L18-shortestPaths

With non negative weights and source s dijkstrag s

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Unformatted text preview: ) 8 for each v ⇤ Adj (u) 9 do if dv > du + w(u, v ) 10 then dv du + w(u, v ) 11 ⇥v u 12 decreasekey(Q, v ) prim(G = (V, E )) 1Q ⇧ Q is a Priority Queue 2 Initialize each v ⇤ V with key kv ⇥, v nil 3 Pick a starting node r and set kr 0 4 Insert all nodes into Q with key kv . 5 while Q ⌅= ⇧ 6 do u extract-min(Q) 7 for each v ⇤ Adj (u) 8 do if v ⇤ Q and w(u, v ) < kv 9 then v u 10 decrease-key(Q, v, w(u, v )) Theorem 4 Given any weighted, directed graph G = (V, E ) with non-negative weights and source s, dijkstra(G, s) terminates with du = (s, v ) for all v ⇤ V . Sets kv w(u, v ) 3 L19 running time Dijkstra(G = (V, E ), s) 1 for all v ⇤ V 2 do du ⇥ 3 ⇥u nil 4 ds 0 5Q makequeue(V ) use du as key 6 while Q ⌅= ⇧ 7 do u extractmin(Q) 8 for each v ⇤ Adj (u) 9 do if dv > du + w(u, v ) 10 then dv du + w(u, v ) 11 ⇥v u 12 decreasekey(Q, v ) Theorem 4 Given any weighted, directed graph G = (V, E ) with non-negative weights and source s, dijkstra(G, s) terminates with du = (s, v ) for all v ⇤ V . why does dijkstra work? triangle inequality: upper bound: ⇤(u, v) ⇥ E, (s, v) dv (s, v) (s, u) + w(u, v) breadth first search input: output: dv = (s, v ) smallest # of edges fr...
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This note was uploaded on 02/25/2014 for the course CS 4102 taught by Professor Horton during the Spring '10 term at UVA.

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