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Shortest Path & Minimum-Spanning Tree Problems

Shortest Path & Minimum-Spanning Tree Problems. Spring 2007 CSE, POSTECH. Shortest Path Problems. Directed weighted graph. Path length is the sum of weights of edges on path. The vertex at which the path begins is the source vertex.

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Shortest Path & Minimum-Spanning Tree Problems

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  1. Shortest Path & Minimum-Spanning Tree Problems Spring 2007 CSE, POSTECH

  2. Shortest Path Problems • Directed weighted graph. • Path length is the sum of weights of edges on path. • The vertex at which the path begins isthe source vertex. • The vertex at which the path ends isthe destination vertex.

  3. 2 8 1 3 6 16 3 7 1 6 5 4 10 4 2 4 7 5 3 14 Example • Find a path from 1 to 7. • The path length is 14.

  4. 2 8 1 3 6 16 3 7 1 6 5 4 10 4 2 4 7 5 3 14 Example • Find another path from 1 to 7. • The path length is 11.

  5. Shortest Path Problems • Single source single destination. • Single source all destinations. • All pairs (every vertex is a source and destination).

  6. Single Source Single Destination • Possible greedy algorithm: • Leave source vertex using the cheapest/shortest edge. • Leave new vertex using the cheapest edge subject to the constraint that a new vertex is reached. • Continue until the destination is reached. • Other method • Breadth-First Search with Greedy

  7. 2 8 1 3 6 16 3 7 1 6 5 4 10 4 2 4 7 5 3 14 Greedy Shortest 1  7 Path • Find the shortest path using the SSSD algorithm

  8. 2 8 1 3 6 16 3 7 1 6 5 4 10 4 2 4 7 5 3 14 Greedy Shortest 1  7 Path • The path length is 12. • Is this the shortest path? • No! This algorithm does not work!

  9. Single Source All Destinations • Need to generate up to n (n is number of vertices) paths (including path from source to itself). • Greedy method: • Construct these up to n paths in order of increasing length. • Assume edge costs (lengths) are >= 0. • So no path has length < 0. • First shortest path is from source vertex to itself.The length of this path is 0.

  10. 2 8 1 3 6 16 3 7 1 6 5 4 10 4 2 4 7 5 3 14 path length 6 1 2 0 1 9 1 3 5 4 2 1 3 10 1 3 6 5 1 3 5 11 1 3 6 7 Single Source All Destinations

  11. path length 0 1 2 1 3 5 1 3 5 6 1 2 9 1 3 5 4 10 1 3 6 11 1 3 6 7 Greedy Single Source All Destinations Each path (other than first)is an one edge extension ofthe previous path. Next shortest path is theshortest one edge extensionof an already generatedshortest path.

  12. Greedy Single Source All Destinations • Let d(i) (distance from source(i)) be the length ofthe shortest one edge extension of an already generated shortest path, the one edge extension ends at vertex i. • The next shortest path is to an yet unreached vertex for which the d() value is the least. • Let p(i) (predecessor(i)) be the vertex just before vertex i on the shortest one edge extension to i.

  13. 2 8 1 3 6 16 3 7 1 6 5 4 10 4 2 4 7 5 3 14 1 [1] [2] [3] [4] [5] [6] [7] d - - - - - - - p - - - - - - - Greedy Single Source All Destinations Start from vertex 1. Update d and p ofadjacent vertices.

  14. 2 8 1 3 6 16 3 7 1 6 5 4 10 4 2 4 7 5 3 14 1 [1] [2] [3] [4] [5] [6] [7] d 0 6 2 16 - - 14 p - 1 1 1 - - 1 Greedy Single Source All Destinations Find a vertex withthe smallest d value.

  15. 2 8 1 3 6 16 3 7 1 6 5 4 10 4 2 4 7 5 3 14 1 [1] [2] [3] [4] [5] [6] [7] d 0 6 2 16 - - 14 p - 1 1 1 - - 1 Greedy Single Source All Destinations Add a path tovertex 3.

  16. 2 8 1 3 6 16 3 7 1 6 5 4 10 4 2 4 7 5 3 14 1 [1] [2] [3] [4] [5] [6] [7] 1 3 d 0 6 2 16 - - 14 p - 1 1 1 - - 1 Greedy Single Source All Destinations Update d and p ofadjacent vertices.

  17. 2 8 1 3 6 16 3 7 1 6 5 4 10 4 2 4 7 5 3 14 1 [1] [2] [3] [4] [5] [6] [7] 1 3 d 0 6 2 16 5 10 14 p - 1 1 1 3 3 1 Greedy Single Source All Destinations Find a vertex withthe smallest d value.

  18. 2 8 1 3 6 16 3 7 1 6 5 4 10 4 2 4 7 5 3 14 1 [1] [2] [3] [4] [5] [6] [7] 1 3 d 0 6 2 16 5 10 14 p - 1 1 1 3 3 1 Greedy Single Source All Destinations Add a path tovertex 5.

  19. 2 8 1 3 6 16 3 7 1 6 5 4 10 4 2 4 7 5 3 14 1 [1] [2] [3] [4] [5] [6] [7] 1 3 d 0 6 2 16 5 10 14 p - 1 1 1 3 3 1 1 3 5 Greedy Single Source All Destinations Update d and p ofadjacent vertices.

  20. 2 8 1 3 6 16 3 7 1 6 5 4 10 4 2 4 7 5 3 14 1 [1] [2] [3] [4] [5] [6] [7] 1 3 d 0 6 295 10 14 p - 1 1 5 3 3 1 1 3 5 Greedy Single Source All Destinations Find a vertex withthe smallest d value.

  21. 2 8 1 3 6 16 3 7 1 6 5 4 10 4 2 4 7 5 3 14 1 [1] [2] [3] [4] [5] [6] [7] 1 3 d 062 9 5 10 14 p - 1 1 5 3 3 1 1 3 5 Greedy Single Source All Destinations Add a path tovertex 2.

  22. 2 8 1 3 6 16 3 7 1 6 5 4 10 4 2 4 7 5 3 14 1 [1] [2] [3] [4] [5] [6] [7] 1 3 d 062 9 5 10 14 p - 1 1 5 3 3 1 1 3 5 1 2 Greedy Single Source All Destinations Update d and p ofadjacent vertices.

  23. 2 8 1 3 6 16 3 7 1 6 5 4 10 4 2 4 7 5 3 14 1 [1] [2] [3] [4] [5] [6] [7] 1 3 d 062 9 5 10 14 p - 1 1 5 3 3 1 1 3 5 1 2 Greedy Single Source All Destinations Find a vertex withthe smallest d value.

  24. 2 8 1 3 6 16 3 7 1 6 5 4 10 4 2 4 7 5 3 14 1 [1] [2] [3] [4] [5] [6] [7] 1 3 d 06295 10 14 p - 1 1 5 3 3 1 1 3 5 1 2 Greedy Single Source All Destinations Add a path tovertex 4.

  25. 2 8 1 3 6 16 3 7 1 6 5 4 10 4 2 4 7 5 3 14 1 [1] [2] [3] [4] [5] [6] [7] 1 3 d 06295 10 14 p - 1 1 5 3 3 1 1 3 5 1 2 1 3 5 4 Greedy Single Source All Destinations Update d and p ofadjacent vertices.

  26. 2 8 1 3 6 16 3 7 1 6 5 4 10 4 2 4 7 5 3 14 1 [1] [2] [3] [4] [5] [6] [7] 1 3 d 06295 10 12 p - 1 1 5 3 3 4 1 3 5 1 2 1 3 5 4 Greedy Single Source All Destinations Find a vertex withthe smallest d value.

  27. 2 8 1 3 6 16 3 7 1 6 5 4 10 4 2 4 7 5 3 14 1 [1] [2] [3] [4] [5] [6] [7] 1 3 d 0629510 12 p - 1 1 5 3 3 4 1 3 5 1 2 1 3 5 4 Greedy Single Source All Destinations Add a path tovertex 6.

  28. 2 8 1 3 6 16 3 7 1 6 5 4 10 4 2 4 7 5 3 14 1 [1] [2] [3] [4] [5] [6] [7] 1 3 d 0629510 12 p - 1 1 5 3 3 4 1 3 5 1 2 1 3 5 4 1 3 6 Greedy Single Source All Destinations Update d and p ofadjacent vertices.

  29. 2 8 1 3 6 16 3 7 1 6 5 4 10 4 2 4 7 5 3 14 1 [1] [2] [3] [4] [5] [6] [7] 1 3 d 062951011 p - 1 1 5 3 3 6 1 3 5 1 2 1 3 5 4 1 3 6 Greedy Single Source All Destinations Find a vertex withthe smallest d value.

  30. 2 8 1 3 6 16 3 7 1 6 5 4 10 4 2 4 7 5 3 14 1 [1] [2] [3] [4] [5] [6] [7] 1 3 d 062951011 p - 1 1 5 3 3 6 1 3 5 1 2 1 3 5 4 1 3 6 Greedy Single Source All Destinations Add a path tovertex 7.

  31. 2 8 1 3 6 16 3 7 1 6 5 4 10 4 2 4 7 5 3 14 1 1 3 [1] [2] [3] [4] [5] [6] [7] 1 3 5 d 062951011 p - 1 1 5 3 3 6 1 2 1 3 5 4 1 3 6 1 3 6 7 Greedy Single Source All Destinations We are finished!

  32. Single Source Single Destination • How can we find the shortest path for SSSD? • Use the SSAD algorithm and terminate as soon as the shortest path to desired vertex has been generated.

  33. Data Structures for Dijkstra’s Algorithm • The greedy single source all destinations algorithm is known as the Dijkstra’s algorithm • Implement d() and p() as 1D arrays. • Keep a linear list L of reachable vertices to which the shortest path is yet to be generated. • Select and remove vertex v in Lthat has the smallest d() value. • Update d() and p() values of vertices adjacent to v.

  34. Complexity • O(n) to select the next destination vertex. • O(out-degree) to update d() and p() values when adjacency lists are used. • O(n) to update d() and p() values when adjacency matrix is used. • Selection and update done once for each vertex to which a shortest path is found. • Total time is O(n2+e) = O(n2).

  35. Complexity • When a min heap of d() values is used in place of the linear list L of reachable vertices, total time is O((n+e) log n),because O(n) remove min operations and O(e) change key (d() value) operations are done. • When e is O(n2), using a min heap is worse than using a chain. • Program code for Dijkstra’s Shortest Path Algorithm is given in Program 13.5.

  36. Minimum-Cost Spanning Tree • weighted connected undirected graph • spanning tree • The cost of spanning tree is sum of edge costs • Find a spanning tree that has the minimum cost

  37. 8 10 14 1 3 5 7 2 7 4 12 6 3 2 4 6 8 9 Example • Network has 10 edges. • Spanning tree has only n-1 edges = 7 edges. • Need to either select 7 edges or discard 3.

  38. Edge Selection Greedy Strategies • Start with an n vertex forest.Consider edges in ascending order of cost.Select edge if it does not form a cycle together with already selected edges. Kruskal’s Algorithm • Start with a 1 vertex tree.Grow it into an n vertex tree by repeatedly adding a vertex and an edge.When there is a choice, add the least cost edge.Prim’s Algorithm

  39. Edge Selection Greedy Strategies • Start with an n vertex forest.Each component selects the least cost edge to connect to another component.Eliminate duplicate selections and possible cycles.Repeat until only 1 component is left.Sollin’s Algorithm

  40. Edge Rejection Greedy Strategies • Start with the connected graph.Repeatedly find a cycle and eliminate the highest cost edge on this cycle.Stop when no cycles remain. • Consider edges in descending order of cost.Eliminate an edge provided this leaves behind a connected graph.

  41. 8 10 14 1 3 5 7 1 3 5 7 2 7 4 12 6 3 2 4 6 8 2 4 6 8 9 Kruskal’s Method • Start with a forest that has no edges. • Consider edges in ascending order of cost. • Edge (1,2) is considered first and added to the forest.

  42. 8 10 14 8 1 3 5 7 1 3 5 7 2 7 4 12 6 3 2 7 4 6 3 2 4 6 8 2 4 6 8 9 Kruskal’s Method • Edge (7,8) is considered next and added. • Edge (3,4) is considered next and added. • Edge (5,6) is considered next and added. • Edge (2,3) is considered next and added. • Edge (1,3) is considered next and rejectedbecause it creates a cycle.

  43. 8 10 14 10 14 1 3 5 7 1 3 5 7 2 7 4 12 6 3 2 7 4 6 3 2 4 6 8 2 4 6 8 9 Kruskal’s Method • Edge (2,4) is considered next and rejectedbecause it creates a cycle. • Edge (3,5) is considered next and added. • Edge (3,6) is considered next and but rejected. • Edge (5,7) is considered next and added.

  44. 8 10 14 10 14 1 3 5 7 1 3 5 7 2 7 4 12 6 3 2 7 4 6 3 2 4 6 8 2 4 6 8 9 Kruskal’s Method • n-1 edges have been selected and no cycle formed. • So we must have a spanning tree with cost 46. • Min-cost spanning tree is uniquewhen all edge costs are different.

  45. 8 10 14 10 14 1 3 5 7 1 3 5 7 2 7 4 12 6 3 2 7 4 6 3 2 4 6 8 2 4 6 8 9 Prim’s Method • Start with a tree T that contains a single vertex. • Add a least cost edge (u,v) to T such that T{(u,v)} is a tree. • Repeat this edge-addition step until T contains n-1 edges. • Notice that edge (u,v) is always such that exactly one ofu and v is in T.

  46. 8 10 14 10 14 1 3 5 7 1 3 5 7 2 7 4 12 6 3 2 7 4 6 3 2 4 6 8 2 4 6 8 9 Sollin’s Method • Start with a forest that has no edges. • Each component select a least cost edge that has exactly one vertex in the component. • Duplicate selections are eliminated. • Cycles are possiblewhen the graph has some edges that have the same cost.

  47. Spanning Tree Example • Use (1) Kruskal’s method, (2) Prim’s method, and (3) Sollin’s method to obtain the spanning tree of the following: • See Figures 13.12, 13.15 & 13.16

  48. Pseudocode for Kruskal’s Method Start with an empty set T of edges. while (E is not empty && |T| != n-1) { Let (u,v) be the least cost edge in E. Delete the edge (u,v) from E. if ((u,v) does not create a cycle in T)Add edge (u,v) to T. } if (|T| == n-1) T is a min-cost spanning tree. else there is no spanning tree.

  49. Data Structures for Kruskal’s Method • Edge Set E • Operations are: • Is E empty? • Select and remove a least cost edge. • Use a min heap of edges. • Initialize. O(e) time. • Remove and return least cost edge. O(log e) time.

  50. Data Structures for Kruskal’s Method • Use an array linear list for the edges of T. • Does T have n-1 edges?Check size of linear list. O(1) time. • Does the addition of an edge (u,v) to T result in a cycle?Not easy. • Add an edge to T.Add at right end of linear list. O(1) time. • READ Chapter 13

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