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DOI: 10.4230/LIPIcs.CPM.2018.20
URL: http://drops.dagstuhl.de/opus/volltexte/2018/8689/
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Abedin, Paniz ; Hooshmand, Sahar ; Ganguly, Arnab ; Thankachan, Sharma V.

The Heaviest Induced Ancestors Problem Revisited

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Abstract

We revisit the heaviest induced ancestors problem, which has several interesting applications in string matching. Let T_1 and T_2 be two weighted trees, where the weight W(u) of a node u in either of the two trees is more than the weight of u's parent. Additionally, the leaves in both trees are labeled and the labeling of the leaves in T_2 is a permutation of those in T_1. A node x in T_1 and a node y in T_2 are induced, iff their subtree have at least one common leaf label. A heaviest induced ancestor query HIA(u_1,u_2) is: given a node u_1 in T_1 and a node u_2 in T_2, output the pair (u_1^*,u_2^*) of induced nodes with the highest combined weight W(u^*_1) + W(u^*_2), such that u_1^* is an ancestor of u_1 and u^*_2 is an ancestor of u_2. Let n be the number of nodes in both trees combined and epsilon >0 be an arbitrarily small constant. Gagie et al. [CCCG' 13] introduced this problem and proposed three solutions with the following space-time trade-offs: - an O(n log^2n)-word data structure with O(log n log log n) query time - an O(n log n)-word data structure with O(log^2 n) query time - an O(n)-word data structure with O(log^{3+epsilon}n) query time. In this paper, we revisit this problem and present new data structures, with improved bounds. Our results are as follows. - an O(n log n)-word data structure with O(log n log log n) query time - an O(n)-word data structure with O(log^2 n/log log n) query time. As a corollary, we also improve the LZ compressed index of Gagie et al. [CCCG' 13] for answering longest common substring (LCS) queries. Additionally, we show that the LCS after one edit problem of size n [Amir et al., SPIRE' 17] can also be reduced to the heaviest induced ancestors problem over two trees of n nodes in total. This yields a straightforward improvement over its current solution of O(n log^3 n) space and O(log^3 n) query time.

BibTeX - Entry

@InProceedings{abedin_et_al:LIPIcs:2018:8689,
  author =	{Paniz Abedin and Sahar Hooshmand and Arnab Ganguly and Sharma V. Thankachan},
  title =	{{The Heaviest Induced Ancestors Problem Revisited}},
  booktitle =	{Annual Symposium on Combinatorial Pattern Matching  (CPM 2018)},
  pages =	{20:1--20:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-074-3},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{105},
  editor =	{Gonzalo Navarro and David Sankoff and Binhai Zhu},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/8689},
  doi =		{10.4230/LIPIcs.CPM.2018.20},
  annote =	{Keywords: Data Structure, String Algorithms, Orthogonal Range Queries}
}

Keywords: Data Structure, String Algorithms, Orthogonal Range Queries
Seminar: Annual Symposium on Combinatorial Pattern Matching (CPM 2018)
Issue Date: 2018
Date of publication: 11.05.2018


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