07-ShortestPaths.pptx - Shortest Path Algorithms Reference Corman Leiserson Rivest Stein[CLRS Introduction to Algorithms MIT Press 2009 Chapter 24

# 07-ShortestPaths.pptx - Shortest Path Algorithms Reference...

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Shortest Path Algorithms Reference: Corman, Leiserson, Rivest, Stein [CLRS], Introduction to Algorithms , MIT Press, 2009, Chapter 24 Outline Basic concepts Bellman-Ford algorithm Relaxation Dijkstra’s algorithm Problem Statement Given an undirected Graph G = (V, E) V = Vertex set = {0, 1, 2, … n-1} E = Edge set = {(v i , v j ) where v i , v j V and an edge exists between v i and v j } w(v i , v j ) = non-negative weight of edge (v i , v j ) Path p is a sequences of distinct vertices (v 0 , v 1 , v 2 … v m-1 ) where (v i , v i+1 ) E A distance d = Σ i=0,m-2 w(v i , v i+1 ) Example: Path 0, 1, 3, 4: d=670 0 0 2 2 1 1 3 3 4 4 420 150 100 400 175 Find the minimum weight path from vertex s 0 to every other vertex Breadth First Search Revisited Breadth first search may not compute shortest path Idea: rather than ignoring marked vertices, check if a better (shorter) path has been found, update if it has 0 0 - - - - 0 1 10 1 3 1 1 10 10 0 0 Update 1, color gray Update 2, color gray 0 becomes black 0 is black 2 is gray No updates done 1 1 10 10 Incorrect distance computed Consider breadth first search, but link distance is any positive value (not necessarily 1) 2 10 1+3 < 10 Update node 2 to 4 Bellman-Ford Algorithm: Relaxation Each vertex v stores: v.d=minimum weight path from source node s 0 computed so far (upper bound on length of shortest path) v.p=predecessor in minimum weight path computed so far Basic idea: progressively compute better and better shortest path upper bounds until no further improvement occurs Final state is referred to as a “fixed point” Algorithm Repeatedly scan through all edges in graph, updating d and p at each vertex by examining d at neighbor vertices Relaxation Step Check if shortest path to node v can be updated by using u as a predecessor v.d is current estimate of shortest path from s 0 to v u.d is current estimate of shortest path from s 0 to u If u.d+w < v.d The path through u is a better (shorter) path to v Update v to use the path through u Node u u.d Node u u.d Node v v.d Node v v.d Node s 0 Node s 0 w Repeatedly apply this relaxation step to all edges, until no further improvement (shorter paths) can be computed The result is we have computed the shortest path to all nodes Relaxation Algorithm // relaxation step for edge from vertex u to v Relax (u,v) if (u.d + w(u,v) < v.d) v.d = u.d + w(u,v);  #### You've reached the end of your free preview.

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