Quasi-fully dynamic algorithms for two-connectivity, cycle equivalence and related problems
In this paper we introduce a new class of dynamic graph algorithms called quasi-fully dynamic algorithms, which are much more general than backtracking algorithms and are much simpler than fully dynamic algorithms. These algorithms are especially suitable for applications in which a certain core connected portion of the graph remains fixed, and fully dynamic updates occur on the remaining edges in the graph.
We present very simple quasi-fully dynamic algorithms with O(log n) worst case time per operation for 2-edge connectivity and cycle equivalence. The former is deterministic while the latter is Monte-Carlo type randomized. For 2-vertex connectivity, we give a randomized Las Vegas algorithm with O(log4n) expected amortized time per operation. We introduce the concept of quasi-k-edge-connectivity, which is a slightly relaxed version of k-edge connectivity, and show that it can be maintained in O(log n) worst case time per operation. We also analyze the performance of a natural extension of our quasi-fully dynamic algorithms to fully dynamic algorithms.
The quasi-fully dynamic algorithm we present for cycle equivalence (which has several applications in optimizing compilers) is of special interest since the algorithm is quite simple, and no special-purpose incremental or backtracking algorithm is known for this problem.
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