Planar reachability under single vertex or edge failures

Abstract

In this paper we present an efficient reachability oracle under single-edge or single-vertex failures for planar directed graphs. Specifically, we show that a planar digraph G can be preprocessed in O(n log2 n/log log n) time, producing an O(n log n)-space data structure that can answer in O(log n) time whether u can reach v in G if the vertex x (the edge f) is removed from G, for any query vertices u, v and failed vertex x (failed edge f). To the best of our knowledge, this is the first data structure for planar directed graphs with nearly optimal preprocessing time that answers all-pairs queries under any kind of failures in polylogarithmic time. We also consider 2-reachability problems, where we are given a planar digraph G and we wish to determine if there are two vertex-disjoint (edge-disjoint) paths from u to v, for query vertices u, v. In this setting we provide a nearly optimal 2-reachability oracle, which is the existential variant of the reachability oracle under single failures, with the following bounds. We can construct in O(n polylog n) time an O(n log3+o(1) n)-space data structure that can check in O(log2+o(1) n) time for any query vertices u, v whether v is 2-reachable from u, or otherwise find some separating vertex (edge) x lying on all paths from u to v in G. To obtain our results, we follow the general recursive approach of Thorup for reachability in planar graphs [J. ACM'04] and we present new data structures which generalize dominator trees and previous data structures for strong-connectivity under failures [Georgiadis et al., SODA'17]. Our new data structures work also for general digraphs and may be of independent interest.