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European Symposium on Algorithms

ESA 2007: Algorithms – ESA 2007 pp 265–274Cite as

Unique Lowest Common Ancestors in Dags Are Almost as Easy as Matrix Multiplication

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4698))

Abstract

We consider the problem of determining for each pair of vertices of a directed acyclic graph (dag) on n vertices whether or not it has a unique lowest common ancestor, and if so, finding such an ancestor. We show that this problem can be solved in time O(n ωlogn), where ω< 2.376 is the exponent of the fastest known algorithm for multiplication of two n×n matrices.

We show also that the problem of determining a lowest common ancestor for each pair of vertices of an arbitrary dag on n vertices is solvable in time \(\widetilde{O}(n^2p+n^{\omega})\), where p is the minimum number of directed paths covering the vertices of the dag. With the help of random bits, we can solve the latter problem in time \(\widetilde{O}(n^2p)\).

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Author information

Authors and Affiliations

  1. Institute of Informatics, Warsaw University, Warsaw,  

    Mirosław Kowaluk

  2. Department of Computer Science, Lund University, S-221 00 Lund,  

    Andrzej Lingas

Authors
  1. Mirosław Kowaluk
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  2. Andrzej Lingas
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Editor information

Lars Arge Michael Hoffmann Emo Welzl

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© 2007 Springer-Verlag Berlin Heidelberg

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Kowaluk, M., Lingas, A. (2007). Unique Lowest Common Ancestors in Dags Are Almost as Easy as Matrix Multiplication. In: Arge, L., Hoffmann, M., Welzl, E. (eds) Algorithms – ESA 2007. ESA 2007. Lecture Notes in Computer Science, vol 4698. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-75520-3_25

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  • DOI: https://doi.org/10.1007/978-3-540-75520-3_25

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-75519-7

  • Online ISBN: 978-3-540-75520-3

  • eBook Packages: Computer ScienceComputer Science (R0)

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