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Unformatted text preview: Algorithms L17.1 Algorithms Professor Ashok Subramanian L ECTURE 17 Shortest Paths I • Properties of shortest paths • Dijkstra’s algorithm • Correctness • Analysis • Breadthfirst search Algorithms L17.2 Paths in graphs Consider a digraph G = ( V , E ) with edgeweight function w : E → R . The weight of path p = v → ∑ = + = 1 1 1 ) , ( ) ( k i i i v v w p w . Algorithms L17.3 Paths in graphs Consider a digraph G = ( V , E ) with edgeweight function w : E → R . The weight of path p = v → ∑ = + = 1 1 1 ) , ( ) ( k i i i v v w p w . v v v v v 4 –2 –5 1 Example: w ( p ) = –2 Algorithms L17.4 Shortest paths A shortest path from u to v is a path of minimum weight from u to v . The shortest path weight from u to v is defined as δ ( u , v ) = min{ w ( p ) : p is a path from u to v } . Note: δ ( u , v ) = ∞ if no path from u to v exists. Algorithms L17.5 Optimal substructure Theorem. A subpath of a shortest path is a shortest path. Algorithms L17.6 Optimal substructure Theorem. A subpath of a shortest path is a shortest path. Proof. Cut and paste: Algorithms L17.7 Optimal substructure Theorem. A subpath of a shortest path is a shortest path. Proof. Cut and paste: Algorithms L17.8 Triangle inequality Theorem. For all u , v , x ∈ V , we have δ ( u , v ) ≤ δ ( u , x ) + δ ( x , v ) . Algorithms L17.9 Triangle inequality Theorem. For all u , v , x ∈ V , we have δ ( u , v ) ≤ δ ( u , x ) + δ ( x , v ) . u Proof. x v δ ( u , v ) δ ( u , x ) δ ( x , v ) Algorithms L17.10 Welldefinedness of shortest paths If a graph G contains a negativeweight cycle, then some shortest paths may not exist. Algorithms L17.11 Welldefinedness of shortest paths If a graph G contains a negativeweight cycle, then some shortest paths may not exist. Example: u v … < 0 Algorithms L17.12 Singlesource shortest paths Problem. From a given source vertex s ∈ V , find the shortestpath weights δ ( s , v ) for all v ∈ V . If all edge weights w ( u , v ) are nonnegative , all shortestpath weights must exist. I DEA : Greedy. 1. Maintain a set S of vertices whose shortest path distances from s are known. 2. At each step add to S the vertex v ∈ V – S whose distance estimate from s is minimal. 3. Update the distance estimates of vertices adjacent to v . Algorithms L17.13 Dijkstra’s algorithm d [ s ] ← for each v ∈ V – { s } do d [ v ] ← ∞ S ← ∅ Q ← V ⊳ Q is a priority queue maintaining V – S Algorithms L17.14 Dijkstra’s algorithm d [ s ] ← for each v ∈ V – { s } do d [ v ] ← ∞ S ← ∅ Q ← V ⊳ Q is a priority queue maintaining V – S while Q ≠ ∅ do u ← E XTRACTM IN ( Q ) S ← S ∪ { u } for each v ∈ Adj [ u ] do if d [ v ] > d [ u ] + w ( u , v ) then d [ v ] ← d [ u ] + w ( u , v ) Algorithms L17.15 Dijkstra’s algorithm...
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This note was uploaded on 04/29/2011 for the course IT 201 taught by Professor K.v.arya during the Spring '11 term at IIT Kanpur.
 Spring '11
 k.v.arya
 Algorithms

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