This preview shows pages 1–15. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: 1 Weighted Graph Algorithms. Weighted Shortest Path Problems Dijkstras algorithm All Pairs Shortest Path Warshalls algorithm Minimum spanning trees Prims algorithm 2 Been there, done that! 3 ShortestPath in Weighted Graphs Weight w(i, j) associated with each edge (v i , v j ). [sometimes youll see c(i,j)] The cost of a path v 1 v 2 ..v n is Find the shortest weighted path from s to every other vertex in G. Single Source Shortest Path 1 = + 1 1 ) 1 , ( n i i i w 4 ShortestPath Problem Solutions Unweighted Breadth First Search Nonnegative Weights Dijkstras Algorithm Negative weights with no negative cycles BellmanFord Algorithm Negative cycles No solution 5 Dijkstras Algorithm Dijkstra (G, v) foreach x V x.dist ; Initially all nodes infinite distance x.p nil ; and no known parent x.known false Q.insert(x) ; Insert into priority queue by dist Q.decreasekey(v, 0) ; Zero distance to first vertex while Q not empty v Q.deletemin() ; Vertex at minimum distance v.known = true foreach x such that (v,x) E and !x.known if v.dist + w(v,x) < x.dist then Q.decreasekey(x, v.dist + w(v,x) ) x.p v 6 Example v 3 2 v 6 v 5 v 4 v 2 v 7 4 10 1 3 2 5 8 1 4 2 6 v d v v 1 v 2 v 3 v 4 v 5 v 6 v 7 Initialization Priority Queue v 7 Example v d v v 4 1 v 2 2 v 3 v 5 v 6 v 7 Priority Queue Process v 1 v 3 2 v 6 v 5 v v v v 7 4 10 1 3 2 5 8 1 4 2 6 2 1 8 Example v d v v 2 2 v 3 3 v 5 3 v 7 5 v 6 9 Priority Queue Process v 4 v 2 v v v v v v 4 10 1 3 2 5 8 1 4 2 6 2 1 3 3 5 9 9 Example v d v v 3 3 v 5 3 v 7 5 v 6 9 Priority Queue Process v 2 v 2 v v v v v v 4 10 1 3 2 5 8 1 4 2 6 2 1 3 3 5 9 V 4 already known V 5 not shorter 10 Example v d v v 5 3 v 7 5 v 6 8 Priority Queue Process v 3 v 2 v v v v v v 4 10 1 3 2 5 8 1 4 2 6 2 1 3 3 5 8 V 1 already known V 6 shorter 11 Example v d v v 7 5 v 6 8 Priority Queue Process v 5 v 2 v v v v v v 4 10 1 3 2 5 8 1 4 2 6 2 1 3 3 5 8 V 7 not shorter 12 Example v d v v 6 6 Priority Queue Process v 7 v 2 v v v v v v 4 10 1 3 2 5 8 1 4 2 6 2 1 3 3 5 6 V 6 shorter 13 Example v d v Priority Queue Process v 6 v 2 v v v v v v 4 10 1 3 2 5 8 1 4 2 6 2 1 3 3 5 6 We be done! 14 Dijkstras Algorithm Dijkstra (G, v) foreach x...
View
Full
Document
 Spring '08
 M.McCullen
 Algorithms, Data Structures

Click to edit the document details