dcs lecture how to solve it patrick prosser. put a different number in each circle (1 to 8) such...
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Put a different number in each circle (1 to 8) suchthat adjacent circles cannot take consecutive numbers
Your Challenge
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Put a different number in each circle (1 to 8) suchthat adjacent circles cannot take consecutive numbers
That’s illegal, okay?
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3
Put a different number in each circle (1 to 8) suchthat adjacent circles cannot take consecutive numbers
That’s illegal, okay?
The Puzzle
• Place numbers 1 through 8 on nodes– Each number appears exactly once
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– No connected nodes have consecutive numbers
You have 4 minutes!
How do we solve it?
Bill Gates asks … how do we solve it?
Heuristic Search
Which nodes are hardest to number?
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Heuristic: a rule of thumb
Heuristic Search
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Heuristic Search
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Which are the least constraining values to use?
Heuristic Search
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1
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8
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Values 1 and 8
Heuristic Search
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8
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Values 1 and 8
Symmetry means we don’t need to consider: 8 1
Inference/propagation
We can now eliminate many values for other nodes
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8
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Inference/propagation: reasoning
Inference/propagation
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8
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{1,2,3,4,5,6,7,8}
Inference/propagation
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1
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8
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{2,3,4,5,6,7}
Inference/propagation
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1
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8
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{3,4,5,6}
Inference/propagation
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1
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8
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{3,4,5,6}
By symmetry
{3,4,5,6}
Inference/propagation
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1
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8
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{3,4,5,6}
{3,4,5,6}
{1,2,3,4,5,6,7,8}
Inference/propagation
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1
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8
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{3,4,5,6}
{3,4,5,6}
{2,3,4,5,6,7}
Inference/propagation
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1
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8
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{3,4,5,6}
{3,4,5,6}
{3,4,5,6}
Inference/propagation
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1
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8
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{3,4,5,6}
By symmetry
{3,4,5,6}
{3,4,5,6}
{3,4,5,6}
Inference/propagation
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1
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8
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{3,4,5,6}
{3,4,5,6,7}
{3,4,5,6}
{3,4,5,6}
{3,4,5,6}
{2,3,4,5,6}
Inference/propagation
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1
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8
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{3,4,5,6}
{3,4,5,6,7}
{3,4,5,6}
{3,4,5,6}
{3,4,5,6}
{2,3,4,5,6}
Value 2 and 7 are left in just one node’s domain
Inference/propagation
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8
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27
{3,4,5,6}
{3,4,5,6,7}
{3,4,5,6}
{3,4,5,6}
{3,4,5,6}
{2,3,4,5,6}
And propagate …
Inference/propagation
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8
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{3,4,5}
{3,4,5,6,7}
{3,4,5}
{3,4,5,6}
{3,4,5,6}
{2,3,4,5,6}
And propagate …
Inference/propagation
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8
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{3,4,5}
{3,4,5,6,7}
{3,4,5}
{4,5,6}
{4,5,6}
{2,3,4,5,6}
And propagate …
Inference/propagation
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8
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{3,4,5}
{3,4,5}
{4,5,6}
{4,5,6}
Guess a value, but be prepared to backtrack … Backtrack?
Inference/propagation
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1
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27
{3,4,5}
{3,4,5}
{4,5,6}
{4,5,6}
Guess a value, but be prepared to backtrack …
Inference/propagation
3
1
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8
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27
{3,4,5}
{3,4,5}
{4,5,6}
{4,5,6}
And propagate …
Inference/propagation
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1
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8
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27
{4,5}
{5,6}
{4,5,6}
And propagate …
Inference/propagation
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1
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8
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27
{4,5}
{5,6}
{4,5,6}
Guess another value …
Inference/propagation
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1
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5
8
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27
{4,5} {4,5,6}
Guess another value …
Inference/propagation
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1
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5
8
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{4,5} {4,5,6}
And propagate …
Inference/propagation
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1
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8
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27
{4} {4,6}
And propagate …
Inference/propagation
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4
5
8
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27
{4} {4,6}
One node has only a single value left …
Inference/propagation
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{6}
Solution!
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How does a computer solve
it?
Bill Gates says … how does a computer solve it?
• Variable, vi for each node• Domain of {1, …, 8}• Constraints
– All values used
Alldifferent(v1 v2 v3 v4 v5 v6 v7 v8)
– No consecutive numbers for adjoining nodes
|v1 - v2 | > 1
|v1 - v3 | > 1…
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A Constraint Satisfaction Problem
How we might input the problem to a program
Viewing the problem as a “graph” with 8 “vertices” and 17 “edges”
Graph Theory?
8 vertices, 17 edges
vertex 0 is adjacent to vertex 1
vertex 3 is adjacent to vertex 7
0
1 2
6 7
5 4
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Our Problem as a Graph
Computer scientists count
from zero
By the way, Bill Gates says …
A Java (Constraint) Programto solve our problem
Read in the name of the input file
Make a “Problem” and attach“variables” to it
Note: variables represent our vertices
Constrain all variables take different values
Read in edges and constrain correspondingvariables/vertices non-consecutive
Solve the problem!
Using constraint propagation and backtracking search
Print out the number of solutions
Why have you read in the
puzzle as a file?
Bill Gates wants to know …
So that we can be more general
0
1 2
8 9
7 6
10
3
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This technology is called
“constraint programming”
Constraint programming
• Model problem by specifying constraints on acceptable solutions– define variables and domains
– post constraints on these variables
• Solve model– choose algorithm
• incremental assignment / backtracking search
• complete assignments / stochastic search
– design heuristics
It is used for solving the following kinds of problems
• Crew scheduling (airlines)• Railway timetabling• Factory/production scheduling• Vehicle routing problems• Network design problems• Design of locks and keys• Spatial layout• workforce management•…
Some sample problems that use constraint programming
BT workforce management
Constraints are everywhere!
• No meetings before 10am• Network traffic < 100
Gbytes/sec• PCB width < 21cm• Salary > 45k Euros…
A Commercial Reality
• First-tier software vendors use CP technology
You know, we’re doing
something on this!
Bill Gates is watching …
So, how do YOU solve it?
Learn to program a computer, learn a bit of discrete maths, algorithmics,learn about hardware, security and data protection, computer graphics, information management, project management, interactive systems, computer networks, operating systems, professional issues, software engineering, machine learning, bioinformatics, grid computing … and of course
constraint programming!
Computing Science at Glasgow
Constraint ProgrammingAn Introduction
by example
with help from Toby Walsh, Chris Beck,Barbara Smith, Peter van Beek, Edward Tsang, ...
That was a 4th year lecture …
That’s all for now folks
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