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We obtain the following results related to dynamic versions of the shortest-paths problem: (i) Reductions that show that the incremental and decremental single-source shortest-paths problems, for weighted directed or undirected graphs, are, in a strong sense, at least as hard as the static all-pairs shortest-paths problem. We also obtain slightly weaker(More)
The diameter and the radius of a graph are fundamental topological parameters that have many important practical applications in real world networks. The fastest combinatorial algorithm for both parameters works by solving the all-pairs shortest paths problem (APSP) and has a running time of ~O(mn) in m-edge, n-node graphs. In a seminal paper, Aingworth,(More)
We present the first dynamic shortest paths algorithms that make any progress beyond a long-standing <i>O</i>(<i>n</i>) update time barrier (while maintaining a reasonable query time), although it is only progress for not-too-sparse graphs. In particular, we obtain new decremental algorithms for two approximate shortest-path problems in unweighted,(More)
We study a network creation game recently proposed by Fabrikant, Luthra, Maneva, Papadimitriou and Shenker. In this game, each player (vertex) can create links (edges) to other players at a cost of &alpha; per edge. The goal of every player is to minimize the sum consisting of (a) the cost of the links he has created and (b) the sum of the distances to all(More)
We obtain a new fully dynamic algorithm for the reachability problem in directed graphs. Our algorithm has an amortized update time of <i>O</i>(<i>m</i>+<i>n</i> log <i>n</i>) and a worst-case query time of <i>O</i>(<i>n</i>), where <i>m</i> is the current number of edges in the graph, and <i>n</i> is the number of vertices in the graph. Each update(More)
Let <i>G</i> = (<i>V,E</i>) be a <i>directed</i> graph and let <i>P</i> be a shortest path from <i>s</i> to <i>t</i> in <i>G</i>. In the <i>replacement paths</i> problem, we are required to find, for every edge <i>e</i> on <i>P</i>, a shortest path from <i>s</i> to <i>t</i> in <i>G</i> that avoids <i>e</i>. The only known algorithm for solving the problem,(More)