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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. Letn denote the number of nodes in the graph. For a sequence of Ω(m 0 ) operations, wherem 0 is the number of edges in the initial graph, the expected time forp updates isO (p log3 n ) (througout the paper the logarithms are based 2) for connectivity and bipartiteness. The worst-case time for one query isO (logn /log logn ). For thek -edge witness problem (“Does the removal ofk given edges disconnect the graph?”) the expected time forp updates isO (p log3 n ) and the expected time forq queries isO (qk log3 n ). Given a graph withk different weights, the minimum spanning tree can be maintained during   a   sequence ofp updates in expected timeO (pk log3 n ). This implies an algorithm to maintain a 1 + ε-approximation of the minimum spanning tree in expected timeO ((p log3 n logU )/ε) forp updates, where the weights of the edges are between 1 andU .

Randomized fully dynamic graph algorithms with polylogarithmic time per operation