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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 111 1 1 1 11 11 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 Los Angeles

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James Tam Example Of An Unweighted Shortest Path Traversal: A -> F C A D G B F E

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James Tam Definition Of The Node Class public class Node { private boolean visited; private int pathLength; private Node predecessor; private List neighbors; : : : }

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James Tam 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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James Tam 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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James Tam The Worse Possible Case For Determining The Shortest Path?

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James Tam Efficiency Of The Shortest Path Algorithm: No Path Weights? O (n 2 ) while ((done == false) && (nodeQueue.isEmpty() == false)) { ::::: while ((done == false) && (temp.hasUnvisitedNeighbor()) { ::::: }

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James Tam Finding The Shortest Weighted Path Origin Destination

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James Tam Finding The Shortest Weighted Path Origin Destination 100 75 60

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James Tam Finding The Shortest Weighted Path Origin Destination 100 75 60 Pick the choice that appears to be the best choice at a particular stage

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James Tam Finding The Shortest Weighted Path Origin Destination 100 75 60 100 25 120

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James Tam Finding The Shortest Weighted Path 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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James Tam An Application Of Finding The Shortest Weighted Path Hotel Local clubs Local malls Arenal volcano Beaches Riding Hiking Canopy tour 10.1 6 4 2 2 8 5

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James Tam An Application Of Finding The Shortest Weighted Path Hotel Local clubs Local malls Arenal volcano Beaches Riding Hiking Canopy tour 10.1 6 4 2 2 8 5 X X X X

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James Tam An Example Of Finding The Shortest Weighted Path: Start C A0 D G B F E 4 2 1 10 2 2 3 5 8 4 6 1 NodeKnownDistancePredecessor AF00 BF∞0 CF∞0 DF∞0 EF∞0 FF∞0 GF∞0

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James Tam An Example Of Finding The Shortest Weighted Path: Declare “A” As Known NodeKnownDistance (from A) Predecessor AT00 BF2A CF∞0 DF1A EF∞0 FF∞0 GF∞0 C A0 D1 G B2 F E 4 2 1 10 2 2 3 5 8 4 6 1

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James Tam An Example Of Finding The Shortest Weighted Path: Declare D As Known NodeKnownDistance (from A) Predecessor AT00 BF2A CF3D DT1A EF3D FF9D GF5D C3 A0 D1 G5 B2 F9 E3 4 2 1 10 2 2 3 5 8 4 6 1

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James Tam An Example Of Finding The Shortest Weighted Path: Declare B As Known NodeKnownDistance (from A) Predecessor AT00 BT2A CF3D DT1A EF3D FF9D GF5D C3 A0 D1 G5 B2 F9 E3 4 2 1 10 2 2 3 5 8 4 6 1

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James Tam An Example Of Finding The Shortest Weighted Path: Declare C As Known NodeKnownDistance (from A) Predecessor AT00 BT2A CT3D DT1A EF3D FF8C GF5D C3 A0 D1 G5 B2 F8 E3 4 2 1 10 2 2 3 5 8 4 6 1

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James Tam An Example Of Finding The Shortest Weighted Path: Declare E As Known NodeKnownDistance (from A) Predecessor AT00 BT2A CT3D DT1A ET3D FF8C GF5D C3 A0 D1 G5 B2 F8 E3 4 2 1 10 2 2 3 5 8 4 6 1

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James Tam An Example Of Finding The Shortest Weighted Path: Declare G As Known NodeKnownDistance (from A) Predecessor AT00 BT2A CT3D DT1A ET3D FF6G GT5D C3 A0 D1 G5 B2 F6 E3 4 2 1 10 2 2 3 5 8 4 6 1

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James Tam An Example Of Finding The Shortest Weighted Path: Declare F As Known NodeKnownDistance (from A) Predecessor AT00 BT2A CT3D DT1A ET3D FT6G GT5D C3 A0 D1 G5 B2 F6 E3 4 2 1 10 2 2 3 5 8 4 6 1

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James Tam An Example Of Finding The Shortest Weighted Path: Algorithm Ends NodeKnownDistance (from A) Predecessor AT00 BT2A CT3D DT1A ET3D FT6G GT5D C3 A0 D1 G5 B2 F6 E3 4 2 1 10 2 2 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 NodeDistancePredecessor A00 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 nodes suc adjacent to temp which are contained in toBeChecked) { if (t [suc].getDistance () > (t [temp].getDistance () + distance from suc to temp)) { t [suc].setDistance (t[temp].getDistance () + distance from suc to temp)); t [suc].setPredecessor (temp); } E.W. Dijkstra

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James Tam Efficiency Of The Shortest Path Algorithm: Weighted O (n 2 ) for (all nodes in the graph) toBeChecked.add (graph.traverse()); while (toBeChecked.empty() == false) { temp = toBeChecked.removeMinDistanceNode (); for (all nodes suc adjacent to temp which are contained in toBeChecked) { ::::: } O (n) O (n 2 )

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James Tam A Related Algorithm: Making Change $1.45 Ideal case: The number of coins returned is minimized $0.55 $0.30 $0.05

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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 shortest combination of paths. -Making change: at each stage in the change making process, give back the largest 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 (Dijkstra’s algorithm) What is a greedy algorithm and how Dijkstra’s algorithm is an example of a greedy approach.

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James Tam Sources Of Lecture Material “Data Structures and Algorithm Analysis in C++” by Mark Allen Weiss “Data Structures and Algorithms with Java” by Frank M. Carrano and Walter Savitch "Data Structures and Algorithms in Java“ by Adam Drozdek CPSC 331 course notes by Marina L. Gavrilova http://pages.cpsc.ucalgary.ca/~marina/331/ http://pages.cpsc.ucalgary.ca/~marina/331/

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Chapter 9: Graphs Shortest Paths

Chapter 9: Graphs Shortest Paths

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