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lecture_16

# lecture_16 - Introduction to Algorithms 6.046J/18.401J...

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Introduction to Algorithms 6.046J/18.401J Prof. Charles E. Leiserson L ECTURE 16 Shortest Paths III All-pairs shortest paths Matrix-multiplication algorithm Floyd-Warshall algorithm Johnson’s algorithm

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Introduction to Algorithms November 8, 2004 L16.2 © 2001–4 by Charles E. Leiserson Shortest paths Single-source shortest paths Nonnegative edge weights Dijkstra’s algorithm: O ( E + V lg V ) General Bellman-Ford algorithm: O ( VE ) DAG One pass of Bellman-Ford: O ( V + E )
Introduction to Algorithms November 8, 2004 L16.3 © 2001–4 by Charles E. Leiserson Shortest paths Single-source shortest paths Nonnegative edge weights Dijkstra’s algorithm: O ( E + V lg V ) General Bellman-Ford: O ( VE ) DAG One pass of Bellman-Ford: O ( V + E ) All-pairs shortest paths Nonnegative edge weights Dijkstra’s algorithm | V | times: O ( VE + V 2 lg V ) General Three algorithms today.

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Introduction to Algorithms November 8, 2004 L16.4 © 2001–4 by Charles E. Leiserson All-pairs shortest paths Input: Digraph G = ( V , E ) , where V = {1, 2, …, n } , with edge-weight function w : E R . Output: n × n matrix of shortest-path lengths δ ( i , j ) for all i , j V .
Introduction to Algorithms November 8, 2004 L16.5 © 2001–4 by Charles E. Leiserson All-pairs shortest paths Input: Digraph G = ( V , E ) , where V = {1, 2, …, n } , with edge-weight function w : E R . Output: n × n matrix of shortest-path lengths δ ( i , j ) for all i , j V . I DEA : Run Bellman-Ford once from each vertex. Time = O( V 2 E ) . Dense graph ( n 2 edges) ⇒ Θ ( n 4 ) time in the worst case. Good first try!

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Introduction to Algorithms November 8, 2004 L16.6 © 2001–4 by Charles E. Leiserson Dynamic programming Consider the n × n adjacency matrix A = ( a ij ) of the digraph, and define d ij (0) = 0 if i = j , if i j ; Claim: We have and for m = 1, 2, …, n – 1 , d ij ( m ) = min k { d ik ( m– 1) + a kj } .
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