Randomized fully dynamic graph algorithms with polylogarithmic time per operation

  • Authors:

  • Monika R. Henzinger;Valerie King

  • Affiliations:

  • Google, Inc., Mountain View, CA;Univ. of Victoria, Victoria, B.C., Canada

  • Venue:
  • Journal of the ACM (JACM)
  • Year:
  • 1999

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Abstract

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 with randomization. They are Las-Vegas type randomized algorithms which use simple data structures and have a small constant factor.Let n denote the number of nodes in the graph. For a sequence of &OHgr;(m0) operations, where m0 is the number of edges in the initial graph, the expected time for p updates is O(p log3 n) (througout the paper the logarithms are based 2) for connectivity and bipartiteness. 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 the expected time for q queries is O(qk log3 n). Given a graph with 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 + &egr;-approximation of the minimum spanning tree in expected time O((p log3 n logU)/&egr;) for p updates, where the weights of the edges are between 1 and U.