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Randomized dynamic graph algorithms with polylogarithmic time per operation

By Monika Rauch Henzinger and Valerie King I

Abstract

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 decomposition with randomization. They are Las-Vegas type randomized algorithms which use simple data structures and have a small constant factor. For a sequence of fl(rno) operations, where no is the number of edges in the initial graph, the expected time for p updates is O(p log3 n) for connectivity and bipartiteness and 0(plog4 n) for 2-edge connectivity. The worst-case time for one query is O(log n / log log n). For the k-edge witness problem (“Does the removal of k given edges disconnect the graph? ” ) the expected time for p updates is O(p log3 n) and expected time for q queries is O(qk log3 n). Note that cycle-equivalence is equivalent to the 2-edge witness problem. Given a graph wit h k different weights, the minimum spanning tree can be maintained during a sequence of p updates in expected time O(pk log3 n). This implies an algorithm to maintain a 1 +e-approximation of the minimum spanning tree in expected time O(p log3 n(log U)/e) for p updates, where the weights of the edges are between 1 and U. We sketch a modification to our connectivity algorithm which reduces the update time for this and othe

Publisher: ACM Press
Year: 1995
OAI identifier: oai:CiteSeerX.psu:10.1.1.192.8615
Provided by: CiteSeerX
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