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 the backtracking algorithms and are much simpler than the fully dynamic algorithms. These algorithms are especially suitable for applications in which a certain core connected portion of the graph remains xed, and fully dynamic updates occur on the remaining edges in the graph. We present very simple quasi-fully dynamic algorithms with O(logn) 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(log n) 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(logn) 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.