LCA Queries in Directed Acyclic Graphs

  • Miroslaw Kowaluk
  • Andrzej Lingas
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3580)

Abstract

We present two methods for finding a lowest common ancestor (LCA) for each pair of vertices of a directed acyclic graph (dag) on n vertices and m edges.

The first method is surprisingly natural and solves the all-pairs LCA problem for the input dag on n vertices and m edges in time O(nm). As a corollary, we obtain an O(n2)-time algorithm for finding genealogical distances considerably improving the previously known O(n2.575) time-bound for this problem.

The second method relies on a novel reduction of the all-pairs LCA problem to the problem of finding maximum witnesses for Boolean matrix product. We solve the latter problem and hence also the all-pairs LCA problem in time \(O(n^{{2}+\frac{1}{4-\omega}})\), where ω =2.376 is the exponent of the fastest known matrix multiplication algorithm. This improves the previously known \(O(n^{\frac{\omega+3}{2}})\) time-bound for the general all-pairs LCA problem in dags.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Miroslaw Kowaluk
    • 1
  • Andrzej Lingas
    • 2
  1. 1.Institute of InformaticsWarsaw UniversityWarsaw
  2. 2.Department of Computer ScienceLund UniversityLund

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