License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.CPM.2016.25
URN: urn:nbn:de:0030-drops-60674
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2016/6067/
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Gawrychowski, Pawel ; Landau, Gad M. ; Mozes, Shay ; Weimann, Oren

The Nearest Colored Node in a Tree

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LIPIcs-CPM-2016-25.pdf (0.5 MB)

Abstract

We start a systematic study of data structures for the nearest colored node problem on trees. Given a tree with colored nodes and weighted edges, we want to answer queries (v,c) asking for the nearest node to node v that has color c. This is a natural generalization of the well-known nearest marked ancestor problem. We give an O(n)-space O(log log n)-query solution and show that this is optimal. We also consider the dynamic case where updates can change a node's color and show that in O(n) space we can support both updates and queries in O(log n) time. We complement this by showing that O(polylog n) update time implies Omega(log n \ log log n) query time. Finally, we consider the case where updates can change the edges of the tree (link-cut operations). There is a known (top-tree based) solution that requires update time that is roughly linear in the number of colors. We show that this solution is probably optimal by showing that a strictly sublinear update time implies a strictly subcubic time algorithm for the classical all pairs shortest paths problem on a general graph. We also consider versions where the tree is rooted, and the query asks for the nearest ancestor/descendant of node v that has color c, and present efficient data structures for both variants in the static and the dynamic setting.

BibTeX - Entry

@InProceedings{gawrychowski_et_al:LIPIcs:2016:6067,
  author =	{Pawel Gawrychowski and Gad M. Landau and Shay Mozes and Oren Weimann},
  title =	{{The Nearest Colored Node in a Tree}},
  booktitle =	{27th Annual Symposium on Combinatorial Pattern Matching (CPM 2016)},
  pages =	{25:1--25:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-012-5},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{54},
  editor =	{Roberto Grossi and Moshe Lewenstein},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2016/6067},
  URN =		{urn:nbn:de:0030-drops-60674},
  doi =		{10.4230/LIPIcs.CPM.2016.25},
  annote =	{Keywords: Marked ancestor, Vertex-label distance oracles, Nearest colored descend- ant, Top-trees}
}

Keywords: Marked ancestor, Vertex-label distance oracles, Nearest colored descend- ant, Top-trees
Collection: 27th Annual Symposium on Combinatorial Pattern Matching (CPM 2016)
Issue Date: 2016
Date of publication: 27.06.2016

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