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# All-Pairs 2-Reachability in O(n^w log n) Time

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LIPIcs.ICALP.2017.74.pdf
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## Cite As

Loukas Georgiadis, Daniel Graf, Giuseppe F. Italiano, Nikos Parotsidis, and Przemyslaw Uznanski. All-Pairs 2-Reachability in O(n^w log n) Time. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 74:1-74:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
https://doi.org/10.4230/LIPIcs.ICALP.2017.74

## Abstract

In the 2-reachability problem we are given a directed graph G and we wish to determine if there are two (edge or vertex) disjoint paths from u to v, for given pair of vertices u and v. In this paper, we present an algorithm that computes 2-reachability information for all pairs of vertices in O(n^w log n) time, where n is the number of vertices and w is the matrix multiplication exponent. Hence, we show that the running time of all-pairs 2-reachability is only within a log factor of transitive closure. Moreover, our algorithm produces a witness (i.e., a separating edge or a separating vertex) for all pair of vertices where 2-reachability does not hold. By processing these witnesses, we can compute all the edge- and vertex-dominator trees of G in O(n^2) additional time, which in turn enables us to answer various connectivity queries in O(1) time. For instance, we can test in constant time if there is a path from u to v avoiding an edge e, for any pair of query vertices u and v, and any query edge e, or if there is a path from u to v avoiding a vertex w, for any query vertices u, v, and w.
##### Keywords
• 2-reachability
• All Dominator Trees
• Directed Graphs
• Boolean Matrix Multiplication

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## References

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