Dijkstra's Algorithm for Shortest Path with Non-negative Edge Weights
This content illustrates Dijkstra's Algorithm for solving the Single-Source Shortest Path problem with non-negative edge weights. The algorithm calculates final distances from a source node to all other nodes in a graph by iteratively selecting the node with the minimum distance and relaxing its outgoing edges. The process continues until all nodes are finalized. Visual representations aid in understanding the step-by-step procedure of finding the shortest path efficiently.
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Presentation Transcript
Dijkstras Algorithm SSSP, non-neg Edge weights = w(x,y) Final distances = d(x,y) = dw(x,y) b 2 2 e 2 s 6 2 c 6 1 d
Dijkstras Algorithm SSSP, non-neg s b c d e 0 L(x) b 2 2 e 2 s 6 2 x <- extractmin c 6 1 // L(x) is the distance of s to x // mark x as final d "relax all the edges out of x L(y) <- min ( L(y), L(x) + w(x,y) )
Dijkstras Algorithm SSSP, non-neg s b c d e 0 L(x) b 2 2 e 2 s 6 2 x <- extractmin c 6 1 // L(x) is the distance of s to x // mark x as final d "relax all the edges out of x L(y) <- min ( L(y), L(x) + w(x,y) )
Dijkstras Algorithm SSSP, non-neg s b c d e 0 2 6 6 L(x) b 2 2 e 2 s 6 2 x <- extractmin c 6 1 // L(x) is the distance of s to x // mark x as final d "relax all the edges out of x L(y) <- min ( L(y), L(x) + w(x,y) )
Dijkstras Algorithm SSSP, non-neg s b c d e 0 2 4 6 4 L(x) b 2 2 e 2 s 6 2 x <- extractmin c 6 1 // L(x) is the distance of s to x // mark x as final d "relax all the edges out of x L(y) <- min ( L(y), L(x) + w(x,y) )
Dijkstras Algorithm SSSP, non-neg s b c d e 0 2 4 6 4 L(x) b 2 2 e 2 s 6 2 x <- extractmin c 6 1 // L(x) is the distance of s to x // mark x as final d "relax all the edges out of x L(y) <- min ( L(y), L(x) + w(x,y) )
Dijkstras Algorithm SSSP, non-neg s b c d e 0 2 4 5 4 L(x) b 2 2 e 2 s 6 2 x <- extractmin c 6 1 // L(x) is the distance of s to x // mark x as final d "relax all the edges out of x L(y) <- min ( L(y), L(x) + w(x,y) )
Dijkstras Algorithm SSSP, non-neg makeheap m decrease-keys n extract-mins s b c d e 0 2 4 5 4 L(x) b 2 2 e 2 s 6 2 x <- extractmin c 6 1 // L(x) is the distance of s to x // mark x as final d "relax all the edges out of x L(y) <- min ( L(y), L(x) + w(x,y) )
Dijkstras Algorithm SSSP, non-neg 1. values in L array correspond to some path length, so never less than SP length 2. when x made final, L(x) = shortest path length
Dijkstras Algorithm SSSP, negative lengths? Change one edge length so that Dijkstra s algorithm gives wrong answer s b c d e 0 L(x) b 2 2 e 2 s 6 2 x <- extractmin c 6 1 // L(x) is the distance of s to x // mark x as final d "relax all the edges out of x L(y) <- min ( L(y), L(x) + w(x,y) )
Bellman-Ford-Moore Algorithm SSSP, neg wts OK!!
Bellman-Ford-Moore Algorithm SSSP, neg wts OK n rounds m time per round O(mn) time s b c d e 0 L(x) // L(x) is an upper bound on d(s,x) b 2 2 for t = 1 to n-1 for all vertices x e 2 s -2 6 c // "relax all the edges out of x L(y) <- min ( L(y), L(x) + w(x,y) ) 6 1 d Claim: if graph has non negative cycles, B-F-M is OK. Proof: induction. At end of round t, L(y) is shortest path using at most t edges.
Bellman-Ford-Moore Algorithm SSSP, neg wts OK n rounds m time per round O(mn) time s b c d e 0 2 2 5 4 L(x) // L(x) is an upper bound on d(s,x) b 2 2 for t = 1 to n-1 for all vertices x e 2 s -2 6 c // "relax all the edges out of x L(y) <- min ( L(y), L(x) + w(x,y) ) 6 1 d Claim: if graph has non negative cycles, B-F-M is OK. Proof: induction. At end of round t, L(y) is shortest path using at most t edges.
Bellman-Ford-Moore Algorithm SSSP, neg wts OK n rounds m time per round O(mn) time s b c d e 0 2 2 5 4 L(x) // L(x) is an upper bound on w(s,x) b 2 2 for t = 1 to n-1 for all vertices x e 2 s -2 6 c // "relax all the edges out of x L(y) <- min ( L(y), L(x) + w(x,y) ) 6 1 d Claim: If at end, some edge xy is over-tight ( it has L(y) > L(x) + w(x,y) ) then graph has negative cycle.
