graphs shortest path

7/29/2019 Graphs Shortest Path

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James Tam

Graph Traversal Algorithms

What you know:-How to traverse a graph

What you will learn:-How to traverse a graph in an optimal fashion


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James Tam

Categories Of Shortest Path Algorithms

Unweighted shortest paths

Weighted shortest paths


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James Tam

Unweighted Graphs

1 1 1

1

1

1

1 1

1 1

1

1

1

A B C D E F G H I

A

B

C

D

E

F

G

H

I


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James Tam

Unweighted Shortest Paths

There is an equal cost incurred from traveling from one node to

another (traversing an arc)

To find the shortest path, traverse the series of nodes which

requires the fewest arcs to be traversed.


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James Tam

An Application Of Finding The Shortest Path

Airplane flights: Minimizing the number of cities to pass

through (stop-overs)

Calgary

Houston

San Jose

Miami

Vancouver

LosAngeles


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James Tam

Example Of An Unweighted ShortestPath Traversal: A -> F

C

A

D

G

B

F

E


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Definition Of The Node Class

public class Node

{

private boolean visited;

private int pathLength;

private Node predecessor;

private List neighbors;

: : :

}


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Unweighted Shortest Path Traversal: Algorithm

unweightedTraversal (Node start, Node end)

{

boolean done = false;

Queue nodeQueue = new Queue ();

Node temp;

Node nextNeighbor;

start.setVisited (true);

start.setPathLength (0);

nodeQueue.enqueue (start);


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Unweighted Shortest Path Traversal: Algorithm (2)

while ((done == false) && (nodeQueue.isEmpty() == false))

{

temp = nodeQueue.dequeue();

while ((done == false) && (temp.hasUnvisitedNeighbor())

{

nextNeighbor = temp.getNextUnvistedNeighbor();

nextNeighbor.setVisited(true);

nextNeighbor.setPathLength (temp.getPathLength() + 1);

nextNeighbor.setPredecessor (temp);

nodeQueue.enqueue (nextNeighbor);if (end.equals (nextNeighbor) == true)

done = true;

}

}

}


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The Worse Possible Case For DeterminingThe Shortest Path?


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Efficiency Of The Shortest Path Algorithm:No Path Weights?

O (n2)

while ((done == false) && (nodeQueue.isEmpty() == false))

{

: : : : :

while ((done == false) && (temp.hasUnvisitedNeighbor())

{

: : : : :

}

}


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Finding The Shortest Weighted Path

Origin

Destination


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Origin

Destination

100

75

60


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Origin

Destination

100

75

60

Pick the choice that appears to be the best choice at aparticular stage


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Origin

Destination

100

75

60

100

25

120


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Origin

Destination

100

75

60

100

25

120

Based on the information received at the next stage: pick

the choice that is currently the best option


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An Application Of Finding The ShortestWeighted Path

Hotel Local clubsLocal malls

Arenal volcano

Beaches

Riding

Hiking

Canopy tour

1 0.1

64

2

2

8 5

X

X X

X
http://www.csbsju.edu/bcll/images/Costa%20Rica/arenal-volcano.jpghttp://www.mauioceanfronthome.com/newimages/Costa%20Rica%20beach%20in%20evening.jpghttp://www.csbsju.edu/bcll/images/Costa%20Rica/arenal-volcano.jpg


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An Example Of Finding The ShortestWeighted Path: Start

C

A0

D

G

B

F

E

4

2

110

22

3

5 8 4 6

1

Node Known Distance Predecessor

A F 0 0

B F 0

C F 0

D F 0

E F 0

F F 0

G F 0


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An Example Of Finding The Shortest WeightedPath: Declare A As Known

Node Known Distance

(from A)

Predecessor

A T 0 0

B F 2 A

C F 0

D F 1 A

E F 0

F F 0

G F 0

C

A0

D1

G

B2

F

E

4

2

110

22

3

5 8 4 6

1


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An Example Of Finding The Shortest WeightedPath: Declare D As Known

Node Known Distance

(from A)

Predecessor

A T 0 0

B F 2 A

C F 3 D

D T 1 A

E F 3 D

F F 9 D

G F 5 D

C3

A0

D1

G5

B2

F9

E3

4

2

110

22

3

5 8 4 6

1


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An Example Of Finding The Shortest WeightedPath: Declare B As Known

Node Known Distance

(from A)

Predecessor

A T 0 0

B T 2 A

C F 3 D

D T 1 A

E F 3 D

F F 9 D

G F 5 D

C3

A0

D1

G5

B2

F9

E3

4

2

110

22

3

5 8 4 6

1


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James Tam

An Example Of Finding The Shortest WeightedPath: Declare C As Known

Node Known Distance

(from A)

Predecessor

A T 0 0

B T 2 A

C T 3 D

D T 1 A

E F 3 D

F F 8 C

G F 5 D

C3

A0

D1

G5

B2

F8

E3

4

2

110

22

3

5 8 4 6

1


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James Tam

An Example Of Finding The Shortest WeightedPath: Declare E As Known

Node Known Distance

(from A)

Predecessor

A T 0 0

B T 2 A

C T 3 D

D T 1 A

E T 3 D

F F 8 C

G F 5 D

C3

A0

D1

G5

B2

F8

E3

4

2

110

22

3

5 8 4 6

1


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James Tam

An Example Of Finding The Shortest WeightedPath: Declare G As Known

Node Known Distance

(from A)

Predecessor

A T 0 0

B T 2 A

C T 3 D

D T 1 A

E T 3 D

F F 6 G

G T 5 D

C3

A0

D1

G5

B2

F6

E3

4

2

110

22

3

5 8 4 6

1


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James Tam

An Example Of Finding The Shortest WeightedPath: Declare F As Known

Node Known Distance

(from A)

Predecessor

A T 0 0

B T 2 A

C T 3 D

D T 1 A

E T 3 D

F T 6 G

G T 5 D

C3

A0

D1

G5

B2

F6

E3

4

2

110

22

3

5 8 4 6

1


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James Tam

An Example Of Finding The Shortest WeightedPath: Algorithm Ends

Node Known Distance

(from A)

Predecessor

A T 0 0

B T 2 A

C T 3 D

D T 1 A

E T 3 D

F T 6 G

G T 5 D

C3

A0

D1

G5

B2

F6

E3

4

2

110

22

3

5 8 4 6

1


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James Tam

Initializing The Starting Values

public void initializeTable (Table t, Graph g,

Node start)

{

for (int i = 1; i < NO_NODES; i++)

{

t[i].setNode (g.getNextNode())

t[i].distance (1);

t[i].setPredecessor (null);

}

t [index of starting node].distance = 0;

}

1 Just pick a distance value that will be greater than any existing distance in the graph

Node Distance Predecessor

A 0 0

B 0

C 0

D 0

E 0

F 0

G 0


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James Tam

Algorithm For Finding The Shortest Path

public void dijkstraShortestPath (Graph g, Table t, Node start)

{

List toBeChecked = new List ();

Node temp;

for (all nodes in the graph)

toBeChecked.add (graph.traverse());

E.W. Dijkstra


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James Tam

Algorithm For Finding The Shortest Path(2)

while (toBeChecked.empty() == false)

{

temp = toBeChecked.removeMinDistanceNode ();

for (all nodessuc adjacent to temp which are contained in toBeChecked)

{

if (t [suc].getDistance () >(t [temp].getDistance () + distance fromsuc to temp))

{

t [suc].setDistance (t[temp].getDistance () +

distance fromsuc to temp));

t [suc].setPredecessor (temp);

}

}

}

E.W. Dijkstra

ffi i Of S A i


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James Tam

Efficiency Of The Shortest Path Algorithm:Weighted

O (n2)

for (all nodes in the graph)

toBeChecked.add (graph.traverse());

while (toBeChecked.empty() == false)

{

temp = toBeChecked.removeMinDistanceNode ();

for (all nodessuc adjacent to temp which are contained in toBeChecked)

{: : : : :

}

}

O (n)

O (n2)


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James Tam

A Related Algorithm: Making Change

Ideal case: The number of coins returned is

minimized

$0.55

$0.30

$0.05
http://www.collectible-treasures.com/catalog/images/item1589.jpghttp://www.collectible-treasures.com/catalog/images/item1591.jpghttp://www.collectible-treasures.com/catalog/images/item1591.jpg


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James Tam

Similarities Of Both Algorithms

At each stage the selection made is what appears to be best at

that point in time:- Shortest path: based on the current paths traversed select the shortestcombination of paths.

- Making change: at each stage in the change making process, give back thelargest possible denomination (without having the total amount of change

given back exceed the original amount of change owed).


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James Tam

You Should Now Know

How to determine the shortest path for traversing a graph when:

- The graph is unweighted

- The graph is weighted (Dijkstras algorithm)

What is a greedy algorithm and how Dijkstras algorithm is an

example of a greedy approach.


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Sources Of Lecture Material

Data Structures and Algorithm Analysis in C++by Mark

Allen Weiss

Data Structures and Algorithms with Javaby Frank M.

Carrano and Walter Savitch

"Data Structures and Algorithms in Javaby Adam DrozdekCPSC 331 course notes by Marina L. Gavrilova

http://pages.cpsc.ucalgary.ca/~marina/331/
http://pages.cpsc.ucalgary.ca/~marina/331/http://pages.cpsc.ucalgary.ca/~marina/331/

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