Average case analysis of fully dynamic connectivity for directed graphs

Abstract

In this paper we consider the problem of maintaining the transitive closure in a directed graph under both edge insertions and deletions from the point of view of average case analysis. Say nthe number of nodes and m the number of edges. We present a data structure that supports the report of a path between two nodes in O(n log n) expected time and O(1) amortized time per update, and connectivity queries in a dense graph in O(1) expected time and O(n log n) expected amortized time per update. If m > n4/3then connectivity queries can be performed in O(1) expected time and O(log3n) expected amortized time per update. These bounds compares favorably with the best bounds known using worst case analysis. Moreover we consider an intermediate model beetween worst case analysis and average case analysis, the semi-random adversary introduced in [2].