A FULLY DYNAMIC REACHABILITY ALGORITHM FOR DIRECTED GRAPHS WITH AN ALMOST LINEAR UPDATE TIME

摘要

We obtain a new fully dynamic algorithm for the reachability problem in directed graphs. Our algorithm has an amortized update time of O(m + n log n) and a worst-case query time of O(n), where m is the current number of edges in the graph, and n is the number of vertices in the graph. Each update operation either inserts a set of edges that touch the same vertex, or deletes an arbitrary set of edges. The algorithm is deterministic and uses fairly simple data structures. One of the ingredients used by this new algorithm may be interesting in its own right. It is a new dynamic algorithm for strong connectivity in directed graphs with an interesting "retrospectiveness" property. Each insert operation creates a new version of the graph. A delete operation deletes edges from all versions. Strong connectivity queries can be made on each version of the graph. The algorithm handles each update in O(m alpha(n)) amortized time, and each query in O(1) worst-case time, where alpha(n) is a functional inverse of Ackermann's function appearing in the analysis of the Union-Find data structure. Note that the update time of O(m alpha(n)), in the case of a delete operation, is the time needed for updating all versions of the graph.