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CS138A 1999 1 Single Source Shortest Paths Peter Schröder

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CS138A 1999 2 Definitions Shortest path problems weighted directed graph G=(V,E) with real valued weight function w weight of a path v 0,v 1,…,v k is sum of edge costs shortest path negative weight edges negative weight cycles representation of shortest path

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CS138A 1999 3 Shortest Paths Predecessor subgraph G p =(V p,E p ) contains all vertices reachable from source and edges induced by predecessor relationship eventually G p will be shortest paths tree shortest paths are not necessarily unique nor are the shortest path trees

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CS138A 1999 4 Relaxation Finding shortest paths by relaxation Lemma 1: optimal substructure property; any subpath of a shortest path is itself a shortest path Corollary 2: suppose a shortest path from s to v can decomposed into a path from s to u and the edge (u,v) then the cost of the shortest path from s to v is the sum of the cost of the shortest path from s to u plus w(u,v)

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CS138A 1999 5 Relaxation Shortest path properties Lemma 3: Let G=(V,E) be a weighted directed graph with weight function w and source s. Then for all edges (u,v) in E we have all algorithms will use relaxation; each vertex contains a field d[] containing an upper bound for the shortest path length; this will repeatedly be reduced

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CS138A 1999 6 Shortest Paths Algorithm 1. Initialize-Single-Source(G,s) 2. for( v in V[G] ) 3. d[v] = infinity; p[v] = nil; 4. d[s] = 0 1. Relax(u,v,w) 2. if( d[v] > d[u] + w(u,v) ) 3. d[v] = d[u] + w(u,v) 4. p[v] = u;

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CS138A 1999 7 Algorithms Main differences order in which edges are relaxed number of times each edge is relaxed Properties Lemma 4: Let G=(V,E) be weighted, directed graph with weight function w; then immediately after relaxing (u,v) in E we have

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CS138A 1999 8 Relaxation Properties Lemma 5: Let G=(V,E) be a weighted, directed graph with weight function w and source s; after intialization for all v in V and this invariant is maintained during relaxation. Moreover, once d[v] achieves its lower bound it never changes again Corollary 6: suppose a vertex is not reachable; then the above still holds

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CS138A 1999 9 Relaxation Properties Lemma 7: Let G=(V,E) bawdgwwfw and source s; let s to u to v be a shortest path in G for some vertices u,v; after initialization suppose a sequence of relaxation steps including Relax(u,v,w) is run; if any time prior to the call then anytime thereafter

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CS138A 1999 10 Dijkstra’s Algorithm Single source shortest paths algorithm for a directed, weighted graph G=(V,E) with all edge weights non-negative 1. Dijkstra(G,w,s) 2. Initialize-Single-Source(G,s) 3. S = {}; Q = V[G]; 4. while( !Q.empty() ) 5. u = Q.extractMin(); S = S + {u} 6. for( v in Adj[u] ) 7. Relax(u,v,w)

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CS138A 1999 11 Analysis Complexity Correctness greedy strategy… optimal substructure property already established Theorem 10: when running Dijkstra on a weighted directed graph with non-negative weight function w from s then at termination for all u in V

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CS138A 1999 12 Bellman-Ford Single source shortest paths algorithm allows for negative weight edges but of course no negative weight cycles 1. Bellmann-Ford(G,s) 2. Initialize-Single-Source(G,s) 3. for( i = 1 to |V[G]|-1 ) 4. for( (u,v) in E[G] ) 5. Relax(u,v,w) 6. for( (u,v) in E[G] ) 7. if( d[v] > d[u] + w(u,v) ) 8. return false 9. return true

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CS138A 1999 13 Analysis Complexity Correctness Lemma 12: Let G=(V,E) be a weighted directed graph with source s and weight function w and assume it contains no negative weight cycles reachable from s. Then upon termination of Bellman-Ford we have for all vertices u reachable from s

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CS138A 1999 14 Analysis Correctness Theorem 14: Let Bellman-Ford be run on a wdgwssawfw and no negative weight cycles reachable from s then the algorithm returns true and for all u in V and the predecessor subgraph G p is a shortest paths tree rooted at s. If G contains a negative weight cycles reachable from s it will return false.

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CS138A 1999 15 Predecessor Graphs Still have to prove things about predecessor trees… Lemma 8: Let G=(V,E) be awdgwssawfw with no negative weight cycles reachable from s. Then after the graph is initialized the predecessor subgraph G p forms a rooted (at s) tree and any sequence of relaxation steps maintains this invariant.

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CS138A 1999 16 Predecessor Subgraphs Correctness Lemma 9: Let G=(V,E) be awdgwssawfw with no negative weight cycles reachable from s. After initialization and any relaxation sequence which produces for all u in V, then the predecessor subgraph G p is a shortest paths tree rooted at s.

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CS138A 1999 17 All Pairs Shortest paths between all pairs of vertices could run single source algorithms |V| times O(VE ln V) for Dijkstra O(V 2 E) for Bellman-Ford which can be very high… mostly use adjacency matrix representation yield predecessor matrices Example: dynamic programming solution

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CS138A 1999 18 Approach Recall dynamic programming characterize the structure of an optimal solution recursively define the value of an optimal solution compute the solution bottom up Structure optimal substructure property (Lemma 1)

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CS138A 1999 19 All Pairs 1. Extend-Shortest-Paths(D,W) 2. n = rows[D] 3. for( i = 1 to n ) 4. for( j = 1 to n ) 5. d’ ij = infinity 6. for( k = 1 to n ) 7. d’ = min( d’ ij, d ik + w kj ) 8. return D’

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Shortest Path Algorithms. Definitions Variants Single-source shortest-paths problem: Given a graph, finding a shortest path from a given source.

Shortest Path Algorithms. Definitions Variants Single-source shortest-paths problem: Given a graph, finding a shortest path from a given source.

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