums are the same.
∑i `(pi) = ∑i ∑e∈pi1 = ∑e ∑pi3e 1 = ∑e c(e)
Shortest path routing minimizes total congestion ! !
![Page 140: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/140.jpg)
Shortest Path Routing.
Minimizes ∑i `(pi).si , ti routed along pi`(pi) is number of edges in pi .
Also minimizes total congestion: ∑e c(e)where c(e) congestion of edge.
Why? Let `(pi) be the length of path pi .
(A) ∑i `(pi) = ∑e c(e)?(B) ∑i `(pi)> ∑e c(e)?(C) ∑i `(pi)< ∑e c(e)?
(A). Proof?Path i uses `(pi) edges. Edge e used by c(e) paths.
Both count “uses”→ Sums are the same.
∑i `(pi) = ∑i ∑e∈pi1 = ∑e ∑pi3e 1 = ∑e c(e)
Shortest path routing minimizes total congestion ! !
![Page 141: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/141.jpg)
Shortest Path Routing.
Minimizes ∑i `(pi).si , ti routed along pi`(pi) is number of edges in pi .
Also minimizes total congestion: ∑e c(e)where c(e) congestion of edge.
Why? Let `(pi) be the length of path pi .
(A) ∑i `(pi) = ∑e c(e)?
(B) ∑i `(pi)> ∑e c(e)?(C) ∑i `(pi)< ∑e c(e)?
(A). Proof?Path i uses `(pi) edges. Edge e used by c(e) paths.
Both count “uses”→ Sums are the same.
∑i `(pi) = ∑i ∑e∈pi1 = ∑e ∑pi3e 1 = ∑e c(e)
Shortest path routing minimizes total congestion ! !
![Page 142: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/142.jpg)
Shortest Path Routing.
Minimizes ∑i `(pi).si , ti routed along pi`(pi) is number of edges in pi .
Also minimizes total congestion: ∑e c(e)where c(e) congestion of edge.
Why? Let `(pi) be the length of path pi .
(A) ∑i `(pi) = ∑e c(e)?(B) ∑i `(pi)> ∑e c(e)?
(C) ∑i `(pi)< ∑e c(e)?
(A). Proof?Path i uses `(pi) edges. Edge e used by c(e) paths.
Both count “uses”→ Sums are the same.
∑i `(pi) = ∑i ∑e∈pi1 = ∑e ∑pi3e 1 = ∑e c(e)
Shortest path routing minimizes total congestion ! !
![Page 143: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/143.jpg)
Shortest Path Routing.
Minimizes ∑i `(pi).si , ti routed along pi`(pi) is number of edges in pi .
Also minimizes total congestion: ∑e c(e)where c(e) congestion of edge.
Why? Let `(pi) be the length of path pi .
(A) ∑i `(pi) = ∑e c(e)?(B) ∑i `(pi)> ∑e c(e)?(C) ∑i `(pi)< ∑e c(e)?
(A). Proof?Path i uses `(pi) edges. Edge e used by c(e) paths.
Both count “uses”→ Sums are the same.
∑i `(pi) = ∑i ∑e∈pi1 = ∑e ∑pi3e 1 = ∑e c(e)
Shortest path routing minimizes total congestion ! !
![Page 144: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/144.jpg)
Shortest Path Routing.
Minimizes ∑i `(pi).si , ti routed along pi`(pi) is number of edges in pi .
Also minimizes total congestion: ∑e c(e)where c(e) congestion of edge.
Why? Let `(pi) be the length of path pi .
(A) ∑i `(pi) = ∑e c(e)?(B) ∑i `(pi)> ∑e c(e)?(C) ∑i `(pi)< ∑e c(e)?
(A).
Proof?Path i uses `(pi) edges. Edge e used by c(e) paths.
Both count “uses”→ Sums are the same.
∑i `(pi) = ∑i ∑e∈pi1 = ∑e ∑pi3e 1 = ∑e c(e)
Shortest path routing minimizes total congestion ! !
![Page 145: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/145.jpg)
Shortest Path Routing.
Minimizes ∑i `(pi).si , ti routed along pi`(pi) is number of edges in pi .
Also minimizes total congestion: ∑e c(e)where c(e) congestion of edge.
Why? Let `(pi) be the length of path pi .
(A) ∑i `(pi) = ∑e c(e)?(B) ∑i `(pi)> ∑e c(e)?(C) ∑i `(pi)< ∑e c(e)?
(A). Proof?
Path i uses `(pi) edges. Edge e used by c(e) paths.
Both count “uses”→ Sums are the same.
∑i `(pi) = ∑i ∑e∈pi1 = ∑e ∑pi3e 1 = ∑e c(e)
Shortest path routing minimizes total congestion ! !
![Page 146: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/146.jpg)
Shortest Path Routing.
Minimizes ∑i `(pi).si , ti routed along pi`(pi) is number of edges in pi .
Also minimizes total congestion: ∑e c(e)where c(e) congestion of edge.
Why? Let `(pi) be the length of path pi .
(A) ∑i `(pi) = ∑e c(e)?(B) ∑i `(pi)> ∑e c(e)?(C) ∑i `(pi)< ∑e c(e)?
(A). Proof?Path i uses `(pi) edges.
Edge e used by c(e) paths.
Both count “uses”→ Sums are the same.
∑i `(pi) = ∑i ∑e∈pi1 = ∑e ∑pi3e 1 = ∑e c(e)
Shortest path routing minimizes total congestion ! !
![Page 147: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/147.jpg)
Shortest Path Routing.
Minimizes ∑i `(pi).si , ti routed along pi`(pi) is number of edges in pi .
Also minimizes total congestion: ∑e c(e)where c(e) congestion of edge.
Why? Let `(pi) be the length of path pi .
(A) ∑i `(pi) = ∑e c(e)?(B) ∑i `(pi)> ∑e c(e)?(C) ∑i `(pi)< ∑e c(e)?
(A). Proof?Path i uses `(pi) edges. Edge e used by c(e) paths.
Both count “uses”→ Sums are the same.
∑i `(pi) = ∑i ∑e∈pi1 = ∑e ∑pi3e 1 = ∑e c(e)
Shortest path routing minimizes total congestion ! !
![Page 148: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/148.jpg)
Shortest Path Routing.
Minimizes ∑i `(pi).si , ti routed along pi`(pi) is number of edges in pi .
Also minimizes total congestion: ∑e c(e)where c(e) congestion of edge.
Why? Let `(pi) be the length of path pi .
(A) ∑i `(pi) = ∑e c(e)?(B) ∑i `(pi)> ∑e c(e)?(C) ∑i `(pi)< ∑e c(e)?
(A). Proof?Path i uses `(pi) edges. Edge e used by c(e) paths.
Both count “uses”
→ Sums are the same.
∑i `(pi) = ∑i ∑e∈pi1 = ∑e ∑pi3e 1 = ∑e c(e)
Shortest path routing minimizes total congestion ! !
![Page 149: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/149.jpg)
Shortest Path Routing.
Minimizes ∑i `(pi).si , ti routed along pi`(pi) is number of edges in pi .
Also minimizes total congestion: ∑e c(e)where c(e) congestion of edge.
Why? Let `(pi) be the length of path pi .
(A) ∑i `(pi) = ∑e c(e)?(B) ∑i `(pi)> ∑e c(e)?(C) ∑i `(pi)< ∑e c(e)?
(A). Proof?Path i uses `(pi) edges. Edge e used by c(e) paths.
Both count “uses”→ Sums are the same.
∑i `(pi) = ∑i ∑e∈pi1 = ∑e ∑pi3e 1 = ∑e c(e)
Shortest path routing minimizes total congestion ! !
![Page 150: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/150.jpg)
Shortest Path Routing.
Minimizes ∑i `(pi).si , ti routed along pi`(pi) is number of edges in pi .
Also minimizes total congestion: ∑e c(e)where c(e) congestion of edge.
Why? Let `(pi) be the length of path pi .
(A) ∑i `(pi) = ∑e c(e)?(B) ∑i `(pi)> ∑e c(e)?(C) ∑i `(pi)< ∑e c(e)?
(A). Proof?Path i uses `(pi) edges. Edge e used by c(e) paths.
Both count “uses”→ Sums are the same.
∑i `(pi)
= ∑i ∑e∈pi1 = ∑e ∑pi3e 1 = ∑e c(e)
Shortest path routing minimizes total congestion ! !
![Page 151: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/151.jpg)
Shortest Path Routing.
Minimizes ∑i `(pi).si , ti routed along pi`(pi) is number of edges in pi .
Also minimizes total congestion: ∑e c(e)where c(e) congestion of edge.
Why? Let `(pi) be the length of path pi .
(A) ∑i `(pi) = ∑e c(e)?(B) ∑i `(pi)> ∑e c(e)?(C) ∑i `(pi)< ∑e c(e)?
(A). Proof?Path i uses `(pi) edges. Edge e used by c(e) paths.
Both count “uses”→ Sums are the same.
∑i `(pi) = ∑i ∑e∈pi1
= ∑e ∑pi3e 1 = ∑e c(e)
Shortest path routing minimizes total congestion ! !
![Page 152: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/152.jpg)
Shortest Path Routing.
Minimizes ∑i `(pi).si , ti routed along pi`(pi) is number of edges in pi .
Also minimizes total congestion: ∑e c(e)where c(e) congestion of edge.
Why? Let `(pi) be the length of path pi .
(A) ∑i `(pi) = ∑e c(e)?(B) ∑i `(pi)> ∑e c(e)?(C) ∑i `(pi)< ∑e c(e)?
(A). Proof?Path i uses `(pi) edges. Edge e used by c(e) paths.
Both count “uses”→ Sums are the same.
∑i `(pi) = ∑i ∑e∈pi1 = ∑e ∑pi3e 1
= ∑e c(e)
Shortest path routing minimizes total congestion ! !
![Page 153: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/153.jpg)
Shortest Path Routing.
Minimizes ∑i `(pi).si , ti routed along pi`(pi) is number of edges in pi .
Also minimizes total congestion: ∑e c(e)where c(e) congestion of edge.
Why? Let `(pi) be the length of path pi .
(A) ∑i `(pi) = ∑e c(e)?(B) ∑i `(pi)> ∑e c(e)?(C) ∑i `(pi)< ∑e c(e)?
(A). Proof?Path i uses `(pi) edges. Edge e used by c(e) paths.
Both count “uses”→ Sums are the same.
∑i `(pi) = ∑i ∑e∈pi1 = ∑e ∑pi3e 1 = ∑e c(e)
Shortest path routing minimizes total congestion ! !
![Page 154: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/154.jpg)
Shortest Path Routing.
Minimizes ∑i `(pi).si , ti routed along pi`(pi) is number of edges in pi .
Also minimizes total congestion: ∑e c(e)where c(e) congestion of edge.
Why? Let `(pi) be the length of path pi .
(A) ∑i `(pi) = ∑e c(e)?(B) ∑i `(pi)> ∑e c(e)?(C) ∑i `(pi)< ∑e c(e)?
(A). Proof?Path i uses `(pi) edges. Edge e used by c(e) paths.
Both count “uses”→ Sums are the same.
∑i `(pi) = ∑i ∑e∈pi1 = ∑e ∑pi3e 1 = ∑e c(e)
Shortest path routing minimizes total congestion
! !
![Page 155: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/155.jpg)
Shortest Path Routing.
Minimizes ∑i `(pi).si , ti routed along pi`(pi) is number of edges in pi .
Also minimizes total congestion: ∑e c(e)where c(e) congestion of edge.
Why? Let `(pi) be the length of path pi .
(A) ∑i `(pi) = ∑e c(e)?(B) ∑i `(pi)> ∑e c(e)?(C) ∑i `(pi)< ∑e c(e)?
(A). Proof?Path i uses `(pi) edges. Edge e used by c(e) paths.
Both count “uses”→ Sums are the same.
∑i `(pi) = ∑i ∑e∈pi1 = ∑e ∑pi3e 1 = ∑e c(e)
Shortest path routing minimizes total congestion !
!
![Page 156: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/156.jpg)
Shortest Path Routing.
Minimizes ∑i `(pi).si , ti routed along pi`(pi) is number of edges in pi .
Also minimizes total congestion: ∑e c(e)where c(e) congestion of edge.
Why? Let `(pi) be the length of path pi .
(A) ∑i `(pi) = ∑e c(e)?(B) ∑i `(pi)> ∑e c(e)?(C) ∑i `(pi)< ∑e c(e)?
(A). Proof?Path i uses `(pi) edges. Edge e used by c(e) paths.
Both count “uses”→ Sums are the same.
∑i `(pi) = ∑i ∑e∈pi1 = ∑e ∑pi3e 1 = ∑e c(e)
Shortest path routing minimizes total congestion ! !
![Page 157: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/157.jpg)
Shortest Path Routing and Congestion.
Minimize each path length minimizes total congestion.
Also minimizes average: 1m ∑e c(e). Just a scaling!
Average load is lower bound on the lowest max congestion!
Shortest path routing minimizes average load.
Does it minimize maximum load?
![Page 158: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/158.jpg)
Shortest Path Routing and Congestion.
Minimize each path length minimizes total congestion.
Also minimizes average: 1m ∑e c(e).
Just a scaling!
Average load is lower bound on the lowest max congestion!
Shortest path routing minimizes average load.
Does it minimize maximum load?
![Page 159: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/159.jpg)
Shortest Path Routing and Congestion.
Minimize each path length minimizes total congestion.
Also minimizes average: 1m ∑e c(e). Just a scaling!
Average load is lower bound on the lowest max congestion!
Shortest path routing minimizes average load.
Does it minimize maximum load?
![Page 160: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/160.jpg)
Shortest Path Routing and Congestion.
Minimize each path length minimizes total congestion.
Also minimizes average: 1m ∑e c(e). Just a scaling!
Average load is lower bound on the lowest max congestion!
Shortest path routing minimizes average load.
Does it minimize maximum load?
![Page 161: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/161.jpg)
Shortest Path Routing and Congestion.
Minimize each path length minimizes total congestion.
Also minimizes average: 1m ∑e c(e). Just a scaling!
Average load is lower bound on the lowest max congestion!
Shortest path routing minimizes average load.
Does it minimize maximum load?
![Page 162: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/162.jpg)
Shortest Path Routing and Congestion.
Minimize each path length minimizes total congestion.
Also minimizes average: 1m ∑e c(e). Just a scaling!
Average load is lower bound on the lowest max congestion!
Shortest path routing minimizes average load.
Does it minimize maximum load?
![Page 163: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/163.jpg)
One problem...
How far from optimal?
Optimal? Factor 2? Factor k?
s1s2sk
t1t2tk
...... > 2k
· · ·
· · ·
··
· ·Value: k .
Opt: 1.
Does minimize average load though, FWIW.Any suggestions?
![Page 164: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/164.jpg)
One problem...
How far from optimal?Optimal?
Factor 2? Factor k?
s1s2sk
t1t2tk
...... > 2k
· · ·
· · ·
··
· ·Value: k .
Opt: 1.
Does minimize average load though, FWIW.Any suggestions?
![Page 165: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/165.jpg)
One problem...
How far from optimal?Optimal? Factor 2?
Factor k?
s1s2sk
t1t2tk
...... > 2k
· · ·
· · ·
··
· ·Value: k .
Opt: 1.
Does minimize average load though, FWIW.Any suggestions?
![Page 166: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/166.jpg)
One problem...
How far from optimal?Optimal? Factor 2? Factor k?
s1s2sk
t1t2tk
...... > 2k
· · ·
· · ·
··
· ·Value: k .
Opt: 1.
Does minimize average load though, FWIW.Any suggestions?
![Page 167: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/167.jpg)
One problem...
How far from optimal?Optimal? Factor 2? Factor k?
s1s2sk
t1t2tk
...... > 2k
· · ·
· · ·
··
· ·
Value: k .
Opt: 1.
Does minimize average load though, FWIW.Any suggestions?
![Page 168: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/168.jpg)
One problem...
How far from optimal?Optimal? Factor 2? Factor k?
s1s2sk
t1t2tk
...... > 2k
· · ·
· · ·
··
· ·
Value: k .
Opt: 1.
Does minimize average load though, FWIW.Any suggestions?
![Page 169: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/169.jpg)
One problem...
How far from optimal?Optimal? Factor 2? Factor k?
s1s2sk
t1t2tk
...... > 2k
· · ·
· · ·
··
· ·
Value: k .
Opt: 1.
Does minimize average load though, FWIW.Any suggestions?
![Page 170: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/170.jpg)
One problem...
How far from optimal?Optimal? Factor 2? Factor k?
s1s2sk
t1t2tk
...... > 2k
· · ·
· · ·
··
· ·
Value: k .
Opt: 1.
Does minimize average load though, FWIW.Any suggestions?
![Page 171: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/171.jpg)
One problem...
How far from optimal?Optimal? Factor 2? Factor k?
s1s2sk
t1t2tk
...... > 2k
· · ·
· · ·
··
· ·
Value: k .
Opt: 1.
Does minimize average load though, FWIW.Any suggestions?
![Page 172: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/172.jpg)
One problem...
How far from optimal?Optimal? Factor 2? Factor k?
s1s2sk
t1t2tk
...... > 2k
· · ·
· · ·
··
· ·Value: k .
Opt: 1.
Does minimize average load though, FWIW.Any suggestions?
![Page 173: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/173.jpg)
One problem...
How far from optimal?Optimal? Factor 2? Factor k?
s1s2sk
t1t2tk
...... > 2k
· · ·
· · ·
··
· ·Value: k .
Opt: 1.
Does minimize average load though, FWIW.Any suggestions?
![Page 174: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/174.jpg)
One problem...
How far from optimal?Optimal? Factor 2? Factor k?
s1s2sk
t1t2tk
...... > 2k
· · ·
· · ·
··
· ·Value: k .
Opt: 1.
Does minimize average load though,
FWIW.Any suggestions?
![Page 175: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/175.jpg)
One problem...
How far from optimal?Optimal? Factor 2? Factor k?
s1s2sk
t1t2tk
...... > 2k
· · ·
· · ·
··
· ·Value: k .
Opt: 1.
Does minimize average load though, FWIW.
Any suggestions?
![Page 176: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/176.jpg)
One problem...
How far from optimal?Optimal? Factor 2? Factor k?
s1s2sk
t1t2tk
...... > 2k
· · ·
· · ·
··
· ·Value: k .
Opt: 1.
Does minimize average load though, FWIW.Any suggestions?
![Page 177: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/177.jpg)
Another problem.
Given G = (V ,E), (s1, t1), . . . ,(sk , tk ), find a set of k paths assign oneunit of “toll” to edges to maximize total toll for connecting pairs.
s1
t1
s2
t2
s3
t3
111
111 1
11
111
111
111
111
111
111
12
12
Assign 111 on each of 11 edges.
Toll paid: 311 + 3
11 + 311 = 9
11
Can we do better?
Assign 1/2 on these two edges.Toll paid: 1
2 +12 +
12 = 3
2
![Page 178: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/178.jpg)
Another problem.
Given G = (V ,E), (s1, t1), . . . ,(sk , tk ), find a set of k paths assign oneunit of “toll” to edges to maximize total toll for connecting pairs.
s1
t1
s2
t2
s3
t3
111
111 1
11
111
111
111
111
111
111
12
12
Assign 111 on each of 11 edges.
Toll paid: 311 + 3
11 + 311 = 9
11
Can we do better?
Assign 1/2 on these two edges.Toll paid: 1
2 +12 +
12 = 3
2
![Page 179: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/179.jpg)
Another problem.
Given G = (V ,E), (s1, t1), . . . ,(sk , tk ), find a set of k paths assign oneunit of “toll” to edges to maximize total toll for connecting pairs.
s1
t1
s2
t2
s3
t3
111
111 1
11
111
111
111
111
111
111
12
12
Assign 111 on each of 11 edges.
Toll paid: 311 + 3
11 + 311 = 9
11
Can we do better?
Assign 1/2 on these two edges.Toll paid: 1
2 +12 +
12 = 3
2
![Page 180: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/180.jpg)
Another problem.
Given G = (V ,E), (s1, t1), . . . ,(sk , tk ), find a set of k paths assign oneunit of “toll” to edges to maximize total toll for connecting pairs.
s1
t1
s2
t2
s3
t3
111
111 1
11
111
111
111
111
111
111
12
12
Assign 111 on each of 11 edges.
Toll paid:
311 + 3
11 + 311 = 9
11
Can we do better?
Assign 1/2 on these two edges.Toll paid: 1
2 +12 +
12 = 3
2
![Page 181: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/181.jpg)
Another problem.
Given G = (V ,E), (s1, t1), . . . ,(sk , tk ), find a set of k paths assign oneunit of “toll” to edges to maximize total toll for connecting pairs.
s1
t1
s2
t2
s3
t3
111
111 1
11
111
111
111
111
111
111
12
12
Assign 111 on each of 11 edges.
Toll paid: 311
+ 311 + 3
11 = 911
Can we do better?
Assign 1/2 on these two edges.Toll paid: 1
2 +12 +
12 = 3
2
![Page 182: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/182.jpg)
Another problem.
Given G = (V ,E), (s1, t1), . . . ,(sk , tk ), find a set of k paths assign oneunit of “toll” to edges to maximize total toll for connecting pairs.
s1
t1
s2
t2
s3
t3
111
111 1
11
111
111
111
111
111
111
12
12
Assign 111 on each of 11 edges.
Toll paid: 311 + 3
11
+ 311 = 9
11
Can we do better?
Assign 1/2 on these two edges.Toll paid: 1
2 +12 +
12 = 3
2
![Page 183: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/183.jpg)
Another problem.
Given G = (V ,E), (s1, t1), . . . ,(sk , tk ), find a set of k paths assign oneunit of “toll” to edges to maximize total toll for connecting pairs.
s1
t1
s2
t2
s3
t3
111
111 1
11
111
111
111
111
111
111
12
12
Assign 111 on each of 11 edges.
Toll paid: 311 + 3
11 + 311
= 911
Can we do better?
Assign 1/2 on these two edges.Toll paid: 1
2 +12 +
12 = 3
2
![Page 184: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/184.jpg)
Another problem.
Given G = (V ,E), (s1, t1), . . . ,(sk , tk ), find a set of k paths assign oneunit of “toll” to edges to maximize total toll for connecting pairs.
s1
t1
s2
t2
s3
t3
111
111 1
11
111
111
111
111
111
111
12
12
Assign 111 on each of 11 edges.
Toll paid: 311 + 3
11 + 311 = 9
11
Can we do better?
Assign 1/2 on these two edges.Toll paid: 1
2 +12 +
12 = 3
2
![Page 185: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/185.jpg)
Another problem.
Given G = (V ,E), (s1, t1), . . . ,(sk , tk ), find a set of k paths assign oneunit of “toll” to edges to maximize total toll for connecting pairs.
s1
t1
s2
t2
s3
t3
111
111 1
11
111
111
111
111
111
111
12
12
Assign 111 on each of 11 edges.
Toll paid: 311 + 3
11 + 311 = 9
11
Can we do better?
Assign 1/2 on these two edges.Toll paid: 1
2 +12 +
12 = 3
2
![Page 186: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/186.jpg)
Another problem.
Given G = (V ,E), (s1, t1), . . . ,(sk , tk ), find a set of k paths assign oneunit of “toll” to edges to maximize total toll for connecting pairs.
s1
t1
s2
t2
s3
t3
111
111 1
11
111
111
111
111
111
111
12
12
Assign 111 on each of 11 edges.
Toll paid: 311 + 3
11 + 311 = 9
11
Can we do better?
Assign 1/2 on these two edges.
Toll paid: 12 +
12 +
12 = 3
2
![Page 187: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/187.jpg)
Another problem.
Given G = (V ,E), (s1, t1), . . . ,(sk , tk ), find a set of k paths assign oneunit of “toll” to edges to maximize total toll for connecting pairs.
s1
t1
s2
t2
s3
t3
111
111 1
11
111
111
111
111
111
111
12
12
Assign 111 on each of 11 edges.
Toll paid: 311 + 3
11 + 311 = 9
11
Can we do better?
Assign 1/2 on these two edges.Toll paid:
12 +
12 +
12 = 3
2
![Page 188: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/188.jpg)
Another problem.
Given G = (V ,E), (s1, t1), . . . ,(sk , tk ), find a set of k paths assign oneunit of “toll” to edges to maximize total toll for connecting pairs.
s1
t1
s2
t2
s3
t3
111
111 1
11
111
111
111
111
111
111
12
12
Assign 111 on each of 11 edges.
Toll paid: 311 + 3
11 + 311 = 9
11
Can we do better?
Assign 1/2 on these two edges.Toll paid: 1
2 +12 +
12
= 32
![Page 189: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/189.jpg)
Another problem.
Given G = (V ,E), (s1, t1), . . . ,(sk , tk ), find a set of k paths assign oneunit of “toll” to edges to maximize total toll for connecting pairs.
s1
t1
s2
t2
s3
t3
111
111 1
11
111
111
111
111
111
111
12
12
Assign 111 on each of 11 edges.
Toll paid: 311 + 3
11 + 311 = 9
11
Can we do better?
Assign 1/2 on these two edges.Toll paid: 1
2 +12 +
12 = 3
2
![Page 190: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/190.jpg)
Toll problem and Routing problem.Given G = (V ,E), (s1, t1), . . . ,(sk , tk ), find a set of k paths assign oneunit of “toll” to edges to maximize total toll paid to connecting pairs.
Possible solution: 1m on each edge.
Toll collected: ≥ ∑i `(pi )m .
Familiar? Consider.
Find d : e→ R with ∑e d(e) = 1 which maximizes
∑i
d(si , ti).
d(si , ti)- shortest path between si and ti under d(·).Digression? d(e) suggests a weighted average.
Remember uniform average congestion is lower bound on congestionof routing!
Any toll solution value (weighted average congestion) is lower boundon path routing value (max congestion).
![Page 191: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/191.jpg)
Toll problem and Routing problem.Given G = (V ,E), (s1, t1), . . . ,(sk , tk ), find a set of k paths assign oneunit of “toll” to edges to maximize total toll paid to connecting pairs.
Possible solution: 1m on each edge.
Toll collected: ≥ ∑i `(pi )m .
Familiar? Consider.
Find d : e→ R with ∑e d(e) = 1 which maximizes
∑i
d(si , ti).
d(si , ti)- shortest path between si and ti under d(·).Digression? d(e) suggests a weighted average.
Remember uniform average congestion is lower bound on congestionof routing!
Any toll solution value (weighted average congestion) is lower boundon path routing value (max congestion).
![Page 192: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/192.jpg)
Toll problem and Routing problem.Given G = (V ,E), (s1, t1), . . . ,(sk , tk ), find a set of k paths assign oneunit of “toll” to edges to maximize total toll paid to connecting pairs.
Possible solution: 1m on each edge.
Toll collected: ≥ ∑i `(pi )m .
Familiar? Consider.
Find d : e→ R with ∑e d(e) = 1 which maximizes
∑i
d(si , ti).
d(si , ti)- shortest path between si and ti under d(·).Digression? d(e) suggests a weighted average.
Remember uniform average congestion is lower bound on congestionof routing!
Any toll solution value (weighted average congestion) is lower boundon path routing value (max congestion).
![Page 193: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/193.jpg)
Toll problem and Routing problem.Given G = (V ,E), (s1, t1), . . . ,(sk , tk ), find a set of k paths assign oneunit of “toll” to edges to maximize total toll paid to connecting pairs.
Possible solution: 1m on each edge.
Toll collected: ≥ ∑i `(pi )m .
Familiar?
Consider.
Find d : e→ R with ∑e d(e) = 1 which maximizes
∑i
d(si , ti).
d(si , ti)- shortest path between si and ti under d(·).Digression? d(e) suggests a weighted average.
Remember uniform average congestion is lower bound on congestionof routing!
Any toll solution value (weighted average congestion) is lower boundon path routing value (max congestion).
![Page 194: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/194.jpg)
Toll problem and Routing problem.Given G = (V ,E), (s1, t1), . . . ,(sk , tk ), find a set of k paths assign oneunit of “toll” to edges to maximize total toll paid to connecting pairs.
Possible solution: 1m on each edge.
Toll collected: ≥ ∑i `(pi )m .
Familiar? Consider.
Find d : e→ R with ∑e d(e) = 1 which maximizes
∑i
d(si , ti).
d(si , ti)- shortest path between si and ti under d(·).Digression? d(e) suggests a weighted average.
Remember uniform average congestion is lower bound on congestionof routing!
Any toll solution value (weighted average congestion) is lower boundon path routing value (max congestion).
![Page 195: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/195.jpg)
Toll problem and Routing problem.Given G = (V ,E), (s1, t1), . . . ,(sk , tk ), find a set of k paths assign oneunit of “toll” to edges to maximize total toll paid to connecting pairs.
Possible solution: 1m on each edge.
Toll collected: ≥ ∑i `(pi )m .
Familiar? Consider.
Find d : e→ R with ∑e d(e) = 1 which maximizes
∑i
d(si , ti).
d(si , ti)- shortest path between si and ti under d(·).
Digression? d(e) suggests a weighted average.
Remember uniform average congestion is lower bound on congestionof routing!
Any toll solution value (weighted average congestion) is lower boundon path routing value (max congestion).
![Page 196: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/196.jpg)
Toll problem and Routing problem.Given G = (V ,E), (s1, t1), . . . ,(sk , tk ), find a set of k paths assign oneunit of “toll” to edges to maximize total toll paid to connecting pairs.
Possible solution: 1m on each edge.
Toll collected: ≥ ∑i `(pi )m .
Familiar? Consider.
Find d : e→ R with ∑e d(e) = 1 which maximizes
∑i
d(si , ti).
d(si , ti)- shortest path between si and ti under d(·).Digression?
d(e) suggests a weighted average.
Remember uniform average congestion is lower bound on congestionof routing!
Any toll solution value (weighted average congestion) is lower boundon path routing value (max congestion).
![Page 197: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/197.jpg)
Toll problem and Routing problem.Given G = (V ,E), (s1, t1), . . . ,(sk , tk ), find a set of k paths assign oneunit of “toll” to edges to maximize total toll paid to connecting pairs.
Possible solution: 1m on each edge.
Toll collected: ≥ ∑i `(pi )m .
Familiar? Consider.
Find d : e→ R with ∑e d(e) = 1 which maximizes
∑i
d(si , ti).
d(si , ti)- shortest path between si and ti under d(·).Digression? d(e) suggests a weighted average.
Remember uniform average congestion is lower bound on congestionof routing!
Any toll solution value (weighted average congestion) is lower boundon path routing value (max congestion).
![Page 198: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/198.jpg)
Toll problem and Routing problem.Given G = (V ,E), (s1, t1), . . . ,(sk , tk ), find a set of k paths assign oneunit of “toll” to edges to maximize total toll paid to connecting pairs.
Possible solution: 1m on each edge.
Toll collected: ≥ ∑i `(pi )m .
Familiar? Consider.
Find d : e→ R with ∑e d(e) = 1 which maximizes
∑i
d(si , ti).
d(si , ti)- shortest path between si and ti under d(·).Digression? d(e) suggests a weighted average.
Remember uniform average congestion is lower bound on congestionof routing!
Any toll solution value (weighted average congestion) is lower boundon path routing value (max congestion).
![Page 199: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/199.jpg)
Toll problem and Routing problem.Given G = (V ,E), (s1, t1), . . . ,(sk , tk ), find a set of k paths assign oneunit of “toll” to edges to maximize total toll paid to connecting pairs.
Possible solution: 1m on each edge.
Toll collected: ≥ ∑i `(pi )m .
Familiar? Consider.
Find d : e→ R with ∑e d(e) = 1 which maximizes
∑i
d(si , ti).
d(si , ti)- shortest path between si and ti under d(·).Digression? d(e) suggests a weighted average.
Remember uniform average congestion is lower bound on congestionof routing!
Any toll solution value (weighted average congestion) is lower boundon path routing value (max congestion).
![Page 200: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/200.jpg)
Proving lower bound: notation.
d(e) - toll assigned to edge e.
d(p) - total toll assigned to path p.d(u,v) - total assigned to shortest path between u and v .d(x) - polymorpic
polymorphic
x could be edge, path, or pair.
![Page 201: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/201.jpg)
Proving lower bound: notation.
d(e) - toll assigned to edge e.d(p) - total toll assigned to path p.
d(u,v) - total assigned to shortest path between u and v .d(x) - polymorpic
polymorphic
x could be edge, path, or pair.
![Page 202: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/202.jpg)
Proving lower bound: notation.
d(e) - toll assigned to edge e.d(p) - total toll assigned to path p.d(u,v) - total assigned to shortest path between u and v .
d(x) - polymorpic
polymorphic
x could be edge, path, or pair.
![Page 203: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/203.jpg)
Proving lower bound: notation.
d(e) - toll assigned to edge e.d(p) - total toll assigned to path p.d(u,v) - total assigned to shortest path between u and v .d(x) - polymorpic
polymorphicx could be edge, path, or pair.
![Page 204: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/204.jpg)
Proving lower bound: notation.
d(e) - toll assigned to edge e.d(p) - total toll assigned to path p.d(u,v) - total assigned to shortest path between u and v .d(x) - polymorpic
polymorphic
x could be edge, path, or pair.
![Page 205: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/205.jpg)
Proving lower bound: notation.
d(e) - toll assigned to edge e.d(p) - total toll assigned to path p.d(u,v) - total assigned to shortest path between u and v .d(x) - polymorpic
polymorpicpolymorphic
x could be edge, path, or pair.
![Page 206: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/206.jpg)
Proving lower bound: notation.
d(e) - toll assigned to edge e.d(p) - total toll assigned to path p.d(u,v) - total assigned to shortest path between u and v .d(x) - polymorpic
polymorpic
polymorphicx could be edge, path, or pair.
![Page 207: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/207.jpg)
Proving lower bound.
Routing solution: pi connects (si , ti) and has length d(pi).
c(e) - congestion on edge e under routing.Max c(e)?maxe c(e)≥ ∑e c(e)d(e) since ∑e d(e) = 1.
∑e
c(e)d(e) = ∑i
d(pi)
∑i
d(pi) = ∑i
∑e∈pi
d(e)
= ∑e
∑i :e3pi
d(e)
= ∑e
d(e) ∑i :e3pi
1
= ∑e
d(e)c(e)
A path uses “volume” d(pi).
Volume on edge is d(e)c(e).
∑i d(pi) = ∑e d(e)c(e).
maxe c(e)≥ ∑e d(e)c(e) = ∑i d(pi)≥ ∑i d(si , ti).Routing solution cost ≥ Any toll solution cost.
![Page 208: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/208.jpg)
Proving lower bound.
Routing solution: pi connects (si , ti) and has length d(pi).
c(e) - congestion on edge e under routing.
Max c(e)?maxe c(e)≥ ∑e c(e)d(e) since ∑e d(e) = 1.
∑e
c(e)d(e) = ∑i
d(pi)
∑i
d(pi) = ∑i
∑e∈pi
d(e)
= ∑e
∑i :e3pi
d(e)
= ∑e
d(e) ∑i :e3pi
1
= ∑e
d(e)c(e)
A path uses “volume” d(pi).
Volume on edge is d(e)c(e).
∑i d(pi) = ∑e d(e)c(e).
maxe c(e)≥ ∑e d(e)c(e) = ∑i d(pi)≥ ∑i d(si , ti).Routing solution cost ≥ Any toll solution cost.
![Page 209: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/209.jpg)
Proving lower bound.
Routing solution: pi connects (si , ti) and has length d(pi).
c(e) - congestion on edge e under routing.Max c(e)?
maxe c(e)≥ ∑e c(e)d(e) since ∑e d(e) = 1.
∑e
c(e)d(e) = ∑i
d(pi)
∑i
d(pi) = ∑i
∑e∈pi
d(e)
= ∑e
∑i :e3pi
d(e)
= ∑e
d(e) ∑i :e3pi
1
= ∑e
d(e)c(e)
A path uses “volume” d(pi).
Volume on edge is d(e)c(e).
∑i d(pi) = ∑e d(e)c(e).
maxe c(e)≥ ∑e d(e)c(e) = ∑i d(pi)≥ ∑i d(si , ti).Routing solution cost ≥ Any toll solution cost.
![Page 210: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/210.jpg)
Proving lower bound.
Routing solution: pi connects (si , ti) and has length d(pi).
c(e) - congestion on edge e under routing.Max c(e)?maxe c(e)≥ ∑e c(e)d(e) since ∑e d(e) = 1.
∑e
c(e)d(e) = ∑i
d(pi)
∑i
d(pi) = ∑i
∑e∈pi
d(e)
= ∑e
∑i :e3pi
d(e)
= ∑e
d(e) ∑i :e3pi
1
= ∑e
d(e)c(e)
A path uses “volume” d(pi).
Volume on edge is d(e)c(e).
∑i d(pi) = ∑e d(e)c(e).
maxe c(e)≥ ∑e d(e)c(e) = ∑i d(pi)≥ ∑i d(si , ti).Routing solution cost ≥ Any toll solution cost.
![Page 211: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/211.jpg)
Proving lower bound.
Routing solution: pi connects (si , ti) and has length d(pi).
c(e) - congestion on edge e under routing.Max c(e)?maxe c(e)≥ ∑e c(e)d(e) since ∑e d(e) = 1.
∑e
c(e)d(e) = ∑i
d(pi)
∑i
d(pi) = ∑i
∑e∈pi
d(e)
= ∑e
∑i :e3pi
d(e)
= ∑e
d(e) ∑i :e3pi
1
= ∑e
d(e)c(e)
A path uses “volume” d(pi).
Volume on edge is d(e)c(e).
∑i d(pi) = ∑e d(e)c(e).
maxe c(e)≥ ∑e d(e)c(e) = ∑i d(pi)≥ ∑i d(si , ti).Routing solution cost ≥ Any toll solution cost.
![Page 212: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/212.jpg)
Proving lower bound.
Routing solution: pi connects (si , ti) and has length d(pi).
c(e) - congestion on edge e under routing.Max c(e)?maxe c(e)≥ ∑e c(e)d(e) since ∑e d(e) = 1.
∑e
c(e)d(e) = ∑i
d(pi)
∑i
d(pi) = ∑i
∑e∈pi
d(e)
= ∑e
∑i :e3pi
d(e)
= ∑e
d(e) ∑i :e3pi
1
= ∑e
d(e)c(e)
A path uses “volume” d(pi).
Volume on edge is d(e)c(e).
∑i d(pi) = ∑e d(e)c(e).
maxe c(e)≥ ∑e d(e)c(e) = ∑i d(pi)≥ ∑i d(si , ti).Routing solution cost ≥ Any toll solution cost.
![Page 213: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/213.jpg)
Proving lower bound.
Routing solution: pi connects (si , ti) and has length d(pi).
c(e) - congestion on edge e under routing.Max c(e)?maxe c(e)≥ ∑e c(e)d(e) since ∑e d(e) = 1.
∑e
c(e)d(e) = ∑i
d(pi)
∑i
d(pi) = ∑i
∑e∈pi
d(e)
= ∑e
∑i :e3pi
d(e)
= ∑e
d(e) ∑i :e3pi
1
= ∑e
d(e)c(e)
A path uses “volume” d(pi).
Volume on edge is d(e)c(e).
∑i d(pi) = ∑e d(e)c(e).
maxe c(e)≥ ∑e d(e)c(e) = ∑i d(pi)≥ ∑i d(si , ti).Routing solution cost ≥ Any toll solution cost.
![Page 214: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/214.jpg)
Proving lower bound.
Routing solution: pi connects (si , ti) and has length d(pi).
c(e) - congestion on edge e under routing.Max c(e)?maxe c(e)≥ ∑e c(e)d(e) since ∑e d(e) = 1.
∑e
c(e)d(e) = ∑i
d(pi)
∑i
d(pi) = ∑i
∑e∈pi
d(e)
= ∑e
∑i :e3pi
d(e)
= ∑e
d(e) ∑i :e3pi
1
= ∑e
d(e)c(e)
A path uses “volume” d(pi).
Volume on edge is d(e)c(e).
∑i d(pi) = ∑e d(e)c(e).
maxe c(e)≥ ∑e d(e)c(e) = ∑i d(pi)≥ ∑i d(si , ti).Routing solution cost ≥ Any toll solution cost.
![Page 215: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/215.jpg)
Proving lower bound.
Routing solution: pi connects (si , ti) and has length d(pi).
c(e) - congestion on edge e under routing.Max c(e)?maxe c(e)≥ ∑e c(e)d(e) since ∑e d(e) = 1.
∑e
c(e)d(e) = ∑i
d(pi)
∑i
d(pi) = ∑i
∑e∈pi
d(e)
= ∑e
∑i :e3pi
d(e)
= ∑e
d(e) ∑i :e3pi
1
= ∑e
d(e)c(e)
A path uses “volume” d(pi).
Volume on edge is d(e)c(e).
∑i d(pi) = ∑e d(e)c(e).
maxe c(e)≥ ∑e d(e)c(e) = ∑i d(pi)≥ ∑i d(si , ti).Routing solution cost ≥ Any toll solution cost.
![Page 216: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/216.jpg)
Proving lower bound.
Routing solution: pi connects (si , ti) and has length d(pi).
c(e) - congestion on edge e under routing.Max c(e)?maxe c(e)≥ ∑e c(e)d(e) since ∑e d(e) = 1.
∑e
c(e)d(e) = ∑i
d(pi)
∑i
d(pi) = ∑i
∑e∈pi
d(e)
= ∑e
∑i :e3pi
d(e)
= ∑e
d(e) ∑i :e3pi
1
= ∑e
d(e)c(e)
A path uses “volume” d(pi).
Volume on edge is d(e)c(e).
∑i d(pi) = ∑e d(e)c(e).
maxe c(e)≥ ∑e d(e)c(e) = ∑i d(pi)≥ ∑i d(si , ti).Routing solution cost ≥ Any toll solution cost.
![Page 217: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/217.jpg)
Proving lower bound.
Routing solution: pi connects (si , ti) and has length d(pi).
c(e) - congestion on edge e under routing.Max c(e)?maxe c(e)≥ ∑e c(e)d(e) since ∑e d(e) = 1.
∑e
c(e)d(e) = ∑i
d(pi)
∑i
d(pi) = ∑i
∑e∈pi
d(e)
= ∑e
∑i :e3pi
d(e)
= ∑e
d(e) ∑i :e3pi
1
= ∑e
d(e)c(e)
A path uses “volume” d(pi).
Volume on edge is d(e)c(e).
∑i d(pi) = ∑e d(e)c(e).
maxe c(e)≥ ∑e d(e)c(e) = ∑i d(pi)≥ ∑i d(si , ti).Routing solution cost ≥ Any toll solution cost.
![Page 218: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/218.jpg)
Proving lower bound.
Routing solution: pi connects (si , ti) and has length d(pi).
c(e) - congestion on edge e under routing.Max c(e)?maxe c(e)≥ ∑e c(e)d(e) since ∑e d(e) = 1.
∑e
c(e)d(e) = ∑i
d(pi)
∑i
d(pi) = ∑i
∑e∈pi
d(e)
= ∑e
∑i :e3pi
d(e)
= ∑e
d(e) ∑i :e3pi
1
= ∑e
d(e)c(e)
A path uses “volume” d(pi).
Volume on edge is d(e)c(e).
∑i d(pi) = ∑e d(e)c(e).
maxe c(e)≥ ∑e d(e)c(e) = ∑i d(pi)≥ ∑i d(si , ti).Routing solution cost ≥ Any toll solution cost.
![Page 219: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/219.jpg)
Proving lower bound.
Routing solution: pi connects (si , ti) and has length d(pi).
c(e) - congestion on edge e under routing.Max c(e)?maxe c(e)≥ ∑e c(e)d(e) since ∑e d(e) = 1.
∑e
c(e)d(e) = ∑i
d(pi)
∑i
d(pi) = ∑i
∑e∈pi
d(e)
= ∑e
∑i :e3pi
d(e)
= ∑e
d(e) ∑i :e3pi
1
= ∑e
d(e)c(e)
A path uses “volume” d(pi).
Volume on edge is d(e)c(e).
∑i d(pi) = ∑e d(e)c(e).
maxe c(e)≥ ∑e d(e)c(e)
= ∑i d(pi)≥ ∑i d(si , ti).Routing solution cost ≥ Any toll solution cost.
![Page 220: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/220.jpg)
Proving lower bound.
Routing solution: pi connects (si , ti) and has length d(pi).
c(e) - congestion on edge e under routing.Max c(e)?maxe c(e)≥ ∑e c(e)d(e) since ∑e d(e) = 1.
∑e
c(e)d(e) = ∑i
d(pi)
∑i
d(pi) = ∑i
∑e∈pi
d(e)
= ∑e
∑i :e3pi
d(e)
= ∑e
d(e) ∑i :e3pi
1
= ∑e
d(e)c(e)
A path uses “volume” d(pi).
Volume on edge is d(e)c(e).
∑i d(pi) = ∑e d(e)c(e).
maxe c(e)≥ ∑e d(e)c(e) = ∑i d(pi)
≥ ∑i d(si , ti).Routing solution cost ≥ Any toll solution cost.
![Page 221: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/221.jpg)
Proving lower bound.
Routing solution: pi connects (si , ti) and has length d(pi).
c(e) - congestion on edge e under routing.Max c(e)?maxe c(e)≥ ∑e c(e)d(e) since ∑e d(e) = 1.
∑e
c(e)d(e) = ∑i
d(pi)
∑i
d(pi) = ∑i
∑e∈pi
d(e)
= ∑e
∑i :e3pi
d(e)
= ∑e
d(e) ∑i :e3pi
1
= ∑e
d(e)c(e)
A path uses “volume” d(pi).
Volume on edge is d(e)c(e).
∑i d(pi) = ∑e d(e)c(e).
maxe c(e)≥ ∑e d(e)c(e) = ∑i d(pi)≥ ∑i d(si , ti).
Routing solution cost ≥ Any toll solution cost.
![Page 222: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/222.jpg)
Proving lower bound.
Routing solution: pi connects (si , ti) and has length d(pi).
c(e) - congestion on edge e under routing.Max c(e)?maxe c(e)≥ ∑e c(e)d(e) since ∑e d(e) = 1.
∑e
c(e)d(e) = ∑i
d(pi)
∑i
d(pi) = ∑i
∑e∈pi
d(e)
= ∑e
∑i :e3pi
d(e)
= ∑e
d(e) ∑i :e3pi
1
= ∑e
d(e)c(e)
A path uses “volume” d(pi).
Volume on edge is d(e)c(e).
∑i d(pi) = ∑e d(e)c(e).
maxe c(e)≥ ∑e d(e)c(e) = ∑i d(pi)≥ ∑i d(si , ti).Routing solution cost ≥ Any toll solution cost.
![Page 223: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/223.jpg)
Toll is lower bound.
From before:Max bigger than minimum weighted average:maxe c(e)≥ ∑e c(e)d(e)Total length is total congestion: ∑e c(e)d(e) = ∑i d(pi)
Each path, pi , in routing has length d(pi)≥ d(si , ti).
maxe
c(e)≥ ∑e
c(e)d(e) =∑i
d(pi)≥∑i
d(si , ti).
A toll solution is lower bound on any routing solution.Any routing solution is an upper bound on a toll solution.
![Page 224: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/224.jpg)
Toll is lower bound.
From before:Max bigger than minimum weighted average:maxe c(e)≥ ∑e c(e)d(e)Total length is total congestion: ∑e c(e)d(e) = ∑i d(pi)Each path, pi , in routing has length d(pi)≥ d(si , ti).
maxe
c(e)≥ ∑e
c(e)d(e) =∑i
d(pi)≥∑i
d(si , ti).
A toll solution is lower bound on any routing solution.Any routing solution is an upper bound on a toll solution.
![Page 225: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/225.jpg)
Toll is lower bound.
From before:Max bigger than minimum weighted average:maxe c(e)≥ ∑e c(e)d(e)Total length is total congestion: ∑e c(e)d(e) = ∑i d(pi)Each path, pi , in routing has length d(pi)≥ d(si , ti).
maxe
c(e)≥ ∑e
c(e)d(e) =∑i
d(pi)≥∑i
d(si , ti).
A toll solution is lower bound on any routing solution.Any routing solution is an upper bound on a toll solution.
![Page 226: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/226.jpg)
Toll is lower bound.
From before:Max bigger than minimum weighted average:maxe c(e)≥ ∑e c(e)d(e)Total length is total congestion: ∑e c(e)d(e) = ∑i d(pi)Each path, pi , in routing has length d(pi)≥ d(si , ti).
maxe
c(e)≥ ∑e
c(e)d(e) =∑i
d(pi)≥∑i
d(si , ti).
A toll solution is lower bound on any routing solution.Any routing solution is an upper bound on a toll solution.
![Page 227: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/227.jpg)
Toll is lower bound.
From before:Max bigger than minimum weighted average:maxe c(e)≥ ∑e c(e)d(e)Total length is total congestion: ∑e c(e)d(e) = ∑i d(pi)Each path, pi , in routing has length d(pi)≥ d(si , ti).
maxe
c(e)≥ ∑e
c(e)d(e) =∑i
d(pi)≥∑i
d(si , ti).
A toll solution is lower bound on any routing solution.Any routing solution is an upper bound on a toll solution.
![Page 228: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/228.jpg)
Toll is lower bound.
From before:Max bigger than minimum weighted average:maxe c(e)≥ ∑e c(e)d(e)Total length is total congestion: ∑e c(e)d(e) = ∑i d(pi)Each path, pi , in routing has length d(pi)≥ d(si , ti).
maxe
c(e)≥ ∑e
c(e)d(e) =∑i
d(pi)≥∑i
d(si , ti).
A toll solution is lower bound on any routing solution.
Any routing solution is an upper bound on a toll solution.
![Page 229: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/229.jpg)
Toll is lower bound.
From before:Max bigger than minimum weighted average:maxe c(e)≥ ∑e c(e)d(e)Total length is total congestion: ∑e c(e)d(e) = ∑i d(pi)Each path, pi , in routing has length d(pi)≥ d(si , ti).
maxe
c(e)≥ ∑e
c(e)d(e) =∑i
d(pi)≥∑i
d(si , ti).
A toll solution is lower bound on any routing solution.Any routing solution is an upper bound on a toll solution.
![Page 230: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/230.jpg)
Shall we continue?
![Page 231: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/231.jpg)
Algorithm.
Assign tolls.
How to route? Shortest paths!Assign routing.How to assign tolls? Higher tolls on congested edges.Toll: d(e) ∝ 2c(e).
Equilibrium:The shortest path routing has d(e) ∝ 2c(e).
The routing does not change, the tolls do not change.
![Page 232: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/232.jpg)
Algorithm.
Assign tolls.How to route?
Shortest paths!Assign routing.How to assign tolls? Higher tolls on congested edges.Toll: d(e) ∝ 2c(e).
Equilibrium:The shortest path routing has d(e) ∝ 2c(e).
The routing does not change, the tolls do not change.
![Page 233: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/233.jpg)
Algorithm.
Assign tolls.How to route? Shortest paths!
Assign routing.How to assign tolls? Higher tolls on congested edges.Toll: d(e) ∝ 2c(e).
Equilibrium:The shortest path routing has d(e) ∝ 2c(e).
The routing does not change, the tolls do not change.
![Page 234: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/234.jpg)
Algorithm.
Assign tolls.How to route? Shortest paths!Assign routing.
How to assign tolls? Higher tolls on congested edges.Toll: d(e) ∝ 2c(e).
Equilibrium:The shortest path routing has d(e) ∝ 2c(e).
The routing does not change, the tolls do not change.
![Page 235: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/235.jpg)
Algorithm.
Assign tolls.How to route? Shortest paths!Assign routing.How to assign tolls?
Higher tolls on congested edges.Toll: d(e) ∝ 2c(e).
Equilibrium:The shortest path routing has d(e) ∝ 2c(e).
The routing does not change, the tolls do not change.
![Page 236: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/236.jpg)
Algorithm.
Assign tolls.How to route? Shortest paths!Assign routing.How to assign tolls? Higher tolls on congested edges.
Toll: d(e) ∝ 2c(e).
Equilibrium:The shortest path routing has d(e) ∝ 2c(e).
The routing does not change, the tolls do not change.
![Page 237: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/237.jpg)
Algorithm.
Assign tolls.How to route? Shortest paths!Assign routing.How to assign tolls? Higher tolls on congested edges.Toll: d(e) ∝ 2c(e).
Equilibrium:The shortest path routing has d(e) ∝ 2c(e).
The routing does not change, the tolls do not change.
![Page 238: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/238.jpg)
Algorithm.
Assign tolls.How to route? Shortest paths!Assign routing.How to assign tolls? Higher tolls on congested edges.Toll: d(e) ∝ 2c(e).
Equilibrium:
The shortest path routing has d(e) ∝ 2c(e).
The routing does not change, the tolls do not change.
![Page 239: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/239.jpg)
Algorithm.
Assign tolls.How to route? Shortest paths!Assign routing.How to assign tolls? Higher tolls on congested edges.Toll: d(e) ∝ 2c(e).
Equilibrium:The shortest path routing has d(e) ∝ 2c(e).
The routing does not change, the tolls do not change.
![Page 240: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/240.jpg)
Algorithm.
Assign tolls.How to route? Shortest paths!Assign routing.How to assign tolls? Higher tolls on congested edges.Toll: d(e) ∝ 2c(e).
Equilibrium:The shortest path routing has d(e) ∝ 2c(e).
The routing does not change, the tolls do not change.
![Page 241: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/241.jpg)
How good is equilibrium?Path is routed along shortest path and d(e) ∝ 2c(e).
For e with c(e)≤ cmax −2logm; 2c(e) ≤ 2cmax−2logm = 2cmax
m2 .
copt ≥ ∑i
d(si , ti) = ∑e
d(e)c(e)
= ∑e
2c(e)
∑e′ 2c(e′) c(e) = ∑e 2c(e)c(e)∑e 2c(e) Let ct = cmax −2logm.
≥∑e:c(e)>ct 2c(e)c(e)
∑e:c(e)>ct 2c(e)+ ∑e:c(e)≤ct 2c(e)
≥(ct)∑e:c(e)>ct 2c(e)
(1+ 1m )∑e:c(e)>ct 2c(e)
≥ (ct)
1+ 1m
=cmax−2logm
(1+ 1m )
Or cmax ≤ (1+ 1m )copt +2logm.
(Almost) within 2 logm of optimal!
![Page 242: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/242.jpg)
How good is equilibrium?Path is routed along shortest path and d(e) ∝ 2c(e).
For e with c(e)≤ cmax −2logm; 2c(e) ≤ 2cmax−2logm = 2cmax
m2 .
copt ≥ ∑i
d(si , ti) = ∑e
d(e)c(e)
= ∑e
2c(e)
∑e′ 2c(e′) c(e) = ∑e 2c(e)c(e)∑e 2c(e) Let ct = cmax −2logm.
≥∑e:c(e)>ct 2c(e)c(e)
∑e:c(e)>ct 2c(e)+ ∑e:c(e)≤ct 2c(e)
≥(ct)∑e:c(e)>ct 2c(e)
(1+ 1m )∑e:c(e)>ct 2c(e)
≥ (ct)
1+ 1m
=cmax−2logm
(1+ 1m )
Or cmax ≤ (1+ 1m )copt +2logm.
(Almost) within 2 logm of optimal!
![Page 243: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/243.jpg)
How good is equilibrium?Path is routed along shortest path and d(e) ∝ 2c(e).
For e with c(e)≤ cmax −2logm; 2c(e) ≤ 2cmax−2logm = 2cmax
m2 .
copt ≥ ∑i
d(si , ti) = ∑e
d(e)c(e)
= ∑e
2c(e)
∑e′ 2c(e′) c(e)
=∑e 2c(e)c(e)
∑e 2c(e) Let ct = cmax −2logm.
≥∑e:c(e)>ct 2c(e)c(e)
∑e:c(e)>ct 2c(e)+ ∑e:c(e)≤ct 2c(e)
≥(ct)∑e:c(e)>ct 2c(e)
(1+ 1m )∑e:c(e)>ct 2c(e)
≥ (ct)
1+ 1m
=cmax−2logm
(1+ 1m )
Or cmax ≤ (1+ 1m )copt +2logm.
(Almost) within 2 logm of optimal!
![Page 244: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/244.jpg)
How good is equilibrium?Path is routed along shortest path and d(e) ∝ 2c(e).
For e with c(e)≤ cmax −2logm; 2c(e) ≤ 2cmax−2logm = 2cmax
m2 .
copt ≥ ∑i
d(si , ti) = ∑e
d(e)c(e)
= ∑e
2c(e)
∑e′ 2c(e′) c(e) = ∑e 2c(e)c(e)∑e 2c(e)
Let ct = cmax −2logm.
≥∑e:c(e)>ct 2c(e)c(e)
∑e:c(e)>ct 2c(e)+ ∑e:c(e)≤ct 2c(e)
≥(ct)∑e:c(e)>ct 2c(e)
(1+ 1m )∑e:c(e)>ct 2c(e)
≥ (ct)
1+ 1m
=cmax−2logm
(1+ 1m )
Or cmax ≤ (1+ 1m )copt +2logm.
(Almost) within 2 logm of optimal!
![Page 245: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/245.jpg)
How good is equilibrium?Path is routed along shortest path and d(e) ∝ 2c(e).
For e with c(e)≤ cmax −2logm; 2c(e) ≤ 2cmax−2logm = 2cmax
m2 .
copt ≥ ∑i
d(si , ti) = ∑e
d(e)c(e)
= ∑e
2c(e)
∑e′ 2c(e′) c(e) = ∑e 2c(e)c(e)∑e 2c(e)
Let ct = cmax −2logm.
≥∑e:c(e)>ct 2c(e)c(e)
∑e:c(e)>ct 2c(e)+ ∑e:c(e)≤ct 2c(e)
≥(ct)∑e:c(e)>ct 2c(e)
(1+ 1m )∑e:c(e)>ct 2c(e)
≥ (ct)
1+ 1m
=cmax−2logm
(1+ 1m )
Or cmax ≤ (1+ 1m )copt +2logm.
(Almost) within 2 logm of optimal!
![Page 246: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/246.jpg)
How good is equilibrium?Path is routed along shortest path and d(e) ∝ 2c(e).
For e with c(e)≤ cmax −2logm; 2c(e) ≤ 2cmax−2logm = 2cmax
m2 .
copt ≥ ∑i
d(si , ti) = ∑e
d(e)c(e)
= ∑e
2c(e)
∑e′ 2c(e′) c(e) = ∑e 2c(e)c(e)∑e 2c(e) Let ct = cmax −2logm.
≥∑e:c(e)>ct 2c(e)c(e)
∑e:c(e)>ct 2c(e)+ ∑e:c(e)≤ct 2c(e)
≥(ct)∑e:c(e)>ct 2c(e)
(1+ 1m )∑e:c(e)>ct 2c(e)
≥ (ct)
1+ 1m
=cmax−2logm
(1+ 1m )
Or cmax ≤ (1+ 1m )copt +2logm.
(Almost) within 2 logm of optimal!
![Page 247: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/247.jpg)
How good is equilibrium?Path is routed along shortest path and d(e) ∝ 2c(e).For e with c(e)≤ cmax −2logm; 2c(e) ≤ 2cmax−2logm = 2cmax
m2 .
copt ≥ ∑i
d(si , ti) = ∑e
d(e)c(e)
= ∑e
2c(e)
∑e′ 2c(e′) c(e) = ∑e 2c(e)c(e)∑e 2c(e) Let ct = cmax −2logm.
≥∑e:c(e)>ct 2c(e)c(e)
∑e:c(e)>ct 2c(e)+ ∑e:c(e)≤ct 2c(e)
≥(ct)∑e:c(e)>ct 2c(e)
(1+ 1m )∑e:c(e)>ct 2c(e)
≥ (ct)
1+ 1m
=cmax−2logm
(1+ 1m )
Or cmax ≤ (1+ 1m )copt +2logm.
(Almost) within 2 logm of optimal!
![Page 248: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/248.jpg)
How good is equilibrium?Path is routed along shortest path and d(e) ∝ 2c(e).For e with c(e)≤ cmax −2logm; 2c(e) ≤ 2cmax−2logm = 2cmax
m2 .
copt ≥ ∑i
d(si , ti) = ∑e
d(e)c(e)
= ∑e
2c(e)
∑e′ 2c(e′) c(e) = ∑e 2c(e)c(e)∑e 2c(e) Let ct = cmax −2logm.
≥∑e:c(e)>ct 2c(e)c(e)
∑e:c(e)>ct 2c(e)+ ∑e:c(e)≤ct 2c(e)
≥(ct)∑e:c(e)>ct 2c(e)
(1+ 1m )∑e:c(e)>ct 2c(e)
≥ (ct)
1+ 1m
=cmax−2logm
(1+ 1m )
Or cmax ≤ (1+ 1m )copt +2logm.
(Almost) within 2 logm of optimal!
![Page 249: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/249.jpg)
How good is equilibrium?Path is routed along shortest path and d(e) ∝ 2c(e).For e with c(e)≤ cmax −2logm; 2c(e) ≤ 2cmax−2logm = 2cmax
m2 .
copt ≥ ∑i
d(si , ti) = ∑e
d(e)c(e)
= ∑e
2c(e)
∑e′ 2c(e′) c(e) = ∑e 2c(e)c(e)∑e 2c(e) Let ct = cmax −2logm.
≥∑e:c(e)>ct 2c(e)c(e)
∑e:c(e)>ct 2c(e)+ ∑e:c(e)≤ct 2c(e)
≥(ct)∑e:c(e)>ct 2c(e)
(1+ 1m )∑e:c(e)>ct 2c(e)
≥ (ct)
1+ 1m
=cmax−2logm
(1+ 1m )
Or cmax ≤ (1+ 1m )copt +2logm.
(Almost) within 2 logm of optimal!
![Page 250: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/250.jpg)
How good is equilibrium?Path is routed along shortest path and d(e) ∝ 2c(e).For e with c(e)≤ cmax −2logm; 2c(e) ≤ 2cmax−2logm = 2cmax
m2 .
copt ≥ ∑i
d(si , ti) = ∑e
d(e)c(e)
= ∑e
2c(e)
∑e′ 2c(e′) c(e) = ∑e 2c(e)c(e)∑e 2c(e) Let ct = cmax −2logm.
≥∑e:c(e)>ct 2c(e)c(e)
∑e:c(e)>ct 2c(e)+ ∑e:c(e)≤ct 2c(e)
≥(ct)∑e:c(e)>ct 2c(e)
(1+ 1m )∑e:c(e)>ct 2c(e)
≥ (ct)
1+ 1m
=cmax−2logm
(1+ 1m )
Or cmax ≤ (1+ 1m )copt +2logm.
(Almost) within 2 logm of optimal!
![Page 251: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/251.jpg)
How good is equilibrium?Path is routed along shortest path and d(e) ∝ 2c(e).For e with c(e)≤ cmax −2logm; 2c(e) ≤ 2cmax−2logm = 2cmax
m2 .
copt ≥ ∑i
d(si , ti) = ∑e
d(e)c(e)
= ∑e
2c(e)
∑e′ 2c(e′) c(e) = ∑e 2c(e)c(e)∑e 2c(e) Let ct = cmax −2logm.
≥∑e:c(e)>ct 2c(e)c(e)
∑e:c(e)>ct 2c(e)+ ∑e:c(e)≤ct 2c(e)
≥(ct)∑e:c(e)>ct 2c(e)
(1+ 1m )∑e:c(e)>ct 2c(e)
≥ (ct)
1+ 1m
=cmax−2logm
(1+ 1m )
Or cmax ≤ (1+ 1m )copt +2logm.
(Almost) within 2 logm of optimal!
![Page 252: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/252.jpg)
The end: sort of.
![Page 253: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/253.jpg)
Getting to equilibrium.
Maybe no equilibrium!
Approximate equilibrium:
Each path is routed along a path with lengthwithin a factor of 3 of the shortest path and d(e) ∝ 2c(e).
Lose a factor of three at the beginning.copt ≥ ∑i d(si , ti)≥ 1
3 ∑e d(pi).
We obtain cmax = 3(1+ 1m )copt +2logm.
This is worse!What do we gain?
![Page 254: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/254.jpg)
Getting to equilibrium.
Maybe no equilibrium!
Approximate equilibrium:
Each path is routed along a path with lengthwithin a factor of 3 of the shortest path and d(e) ∝ 2c(e).
Lose a factor of three at the beginning.copt ≥ ∑i d(si , ti)≥ 1
3 ∑e d(pi).
We obtain cmax = 3(1+ 1m )copt +2logm.
This is worse!What do we gain?
![Page 255: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/255.jpg)
Getting to equilibrium.
Maybe no equilibrium!
Approximate equilibrium:
Each path is routed along a path with lengthwithin a factor of 3 of the shortest path and d(e) ∝ 2c(e).
Lose a factor of three at the beginning.copt ≥ ∑i d(si , ti)≥ 1
3 ∑e d(pi).
We obtain cmax = 3(1+ 1m )copt +2logm.
This is worse!What do we gain?
![Page 256: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/256.jpg)
Getting to equilibrium.
Maybe no equilibrium!
Approximate equilibrium:
Each path is routed along a path with lengthwithin a factor of 3 of the shortest path and d(e) ∝ 2c(e).
Lose a factor of three at the beginning.
copt ≥ ∑i d(si , ti)≥ 13 ∑e d(pi).
We obtain cmax = 3(1+ 1m )copt +2logm.
This is worse!What do we gain?
![Page 257: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/257.jpg)
Getting to equilibrium.
Maybe no equilibrium!
Approximate equilibrium:
Each path is routed along a path with lengthwithin a factor of 3 of the shortest path and d(e) ∝ 2c(e).
Lose a factor of three at the beginning.copt ≥ ∑i d(si , ti)≥ 1
3 ∑e d(pi).
We obtain cmax = 3(1+ 1m )copt +2logm.
This is worse!What do we gain?
![Page 258: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/258.jpg)
Getting to equilibrium.
Maybe no equilibrium!
Approximate equilibrium:
Each path is routed along a path with lengthwithin a factor of 3 of the shortest path and d(e) ∝ 2c(e).
Lose a factor of three at the beginning.copt ≥ ∑i d(si , ti)≥ 1
3 ∑e d(pi).
We obtain cmax = 3(1+ 1m )copt +2logm.
This is worse!What do we gain?
![Page 259: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/259.jpg)
Getting to equilibrium.
Maybe no equilibrium!
Approximate equilibrium:
Each path is routed along a path with lengthwithin a factor of 3 of the shortest path and d(e) ∝ 2c(e).
Lose a factor of three at the beginning.copt ≥ ∑i d(si , ti)≥ 1
3 ∑e d(pi).
We obtain cmax = 3(1+ 1m )copt +2logm.
This is worse!
What do we gain?
![Page 260: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/260.jpg)
Getting to equilibrium.
Maybe no equilibrium!
Approximate equilibrium:
Each path is routed along a path with lengthwithin a factor of 3 of the shortest path and d(e) ∝ 2c(e).
Lose a factor of three at the beginning.copt ≥ ∑i d(si , ti)≥ 1
3 ∑e d(pi).
We obtain cmax = 3(1+ 1m )copt +2logm.
This is worse!What do we gain?
![Page 261: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/261.jpg)
An algorithm!
Algorithm: reroute paths that are off by a factor of three.(Note: d(e) recomputed every rerouting.)
si ti
p: w(p) = X
=⇒ w ′(p) = X/2−1 for c(e)
+1 for c(e)
p′: w(p′)≤ X/3
=⇒ w ′(p′)≤ 2X/3
Potential function: ∑e w(e), w(e) = 2c(e)
Moving path:Divides w(e) along long path (with w(p) of X ) by two.Multiplies w(e) along shorter (w(p)≤ X/3) path by two.
−X2 + X
3 =−X6 .
Potential function decreases. =⇒ termination and existence.
![Page 262: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/262.jpg)
An algorithm!
Algorithm: reroute paths that are off by a factor of three.(Note: d(e) recomputed every rerouting.)
si ti
p: w(p) = X
=⇒ w ′(p) = X/2−1 for c(e)
+1 for c(e)
p′: w(p′)≤ X/3
=⇒ w ′(p′)≤ 2X/3
Potential function: ∑e w(e), w(e) = 2c(e)
Moving path:Divides w(e) along long path (with w(p) of X ) by two.Multiplies w(e) along shorter (w(p)≤ X/3) path by two.
−X2 + X
3 =−X6 .
Potential function decreases. =⇒ termination and existence.
![Page 263: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/263.jpg)
An algorithm!
Algorithm: reroute paths that are off by a factor of three.(Note: d(e) recomputed every rerouting.)
si ti
p: w(p) = X
=⇒ w ′(p) = X/2−1 for c(e)
+1 for c(e)
p′: w(p′)≤ X/3
=⇒ w ′(p′)≤ 2X/3
Potential function: ∑e w(e), w(e) = 2c(e)
Moving path:Divides w(e) along long path (with w(p) of X ) by two.Multiplies w(e) along shorter (w(p)≤ X/3) path by two.
−X2 + X
3 =−X6 .
Potential function decreases. =⇒ termination and existence.
![Page 264: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/264.jpg)
An algorithm!
Algorithm: reroute paths that are off by a factor of three.(Note: d(e) recomputed every rerouting.)
si ti
p: w(p) = X
=⇒ w ′(p) = X/2−1 for c(e)
+1 for c(e)
p′: w(p′)≤ X/3
=⇒ w ′(p′)≤ 2X/3
Potential function: ∑e w(e), w(e) = 2c(e)
Moving path:
Divides w(e) along long path (with w(p) of X ) by two.Multiplies w(e) along shorter (w(p)≤ X/3) path by two.
−X2 + X
3 =−X6 .
Potential function decreases. =⇒ termination and existence.
![Page 265: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/265.jpg)
An algorithm!
Algorithm: reroute paths that are off by a factor of three.(Note: d(e) recomputed every rerouting.)
si ti
p: w(p) = X=⇒ w ′(p) = X/2−1 for c(e)
+1 for c(e)
p′: w(p′)≤ X/3
=⇒ w ′(p′)≤ 2X/3
Potential function: ∑e w(e), w(e) = 2c(e)
Moving path:Divides w(e) along long path (with w(p) of X ) by two.
Multiplies w(e) along shorter (w(p)≤ X/3) path by two.
−X2 + X
3 =−X6 .
Potential function decreases. =⇒ termination and existence.
![Page 266: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/266.jpg)
An algorithm!
Algorithm: reroute paths that are off by a factor of three.(Note: d(e) recomputed every rerouting.)
si ti
p: w(p) = X=⇒ w ′(p) = X/2−1 for c(e)
+1 for c(e)p′: w(p′)≤ X/3
=⇒ w ′(p′)≤ 2X/3
Potential function: ∑e w(e), w(e) = 2c(e)
Moving path:Divides w(e) along long path (with w(p) of X ) by two.Multiplies w(e) along shorter (w(p)≤ X/3) path by two.
−X2 + X
3 =−X6 .
Potential function decreases. =⇒ termination and existence.
![Page 267: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/267.jpg)
An algorithm!
Algorithm: reroute paths that are off by a factor of three.(Note: d(e) recomputed every rerouting.)
si ti
p: w(p) = X=⇒ w ′(p) = X/2−1 for c(e)
+1 for c(e)p′: w(p′)≤ X/3
=⇒ w ′(p′)≤ 2X/3
Potential function: ∑e w(e), w(e) = 2c(e)
Moving path:Divides w(e) along long path (with w(p) of X ) by two.Multiplies w(e) along shorter (w(p)≤ X/3) path by two.
−X2 + X
3 =−X6 .
Potential function decreases. =⇒ termination and existence.
![Page 268: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/268.jpg)
An algorithm!
Algorithm: reroute paths that are off by a factor of three.(Note: d(e) recomputed every rerouting.)
si ti
p: w(p) = X=⇒ w ′(p) = X/2−1 for c(e)
+1 for c(e)p′: w(p′)≤ X/3
=⇒ w ′(p′)≤ 2X/3
Potential function: ∑e w(e), w(e) = 2c(e)
Moving path:Divides w(e) along long path (with w(p) of X ) by two.Multiplies w(e) along shorter (w(p)≤ X/3) path by two.
−X2 + X
3 =−X6 .
Potential function decreases.
=⇒ termination and existence.
![Page 269: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/269.jpg)
An algorithm!
Algorithm: reroute paths that are off by a factor of three.(Note: d(e) recomputed every rerouting.)
si ti
p: w(p) = X=⇒ w ′(p) = X/2−1 for c(e)
+1 for c(e)p′: w(p′)≤ X/3
=⇒ w ′(p′)≤ 2X/3
Potential function: ∑e w(e), w(e) = 2c(e)
Moving path:Divides w(e) along long path (with w(p) of X ) by two.Multiplies w(e) along shorter (w(p)≤ X/3) path by two.
−X2 + X
3 =−X6 .
Potential function decreases. =⇒ termination and existence.
![Page 270: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/270.jpg)
Tuning...
Replace d(e) = (1+ ε)c(e).
Replace factor of 3 by (1+2ε)
cmax ≤ (1+2ε)copt +2logm/ε.. (Roughly)
Fractional paths?
![Page 271: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/271.jpg)
Tuning...
Replace d(e) = (1+ ε)c(e).
Replace factor of 3 by (1+2ε)
cmax ≤ (1+2ε)copt +2logm/ε.. (Roughly)
Fractional paths?
![Page 272: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/272.jpg)
Tuning...
Replace d(e) = (1+ ε)c(e).
Replace factor of 3 by (1+2ε)
cmax ≤ (1+2ε)copt +2logm/ε.. (Roughly)
Fractional paths?
![Page 273: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/273.jpg)
Tuning...
Replace d(e) = (1+ ε)c(e).
Replace factor of 3 by (1+2ε)
cmax ≤ (1+2ε)copt +2logm/ε.. (Roughly)
Fractional paths?
![Page 274: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/274.jpg)
Tuning...
Replace d(e) = (1+ ε)c(e).
Replace factor of 3 by (1+2ε)
cmax ≤ (1+2ε)copt +2logm/ε.. (Roughly)
Fractional paths?
![Page 275: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/275.jpg)
Wrap up.
Dueling players:
Toll player raises tolls on congested edges.Congestion player avoids tolls.
Converges to near optimal solution!
A lower bound is “necessary” (natural),and helpful (mysterious?)!
![Page 276: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/276.jpg)
Wrap up.
Dueling players:Toll player raises tolls on congested edges.
Congestion player avoids tolls.
Converges to near optimal solution!
A lower bound is “necessary” (natural),and helpful (mysterious?)!
![Page 277: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/277.jpg)
Wrap up.
Dueling players:Toll player raises tolls on congested edges.Congestion player avoids tolls.
Converges to near optimal solution!
A lower bound is “necessary” (natural),and helpful (mysterious?)!
![Page 278: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/278.jpg)
Wrap up.
Dueling players:Toll player raises tolls on congested edges.Congestion player avoids tolls.
Converges to near optimal solution!
A lower bound is “necessary” (natural),and helpful (mysterious?)!
![Page 279: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/279.jpg)
Wrap up.
Dueling players:Toll player raises tolls on congested edges.Congestion player avoids tolls.
Converges to near optimal solution!
A lower bound is “necessary” (natural),
and helpful (mysterious?)!
![Page 280: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/280.jpg)
Wrap up.
Dueling players:Toll player raises tolls on congested edges.Congestion player avoids tolls.
Converges to near optimal solution!
A lower bound is “necessary” (natural),and helpful (mysterious?)!
![Page 281: CS270: Lecture 1. - People](https://reader033.vdocument.in/reader033/viewer/2022052017/628674e63ee2c755cb513726/html5/thumbnails/281.jpg)
Done for the day.....
...see you on Thursday.