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Cartesian Tree Subsequence Matching

Authors: Tsubasa Oizumi, Takeshi Kai, Takuya Mieno, Shunsuke Inenaga, and Hiroki Arimura

Published in: LIPIcs, Volume 223, 33rd Annual Symposium on Combinatorial Pattern Matching (CPM 2022)

Park et al. [TCS 2020] observed that the similarity between two (numerical) strings can be captured by the Cartesian trees: The Cartesian tree of a string is a binary tree recursively constructed by picking up the smallest value of the string as the root of the tree. Two strings of equal length are said to Cartesian-tree match if their Cartesian trees are isomorphic. Park et al. [TCS 2020] introduced the following Cartesian tree substring matching (CTMStr) problem: Given a text string T of length n and a pattern string of length m, find every consecutive substring S = T[i..j] of a text string T such that S and P Cartesian-tree match. They showed how to solve this problem in Õ(n+m) time. In this paper, we introduce the Cartesian tree subsequence matching (CTMSeq) problem, that asks to find every minimal substring S = T[i..j] of T such that S contains a subsequence S' which Cartesian-tree matches P. We prove that the CTMSeq problem can be solved efficiently, in O(m n p(n)) time, where p(n) denotes the update/query time for dynamic predecessor queries. By using a suitable dynamic predecessor data structure, we obtain O(mn log log n)-time and O(n log m)-space solution for CTMSeq. This contrasts CTMSeq with closely related order-preserving subsequence matching (OPMSeq) which was shown to be NP-hard by Bose et al. [IPL 1998].

Cite as

Tsubasa Oizumi, Takeshi Kai, Takuya Mieno, Shunsuke Inenaga, and Hiroki Arimura. Cartesian Tree Subsequence Matching. In 33rd Annual Symposium on Combinatorial Pattern Matching (CPM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 223, pp. 14:1-14:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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  author =	{Oizumi, Tsubasa and Kai, Takeshi and Mieno, Takuya and Inenaga, Shunsuke and Arimura, Hiroki},
  title =	{{Cartesian Tree Subsequence Matching}},
  booktitle =	{33rd Annual Symposium on Combinatorial Pattern Matching (CPM 2022)},
  pages =	{14:1--14:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-234-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{223},
  editor =	{Bannai, Hideo and Holub, Jan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2022.14},
  URN =		{urn:nbn:de:0030-drops-161414},
  doi =		{10.4230/LIPIcs.CPM.2022.14},
  annote =	{Keywords: string algorithms, pattern matching, Cartesian tree subsequence matching, order preserving matching, episode matching}
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