Testing 2-vertex connectivity and computing pairs of vertex-disjoint s-t paths in digraphs
ICALP'10 Proceedings of the 37th international colloquium conference on Automata, languages and programming
Linear time algorithms for two disjoint paths problems on directed acyclic graphs
Theoretical Computer Science
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This dissertation deals with several topics related to the problem of finding dominators in flowgraphs. The concept of dominators has applications in various fields, including program optimization, circuit testing and theoretical biology. We are interested both in asymptotically fast algorithms and in algorithms that are practical. We begin with an experimental study of various algorithms that compute dominators efficiently in practice. We describe two practical algorithms that have been proposed in the related literature: an iterative algorithm initially presented by Allen and Cocke and later refined by Cooper, Harvey and Kennedy, and the well-known algorithm of Lengauer and Tarjan. We discuss how to achieve efficient implementations, and furthermore, introduce a new practical algorithm. We present a thorough empirical analysis using real as well as artificial data. Then we present a linear-time algorithm for dominators implementable on the pointer machine model of computation. Previously, Alstrup, Harel, Lauridsen and Thorup gave a complicated linear-time algorithm for the random-access model. Buchsbaum, Kaplan, Rogers and Westbrook presented a simpler dominators algorithm, implementable on a pointer machine and claimed linear running time. However, as we show, one of their techniques cannot be applied to the dominators problem and, consequently, their algorithm does not run in linear time. Nonetheless, based on this algorithm, we show how to achieve linear running time on a pointer machine. Next we address the question of how to verify dominators. We derive a linear-time verification algorithm, which is much simpler than the known algorithms that compute dominators in linear time. Still, this algorithm is non-trivial and we believe it provides some new intuition and ideas towards a simpler dominators algorithm. Finally we study the relation of dominators to spanning trees. Our central result is a linear-time algorithm that constructs two spanning trees of any input flowgraph G, such that corresponding paths in the two trees satisfy a vertex-disjointness property we call ancestor-dominance . This result is related to the concepts of independent spanning trees and directed st-numberings, previously studied by various authors, and implies linear-time algorithms for these constructions.