Planar Reachability Under Single Vertex or Edge Failures
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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(nlog^2n/loglogn) time, producing an O(nlogn)-space data structure that can answer in O(logn) 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(nlog^O(1)n) time an O(nlog^3+o(1)n)-space data structure that can check in O(log^2+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.
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