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Algorithms

Algorithms. Shortest Path Problems. G = (V, E) weighted directed graph w: E-> R weight function Weight of a path p = <v 0 , v 1 ,. . ., v n > Shortest path weight from u to v Shortest path from u to v: Any path from u to v with w(p) =  (u,v)  [v] predecessor of v on a path. .

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Algorithms

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  1. Algorithms Shortest Path Problems

  2. G = (V, E) weighted directed graph w: E-> R weight function Weight of a path p = <v0, v1,. . ., vn> Shortest path weight from u to v Shortest path from u to v: Any path from u to v with w(p) = (u,v) [v] predecessor of v on a path

  3.            

  4. Variants • Single-source shortest paths: • find shortest paths from source vertex to every other vertex • Single-destination shortest paths: • find shortest paths to a destination from every vertex • Single-pair shortest-path • find shortest path from u to v • All pairs shortest paths

  5. Lemma 25.1 • Subpaths of shortest paths are shortest paths. Given G=(G,E) w: E -> R Let p = p = <v1, v2,. . ., vk> be a shortest path from v1 to vk For any i,j such that 1 i j k, let pij be a subpath from vi to vj.. Then pij is a shortest path from vi to vj.

  6. p 1 i j k

  7. Corollary 25.2 Let G = (V,E) w: E -> R Suppose shortest path p from a source s to vertex v can be decomposed into p’ s u v for vertex u and path p’. Then weight of the shortest path from s to v is (s,v) = (s,u) + w(u,v)

  8. Lemma 25.3 Let G = (V,E) w: E -> R Source vertex s For all edges (u,v)E (s,v) (s,u) + w(u,v)

  9. u1 u2 u3 s v u4 un

  10. Relaxation • Shortest path estimate d[v] is an attribute of each vertex which is an upper bound on the weight of the shortest path from s to v • Relaxation is the process of incrementally reducing d[v] until is is an exact weight of the shortest path from s to v

  11. INITIALIZE-SINGLE-SOURCE(G, s) 1. for each vertex v V(G) 2. do d[v] 3. [v]  nil 4. d[s]  0

  12.            

  13. Relaxing an Edge (u,v) • Question: Can we improve the shortest path to v found so far by going through u? • If yes, update d[v] and [v]

  14. RELAX(u,v,w) 1. if d[v] > d[u] + w(u,v) 2. then d[v]  d[u] + w(u,v) 3. [v]  u

  15. EXAMPLE 1 s  s  Relax       v u v u

  16. EXAMPLE 2 s  s  Relax       v u v u

  17. Dijkstra’s Algorithm • Problem: • Solve the single source shortest-path problem on a weighted, directed graph G(V,E) for the cases in which edge weights are non-negative

  18. Dijkstra’s Algorithm • Approach • maintain a set S of vertices whose final shortest path weights from the source s have been determined. • repeat • select vertex from V-S with the minimum shortest path estimate • insert u in S • relax all edges leaving u

  19. DIJKSTRA(G,w,s) 1. INITIALIZE-SINGLE-SOURCE(G,s) 2. S 3. Q  V[G] 4. while Q  5. do u  EXTRACT-MIN(Q) 6. S  S {u} 7. for each vertex v Adj[u] 8. do RELAX(u,v,w)

  20.            

  21. Analysis of Dijkstra’s Algorithm • Suppose priority Q is: • an ordered (by d) linked list • Building the Q O(V lg V) • Each EXTRACT-MIN O(V) • This is done V times O(V2) • Each edge is relaxed one time O(E) • Total time O(V2 + E) = O(V2)

  22. Analysis of Dijkstra’s Algorithm • Suppose priority Q is: • a binary heap • BUILD-HEAP O(V) • Each EXTRACT-MIN O(lg V) • This is done V times O(V lg V) • Each edge is relaxation O(lg V) • Each edge relaxed one time O(E lg V) • Total time O(Vlg V + E lg V))

  23. Properties of Relaxation • Lemma 25.4 G=(V,E) w: E -> R (u,v)  E After relaxing edge (u,v) by executing RELAX(u,v,w) we have d[v]  d[u] + w(u,v)

  24. Lemma 25.5 • Given: G=(V,E) w: E -> R source s  V Graph initialized by INITIALIZE-SINGLE-SOURCE(G,s) • then d[v] (s,v) for all v V and this invariant is maintained over all relaxation steps Once d[v] achieves a lower bound (s,v), it never changes

  25. Corollary 25.6 • Given: • G=(V,E) w: E -> R source s  V • No path connects s to given v • then • after initialization • d[v] (s,v) • and this inequality is maintained over all relaxation steps.

  26. Lemma 25.7 • Given: • G=(V,E) w: E -> R source s  V • Let s - - ->u ->v be the shortest path in G for all vertices u and v. • Suppose G initialized by INITIALIZE-SINGLE-SOURCE is followed by a sequence of relaxations including RELAX(u,v,w) • Then d[u] = (s,u) prior to call implies that d[u] = (s,u) after the call

  27. Bottom Line • Therefore, relaxation causes the shortest path estimates to descend monotonically toward the actual shortest-path weights.

  28. Shortest-Paths Tree of G(V,E) • The shortest-paths tree at S of G(V,E) is a directed subgraph G’-(V’,E’), where V’ V, E’E, such that • V’ is the set of vertices reachable from S in G • G’ forms a rooted tree with root s, and • for all v V’, the unique simple path from s to v in G’ is a shortest path from s to v in G

  29. Goal • We want to show that successive relaxations will yield a shortest-path tree

  30. Lemma 25.8 • Given: • G=(V,E) w: E -> R source s  V • Assume that G contains no negative-weight cycles reachable from s. • Then after the graph is initialized with INITIALIZE-SINGLE-SOURCE • • the predecessor subgraph G forms a rooted tree with root s, and • • any sequence of relaxation steps on edges in G maintains this property as an invariant.

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