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Algorithms

Algorithms. Bellman-Ford Algorithm Directed-Acyclic Graphs All Pairs-Shortest Path Algorithm. Why does Dijkstra’s greedy algorithm work?. Because we know that when we add a node u to the set S, the value d is the length of the shortest path from s to u.

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Algorithms

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  1. Algorithms Bellman-Ford Algorithm Directed-Acyclic Graphs All Pairs-Shortest Path Algorithm

  2. Why does Dijkstra’s greedy algorithm work? • Because we know that when we add a node u to the set S, the value d is the length of the shortest path from s to u. • But, this only works if the edges of the graph are nonnegative.

  3. a 10 7 b c -4 Would Dikjstra’s Algorithm work with this graph?

  4. a 6 7 b c -14 What is the length of the shortest path from a to c in this graph?

  5. Bellman-Ford Algorithm • The Bellman-Ford algorithm can be used to solve the general single source shortest path problem • Negative weights are allowed • The algorithm detects negative cycles and returns false if the graph contains one.

  6. BELLMAN-FORD(G,w,s) 1 INITIALIZE-SINGLE-SOURCE(G,s) 2 for i  1 to |V[G]| -1 3 do for each edge (u,v)  E[G] 4 do RELAX(u,v,w) 5 for each edge (u,v)  E[G] 6 do if d[v] > d[u] + w(u,v) 7 then return false 8 return true

  7. When there are no cycles of negative length, what is maximum number of edges in a shortest path when the graph has |V[G]| vertices and |E[G]| edges?

  8. Dynamic Programming Formulation • The following recurrence shows how the Bellman Ford algorithm computes the d values for paths of length k.

  9. a 10 Processing order 1 2 3 (a,c) (b,a) (c,b) (c,d) (d,b) 7 b c -4 -5 9 d

  10. Complexity of the Bellman Ford Algorithm • Time complexity: • Performance can be improved by • stopping adding a test to the loop to see if any d values were updated on the previous iteration, or • maintain a queue of vertices whose d value changed on the previous iteration-only process these on the next iteration

  11. Single-source shortest paths in directed acyclic graphs • Topological sorting is the key to efficient algorithms for many DAG applications. • A topological sort of a dag is a linear ordering of all of its vertices such that if G contains an edge (u,v), then u appears before v in the ordering.

  12. e g d f b c a a d c b e f g

  13. DAG-SHORTEST-PATHS(G,w,s) 1 topologically sort the vertices of G 2 INITIALIZE-SINGLE-SOURCE(G,s) 3 for each vertex u taken in topological order 4 do for each vertex v  Adj[u] 5 do RELAX(u,v,w)

  14. Topological Sorting TOPOLOGICAL-SORT(G) 1 call DFS(G) to compute finishing times f[v] for each vertex v 2 as each vertex is finished, insert it onto the front of a linked list 3 return the linked list of vertices

  15. Depth-first search • Goal: search all edges in the graph one time • Strategy: Search deeper in the graph whenever possible • Edges are explored out of the most recently discovered vertex v that still has unexplored edges leaving it. • Backtrack when a dead end is encountered

  16. Predecessor subgraph • The predecessor subgraph of a depth first search • forms a depth-first forest • composed of depth-first trees • The edges in E are called tree edges

  17. Vertex coloring scheme • All vertices are initially white • A vertex is colored gray when it is discovered • A vertex is colored black when it is finished (all vertices adjacent to the vertex have been examined completely)

  18. Time Stamps • Each vertex v has two time-stamps • d[v] records when v is first discovered (and grayed) • f[v] records when the search finishes examining its adjacency list (and is blackened) • For every vertex u • d[u] < f[u]

  19. Color and Time Stamp Summary • Vertex u is • white before d[u] • gray between d[u] and f[u] • black after f[u] • Time is a global variable in the pseudocode

  20. DFS(G) 1 for each vertex u  V[G] 2 do color[u] white 3 [u] nil 4 time 0 5 do for each vertex u  Adj[u] 6 do if color[u] = white 7 then DFS-VISIT(u)

  21. DFS-VISIT(u) 1 color[u] gray 2 d[u] time time time +1 3 for each vertex v  Adj[u] 4 do if color[v] = white 5 then [v] u 6 DFS-VISIT(v) 7 color[u] black 8 f[u] time time time +1

  22. b e 1/ c a g f d

  23. b 2/ e 1/ c a g f d

  24. b 2/ e 3/ 1/ c a g f d

  25. b 2/ e 3/ 4/ 1/ c a g f d

  26. b 2/ e 3/ 4/5 1/ c a g f d

  27. b 2/ e 3/6 4/5 1/ c a g f d

  28. b 2/7 e 3/6 4/5 1/ c a g f d

  29. b 2/7 e 3/6 8/ 4/5 1/ c a g f d

  30. b 2/7 e 3/6 8/9 4/5 1/ c a g f d

  31. b 2/7 e 3/6 8/9 4/5 1/ c a g 10/ f d

  32. b 2/7 e 3/6 8/9 4/5 1/ c a g 10/ 11/ f d

  33. b 2/7 e 3/6 8/9 4/5 1/ c a g 11/12 10/ f d

  34. b 2/7 e 3/6 8/9 4/5 1/ c a g 11/12 10/13 f d

  35. b 2/7 e 3/6 8/9 4/5 1/14 c a g 11/12 10/13 f d

  36. Running time of DFS • lines 1-3 of DFS • lines 5-7 of DFS • lines 2-6 of DFS-VISIT

  37. Running Time of Topological Sort • DFS • Insertion in linked list

  38. 8 b 7 e 6 4 7 7 10 2 9 5 14 c 6 a g 1 3 12 13 9 f d

  39. Running Time for DAG-SHORTEST-PATHS • Topological sort • Initialize-single source • 3-5 each edge examined one time

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