License: Creative Commons Attribution-NoDerivs 3.0 Unported license (CC BY-ND 3.0)
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DOI: 10.4230/LIPIcs.STACS.2010.2456
URN: urn:nbn:de:0030-drops-24563
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2010/2456/
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Das, Bireswar ; Datta, Samir ; Nimbhorkar, Prajakta

Log-space Algorithms for Paths and Matchings in k-trees

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Abstract

Reachability and shortest path problems are \NLC\ for general graphs. They are known to be in \Log\ for graphs of tree-width $2$ \cite{JT07}. However, for graphs of tree-width larger than $2$, no bound better than \NL\ is known.

In this paper, we improve these bounds for $k$-trees, where $k$ is a constant. In particular, the main results of our paper are log-space algorithms for reachability in directed $k$-trees, and for computation of shortest and longest paths in directed acyclic $k$-trees.

Besides the path problems mentioned above, we consider the problem of deciding whether a $k$-tree has a perfect macthing (decision version), and if so, finding a perfect matching (search version), and prove that these problems are \Log-complete.
These problems are known to be in \Ptime\ and in \RNC\ for general graphs, and in \SPL\ for planar bipartite graphs \cite{DKR08}.

Our results settle the complexity of these problems for the class of $k$-trees. The results are also applicable for bounded tree-width graphs, when a tree-decomposition is given as input. The technique
central to our algorithms is a careful implementation of divide-and-conquer approach in log-space, along with some ideas from \cite{JT07} and \cite{LMR07}.

BibTeX - Entry

@InProceedings{das_et_al:LIPIcs:2010:2456,
  author =	{Bireswar Das and Samir Datta and Prajakta Nimbhorkar},
  title =	{{Log-space Algorithms for Paths and Matchings in k-trees}},
  booktitle =	{27th International Symposium on Theoretical Aspects of Computer Science},
  pages =	{215--226},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-16-3},
  ISSN =	{1868-8969},
  year =	{2010},
  volume =	{5},
  editor =	{Jean-Yves Marion and Thomas Schwentick},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2010/2456},
  URN =		{urn:nbn:de:0030-drops-24563},
  doi =		{10.4230/LIPIcs.STACS.2010.2456},
  annote =	{Keywords: k-trees, reachability, matching, log-space}
}

Keywords: k-trees, reachability, matching, log-space
Collection: 27th International Symposium on Theoretical Aspects of Computer Science
Issue Date: 2010
Date of publication: 09.03.2010

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