Fast lowest common ancestor computations in dags

  • Authors:

  • Stefan Eckhardt;Andreas Michael Mühling;Johannes Nowak

  • Affiliations:

  • Fakultät für Informatik, Technische Universität München, Garching, Germany;Fakultät für Informatik, Technische Universität München, Garching, Germany;Fakultät für Informatik, Technische Universität München, Garching, Germany

  • Venue:
  • ESA'07 Proceedings of the 15th annual European conference on Algorithms
  • Year:
  • 2007

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Abstract

This work studies lowest common ancestor computations in directed acyclic graphs. We present fast algorithms for solving the ALL-PAIRS REPRESENTATIVE LCA and ALL-PAIRS ALL LCA problems with expected running time of O(n2 log n) and O(n3 log log n) respectively, where the expectation is taken over a distribution of input graphs. The speed-ups over recently developed methods are achieved by applying transitive reduction on the input dags. The algorithms are experimentally evaluated against previous approaches demonstrating a significant improvement. On the purely theoretical side, we improve the upper bound for ALL-PAIRS ALL LCA to O(n3.3399). We give first fully dynamic algorithms for both ALL-PAIRS REPRESENTATIVE LCA and ALL-PAIRS ALL LCA. Here, the non-trivial update complexities are O(n2.5) and O(n3) respectively, with constant query times.