Approximate Cartesian Tree Matching with Substitutions

2026-02-09

AI Summary

This work addresses the sensitivity and rigidity of exact Cartesian tree matching in time series by studying approximate Cartesian tree matching with substitution errors—specifically, identifying all substrings whose Cartesian trees are within Hamming distance $k$ of a given pattern. We introduce, for the first time, the notion of periodicity in Cartesian trees, extending classical string periodicity theory to this structural domain, and leverage this insight to design a segmented optimization strategy tailored to the properties of Cartesian trees. The proposed algorithm achieves a time complexity of $O(n\sqrt{m}\,k^{2.5})$ when $k \leq m^{1/5}$ and $O(nk^5)$ when $k \geq m^{1/5}$, significantly improving upon the state-of-the-art $O(nmk)$ approach for $k = o(m^{1/4})$, thereby enabling more efficient approximate matching.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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 $\mathcal O(n \sqrt{m} \cdot k^{2.5})$ for $k \leq m^{1/5}$ and in time $\mathcal O(nk^5)$ for $k \geq m^{1/5}$, thereby improving upon the state-of-the-art $\mathcal O(nmk)$-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.

Problem

• Cartesian tree matching
• approximate matching
• Hamming distance
• time series analysis
• string algorithms

Innovation

• Cartesian tree matching
• approximate pattern matching
• Hamming distance
• periodicity in Cartesian trees
• time series analysis