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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)