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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(n 2)-time algorithm for finding genealogical distances considerably improving the previously known O(n 2.575) 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.

Keywords

Common Ancestor Direct Acyclic Graph Transitive Closure Source Vertex Lower Common Ancestor 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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