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 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. Let n denote the number of nodes in the graph. For a sequence of \Omega\Gamma m 0 ) operations, where m 0 is the number of edges in the initial graph, the expected time for p updates is O(p log 3 n) 1 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 log 3 n) and expected time for q queries is O(qk log 3 n). Given a grap...

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