All-Pairs Ancestor Problems in Weighted Dags

  • Matthias Baumgart
  • Stefan Eckhardt
  • Jan Griebsch
  • Sven Kosub
  • Johannes Nowak
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4614)

Abstract

This work studies (lowest) common ancestor problems in (weighted) directed acyclic graphs. We improve previous algorithms for the all-pairs representative LCA problem to O(n2.575) by using fast rectangular matrix multiplication. We prove a first non-trivial upper bound of O( min {n2m, n3.575 }) for the all-pairs all lowest common ancestors problem. Furthermore, classes of dags are identified for which the problem can be solved considerably faster. Our algorithms scale with the maximal number of LCAs for one pair and—based on the famous Dilworth’s theorem—with the size of a maximum antichain (i.e., width) of the dag. We extend and generalize previous results on computing shortest ancestral distances. It is shown that finding shortest distance common ancestors in weighted dags is not harder than computing all-pairs shortest distances, up to a polylogarithmic factor. Finally, we present a solution for the general all-pairs shortest distance LCA problem based on computing all-pairs all LCAs.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Matthias Baumgart
    • 1
  • Stefan Eckhardt
    • 1
  • Jan Griebsch
    • 1
  • Sven Kosub
    • 1
  • Johannes Nowak
    • 1
  1. 1.Fakultät für Informatik, Technische Universität München, Boltzmannstraße 3, D-85748 Garching bei MünchenGermany

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