Average case analysis of fully dynamic reachability for directed graphs

We consider the problem of maintaining the transitive closure in a directed graph under edge insertions and deletions from the point of view of average case analysis. Say n the 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) worst case time per update, and reachability queries in O (1) expected time and O (n . log n) expected amortized time per update. If m > n(4/3) then reachability queries can be performed in O (1) expected time and O (log(3) n) expected amortized time per update. These bounds compare favorably with the best bounds known using worst case analysis. Furthermore we consider an intermediate model between worst case analysis and average case analysis: the semi-random adversary introduced in [3].