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This paper presents the first fully dynamic algorithms for maintaining all-pairs shortest paths in digraphs with positive integer weights less than b. For approximate shortest paths with an error factor of (2 +), for any positive constant , the amortized update time is O(n 2 log 2 n= log log n); for an error factor of (1 +) the amortized update time is O(n(More)
This paper presents an eecient fully dynamic graph algorithm for maintaining the transitive closure of a directed graph. The algorithm updates the adjacency matrix of the transitive closure with each update to the graph. Hence, each reachability query of the form \Is there a directed path from i to j?" can be answered in O(1) time. The algorithm is(More)
This paper solves a longstanding open problem in fully dynamic algorithms: We present the first fully dynamic algorithms that maintain connectivity, bipartiteness, and approximate minimum spanning trees in polylogarithmic time per edge insertion or deletion. The algorithms are designed using a new dynamic technique that combines a novel graph decomposition(More)
We are given a set T = {T 1 , T 2 ,. .. , T k } of rooted binary trees, each T i leaf-labeled by a subset L(T i) ⊂ {1, 2,. .. , n}. If T is a tree on {1, 2,. .. , n}, we let T |L denote the minimal subtree of T induced by the nodes of L and all their ancestors. The consensus tree problem asks whether there exists a tree T * such that, for every i, T * |L(T(More)
This paper solves a longstanding open problem in dynamic algorithms: We present the first dynamic algorithms that maintain connectivity, 2-edge connectivity, bipartiteness, cycle-equivalence, and approximate minimum spanning trees in polylogarithmic time per operation. The algorithms are designed using a new dynamic technique which combines a novel graph(More)
We consider a wireless sensor network in which each sensor is able to transmit within a disk of radius one. We show with elementary techniques that there exists a constant c such that if we throw cN sensors into an n × n square (of area N) independently at random, then with high probability there will be exactly one connected component which reaches all(More)
We describe an algorithm for Byzantine agreement that is scalable in the sense that each processor sends only˜O(√ n) bits, where n is the total number of processors. Our algorithm succeeds with high probability against an adaptive adversary, which can take over processors at any time during the protocol, up to the point of taking over arbitrarily close to a(More)