3 Search Results for "Oizumi, Tsubasa"

Document
DOI: 10.4230/LIPIcs.STACS.2026.26
Approximate Cartesian Tree Matching with Substitutions

Authors: Panagiotis Charalampopoulos, Jonas Ellert, and Manal Mohamed

Published in: LIPIcs, Volume 364, 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)

Abstract
The Cartesian tree of a sequence captures the relative order of the sequence’s elements. In recent years, Cartesian tree matching has attracted considerable attention, particularly due to its applications in time series analysis. Consider a text T of length n and a pattern P of length m. In the exact Cartesian tree matching problem, the task is to find all length-m fragments of T whose Cartesian tree coincides with the Cartesian tree CT(P) of the pattern. Although the exact version of the problem can be solved in linear time [Park et al., TCS 2020], it remains rather restrictive; for example, it is not robust to outliers in the pattern. To overcome this limitation, we consider the approximate setting, where the goal is to identify all fragments of T that are close to some string whose Cartesian tree matches CT(P). In this work, we quantify closeness via the widely used Hamming distance metric. For a given integer parameter k > 0, we present an algorithm that computes all fragments of T that are at Hamming distance at most k from a string whose Cartesian tree matches CT(P). Our algorithm runs in time 𝒪(n √m ⋅ k^{2.5}) for k ≤ m^{1/5} and in time 𝒪(nk⁵) for k ≥ m^{1/5}, thereby improving upon the state-of-the-art 𝒪(nmk)-time algorithm of Kim and Han [TCS 2025] in the regime k = o(m^{1/4}). On the way to our solution, we develop a toolbox of independent interest. First, we introduce a new notion of periodicity in Cartesian trees. Then, we lift multiple well-known combinatorial and algorithmic results for string matching and periodicity in strings to Cartesian tree matching and periodicity in Cartesian trees.
Cite as

Panagiotis Charalampopoulos, Jonas Ellert, and Manal Mohamed. Approximate Cartesian Tree Matching with Substitutions. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 26:1-26:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{charalampopoulos_et_al:LIPIcs.STACS.2026.26,
  author =	{Charalampopoulos, Panagiotis and Ellert, Jonas and Mohamed, Manal},
  title =	{{Approximate Cartesian Tree Matching with Substitutions}},
  booktitle =	{43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
  pages =	{26:1--26:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-412-3},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{364},
  editor =	{Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.26},
  URN =		{urn:nbn:de:0030-drops-255151},
  doi =		{10.4230/LIPIcs.STACS.2026.26},
  annote =	{Keywords: Cartesian tree, Hamming distance, approximate pattern matching}
}
Document
DOI: 10.4230/LIPIcs.CPM.2025.26
Extending the Burrows-Wheeler Transform for Cartesian Tree Matching and Constructing It

Authors: Eric M. Osterkamp and Dominik Köppl

Published in: LIPIcs, Volume 331, 36th Annual Symposium on Combinatorial Pattern Matching (CPM 2025)

Abstract
Cartesian tree matching is a form of generalized pattern matching where a substring of the text matches with the pattern if they share the same Cartesian tree. This form of matching finds application for time series of stock prices and can be of interest for melody matching between musical scores. For the indexing problem, the state-of-the-art data structure is a Burrows-Wheeler transform based solution due to [Kim and Cho, CPM'21], which uses nearly succinct space and can count the number of substrings that Cartesian tree match with a pattern in time linear in the pattern length. The authors address the construction of their data structure with a straight-forward solution that, however, requires pointer-based data structures, resulting in O(n lg n) bits of space, where n is the text length [Kim and Cho, CPM'21, Section A.4]. We address this bottleneck by a construction that requires O(n lg σ) bits of space and has a time complexity of O(n (lg σ lg n)/(lg lg n)), where σ is alphabet size. Additionally, we can extend this index for indexing multiple circular texts in the spirit of the extended Burrows-Wheeler transform without sacrificing the time and space complexities. We present this index in a dynamic variant, where we pay a logarithmic slowdown and need space linear in the input texts in bits for the extra functionality that we can incrementally add texts. Our extended setting is of interest for finding repetitive motifs common in the aforementioned applications, independent of offsets and scaling.
Cite as

Eric M. Osterkamp and Dominik Köppl. Extending the Burrows-Wheeler Transform for Cartesian Tree Matching and Constructing It. In 36th Annual Symposium on Combinatorial Pattern Matching (CPM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 331, pp. 26:1-26:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{osterkamp_et_al:LIPIcs.CPM.2025.26,
  author =	{Osterkamp, Eric M. and K\"{o}ppl, Dominik},
  title =	{{Extending the Burrows-Wheeler Transform for Cartesian Tree Matching and Constructing It}},
  booktitle =	{36th Annual Symposium on Combinatorial Pattern Matching (CPM 2025)},
  pages =	{26:1--26:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-369-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{331},
  editor =	{Bonizzoni, Paola and M\"{a}kinen, Veli},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2025.26},
  URN =		{urn:nbn:de:0030-drops-231201},
  doi =		{10.4230/LIPIcs.CPM.2025.26},
  annote =	{Keywords: Cartesian tree matching, extended Burrows-Wheeler transform, construction algorithm, generalized pattern matching}
}
Document
DOI: 10.4230/LIPIcs.CPM.2022.14
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)

Abstract
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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@InProceedings{oizumi_et_al:LIPIcs.CPM.2022.14,
  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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