Unformatted text preview: T that end at C 2 , respectively. It is not hard to see that for every i ∈ C 1 and j ∈ C 2 , there exists a shortest itoj path in G that does not cross both P ‘ and P r . Let A ‘ and A r be the dense distance matrices that correspond to the graphs obtained by cutting G open along P ‘ and P r , respectively. Both A ‘ and A r are Monge, and A i,j ≤ A ‘ i,j and A i,j ≤ A r i,j . It follows that the column minima of A are the minimum between the column minima of A ‘ and A r which can be found in O (  C 1  +  C 2  ) time each using SMAWK....
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 Fall '11
 ErikDemaine
 Algorithms, Graph Theory, Shortest path problem, Monge, shortest path tree, shortest itoj path

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