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09 - Dijkstra's Shortest Path Algorithm

# 09 - Dijkstra's Shortest Path Algorithm - Part II Graph...

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Part II: Graph Algorithms Lecture 9: Dijkstra’s Shortest Path Algorithm Lecture 9: Dijkstra’s Shortest Path Algorithm Part II: Graph Algorithms

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Objective and Outline Objective : Discuss a third graph problem: single-source shortest parths Reference : Section 24.3 of CLRS Outline : The single-source shortest paths problem Dijkstra’s algorithm Example Correctness Running time Lecture 9: Dijkstra’s Shortest Path Algorithm Part II: Graph Algorithms
Shortest Path Problem for Weighted Graphs Let G = ( V , E ) be a weighted digraph, with weight function w : E R mapping edges to real-valued weights If e = ( u , v ), we write w ( u , v ) for w ( e ). Definition The length of a path p = v 0 , v 1 , ..., v k is the sum of the weights of its constituent edges: length( p ) = k i =1 w ( v i - 1 , v i ) . Lecture 9: Dijkstra’s Shortest Path Algorithm Part II: Graph Algorithms

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Distance Definition The distance from u to v , denoted δ ( u , v ), is the length of the minimum length path if there is a path from u to v ; and is otherwise. Example length( a , b , c , e ) = 6; distance from a to e is 6 Lecture 9: Dijkstra’s Shortest Path Algorithm Part II: Graph Algorithms
Single-Source Shortest-Paths Problem Definition (Single-source shortest-paths problem) Given a digraph with non-negative edge weights G = ( V , E ) and a designated source vertex , s V , determine the distance and a shortest path from the source vertex to every vertex in the digraph. Question How do you design an efficient algorithm for this problem? Lecture 9: Dijkstra’s Shortest Path Algorithm Part II: Graph Algorithms

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Outline The single-source shortest paths problem Dijkstra’s algorithm Example Correctness Running time Lecture 9: Dijkstra’s Shortest Path Algorithm Part II: Graph Algorithms
The Rough Idea of Dijkstra’s Algorithm Maintain d [ v ] and S : d [ v ] is an estimate of the length δ ( s , v ) of the shortest path for each vertex v . S V is a subset of vertices for which we know the true distance , that is d [ v ] = δ ( s , v ). Initially S = d [ s ] = 0 and d [ v ] = for all others vertices v . One by one we select vertices from V \ S to add to S . Questions to answer at each step: 1 Which vertex do we select? 2 How do we update the distance estimates after a vertex is added to S ? Lecture 9: Dijkstra’s Shortest Path Algorithm Part II: Graph Algorithms

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Selection of vertex Question How does the algorithm select which vertex among the vertices of V \ S to process next?
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