Quasi-Fully Dynamic Algorithms for Two-Connectivity and Cycle Equivalence

Abstract

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 O(log  n) amortized time per operation for cycle equivalence. The former is deterministic while the latter is Monte-Carlo-type randomized. For 2-vertex connectivity, we give a deterministic quasi-fully dynamic algorithm with O(log ³ n) amortized time per operation. 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.