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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)
We obtain three dynamic algorithms for the approximate all-pairs shortest paths problem in unweighted undirected graphs: 1) For any fixed /spl epsiv/ > 0, a decremental algorithm with an expected total running time of O(mn), where m is the number of edges and n is the number of vertices in the initial graph. Each distance query is answered in O(1)(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 give the first improvement to the space/approximation trade-off of distance oracles since the seminal result of Thorup and Zwick [STOC'01]. For unweighted graphs, our distance oracle has size $O(n^{5/3}) = O(n^{1.66\cdots})$ and, when queried about vertices at distance $d$, returns a path of length $2d+1$. For weighted graphs with $m=n^2/\alpha$ edges,(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 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)