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Lecture 34: Relaxation-Based Approach

CSC 213 – Large Scale Programming. Lecture 34: Relaxation-Based Approach. Today’s Goals. Discuss what is meant by weighted graphs Where weights placed within Graph How to use Graph ’s weights to model problems How to solve problems once Graph is set up

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Lecture 34: Relaxation-Based Approach

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  1. CSC 213 – Large Scale Programming Lecture 34:Relaxation-Based Approach

  2. Today’s Goals • Discuss what is meant by weighted graphs • Where weights placed within Graph • How to use Graph’s weights to model problems • How to solve problems once Graph is set up • Learn about myth & legend of EdsgarDijkstra • Who was he? Why should we care? How is it related? • What was his largest contribution to graph theory? • How does Dijkstra’s algorithm find smallest path?

  3. Weighted Graphs • Edge’s weight is cost of using edge • Distance, cost, travel time, &c. usable as the weight • Weights below are distance in miles PVD 849 ORD 1843 142 SFO 802 LGA 1743 337 1387 HNL 2555 1099 LAX 1233 DFW 1120 MIA

  4. Cheapest Path Problem • Find path with min. weight between 2 vertices • Sum of edge weights is the path weight • Consider the cheapest path from PVD to HNL • None of edges is cheapest in this example PVD 849 ORD 1843 142 SFO 802 LGA 1743 337 1387 HNL 2555 1099 LAX 1233 DFW 1120 MIA

  5. Cheapest Path Problem • Subpath on shortest path is shortestpath also • Otherwise we would use shorter subpath • Tree made by all shortest paths from vertex • Consider all shortest paths from PVD PVD 849 ORD 1843 142 SFO 802 LGA 1743 337 1387 HNL 2555 1099 LAX 1233 DFW 1120 MIA

  6. Dijkstra’s Algorithm • Finds cheapest paths from single vertex • Normally, computes cheapest path to all vertices • Stop once vertex computedfor single target vertex • Makes several fundamental assumptions • Connected graph needed when targeting all vertices • Only works if edge weights must be nonnegative

  7. Dijkstra’s Algorithm • Grows cloud of vertices as it goes • Cloud starts with source vetex • Add vertex to cloud with each step • Tracks distances to each vertexnot in cloud • For each vertex, considers only cheapest path • Only uses 1 edge from cloud to vertex not in cloud • Each step uses vertex with smallest distance • Adds this vertex to cloud, if not done yet • Checks if creates smaller path to any vertices

  8. Edge Relaxation • Consider e from u to z • When u added to cloud • Check adjacent vertices • Assume z not in cloud • Found faster path! • Update via relaxation • New minimum selected: d(u) = 50 10 u e z d(z) = 75 s

  9. Edge Relaxation • Consider e from u to z • When u added to cloud • Check adjacent vertices • Assume z not in cloud • Found faster path! • Update via relaxation • New minimum selected: d(u) = 50 d(z) =60 10 u e z d(z) = 75 s

  10. Edge Relaxation • Consider e from u to z • When u added to cloud • Check adjacent vertices • Assume z not in cloud • Found faster path! • Update via relaxation • New minimum selected: d(u) = 50 d(z) =60 10 u e z d(z) = 75 s

  11. Edge Relaxation • Consider e from u to z • When u added to cloud • Check adjacent vertices • Assume z not in cloud • Found faster path! • Update via relaxation • New minimum selected: d(u) = 50 d(z) =60 10 u e z s

  12. Dijkstra Example 0 A 4 8 2 8 2 4 1 7 B C D 9 3 2 5   E F

  13. Dijkstra Example 0 A 4 8 2 8 2 3 1 7 B C D 9 3 2 5 11 5 E F

  14. Dijkstra Example 0 A 4 8 2 8 2 3 1 7 B C D 9 3 2 5 8 5 E F

  15. Dijkstra Example 0 A 4 8 2 7 2 3 1 7 B C D 9 3 2 5 8 5 E F

  16. Dijkstra Example 0 A 4 8 2 7 2 3 1 7 B C D 9 3 2 5 8 5 E F

  17. Dijkstra Example 0 A 4 8 2 7 2 3 1 7 B C D 9 3 2 5 8 5 E F

  18. Why Dijkstra’s Algorithm Works • Ultimately, Dijkstrawas smart • Smarter than me, if that is possible

  19. Why Dijkstra’s Algorithm Works • Ultimately, Dijkstrawas smart • Smarter than me, if that is possible

  20. Why Dijkstra’s Algorithm Works • Ultimately, Dijkstrawas smart • Smarter than me, if that is possible • Example of a greedy algorithm • Takes best choice at each point in time • Vertices added in increasing distance • Brings vertices closer at each step • Stops when vertex cannot move closer

  21. Why No Negative-Weight Edges? 0 A 4 8 • Assume edge has negative weight • Greedily chose vertex before finding edge • Cloud will include only one endpoint • Negative weight changes everything, however • Vertices not added in order • Negative weight cycles? • Repeat cycle to optimize 6 5 7 1 4 7 B C D 0 -8 5 2 5 9 E F

  22. Why No Negative-Weight Edges? 0 A 4 8 • Assume edge has negative weight • Greedily chose vertex before finding edge • Cloud will include only one endpoint • Negative weight changes everything, however • Vertices not added in order • Negative weight cycles? • Repeat cycle to optimize 6 5 7 1 4 7 B C D 0 -8 5 2 5 9 E F C added when distance was 5, but cheapest distance is 1!

  23. Spanning Tree • Subgraphthat is both spanning subgraph& tree • Contains all vertices in graph spanning subgraph • Tree  connected without any cycles Graph

  24. Spanning Tree • Subgraphthat is both spanning subgraph& tree • Contains all vertices in graph spanning subgraph • Tree  connected without any cycles Tree

  25. Spanning Tree • Subgraphthat is both spanning subgraph& tree • Contains all vertices in graph spanning subgraph • Tree  connected without any cycles Spanning subgraph

  26. Spanning Tree • Subgraphthat is both spanning subgraph& tree • Contains all vertices in graph spanning subgraph • Tree  connected without any cycles Spanning tree

  27. Prim-Jarnik’s Algorithm • Similar to Dijkstra’salgorithm but for MST • Processing must start with some vertex s • Grow MST using “cloud” of vertices • Label vertices with least Edge weight to cloud • At each step: • Find and add vertex closest to cloud • Update adjacent vertices to vertex just added

  28. Prim-Jarnik’s Algorithm • Priority queue stores vertices outside of cloud • You all should be reminded of Dijkstra's algorithm • Three decorations used for each Vertex • Distance from cloud • Edge connecting vertex to cloud • Entry for Vertexin the priority queue

  29. Prim-Jarnik Example  D 7 2 B 4 9  5 F 2 8 C 8 3 8 E A 7 7 0

  30. Prim-Jarnik Example  D 7 2 B 4 9  5 F 2 8 C 8 3 8 E A 7 7 0

  31. Prim-Jarnik Example 7 D 7 2 B 4 9  5 F 2 8 C 8 3 8 E A 7 7 0

  32. Prim-Jarnik Example 7 D 7 2 B 4 9  5 F 2 5 C 8 3 8 E A 7 7 0

  33. Prim-Jarnik Example 7 D 7 2 B 4 9  5 F 2 5 C 8 3 8 E A 7 7 0

  34. Prim-Jarnik Example 7 D 7 2 B 4 9  5 F 2 5 C 8 3 8 E A 7 7 0

  35. Prim-Jarnik Example 7 D 7 2 B 4 9  5 F 2 5 C 8 3 8 E A 7 7 0

  36. Prim-Jarnik Example 7 D 7 2 B 4 9  5 F 2 5 C 8 3 8 E A 7 7 0

  37. Prim-Jarnik Example 7 D 7 2 B 4 9 4 5 F 2 5 C 8 3 8 E A 7 7 0

  38. Prim-Jarnik Example 7 D 7 2 B 4 9 4 5 F 2 5 C 8 3 8 E A 7 7 0

  39. Prim-Jarnik Example 7 D 7 2 B 4 9 4 5 F 2 5 C 8 3 8 E A 3 7 0

  40. Prim-Jarnik Example 7 D 7 2 B 4 9 4 5 F 2 5 C 8 3 8 E A 3 7 0

  41. Prim-Jarnik’s Analysis • Each connected vertex is: • Decorated O(deg(v)) times going through algorithm • Priority queue will have added & removed once • Takes O((n + m) log n) time using adjacency list • Each operation on priority queue takes O(logn) time • Takes O(logn) time to decorate Vertex each time

  42. For Next Lecture • Weekly assignment available on Angel • Due at special time: before next Monday’s quiz • Programming assignment #3 designs due Friday • Reading on more cheap paths for Friday • Why does everything need to be connected? • Algorithms for the uptight who do not want to relax?

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