Approximate Cartesian Tree Matching with Substitutions

TL;DR

This paper introduces an efficient algorithm for approximate Cartesian tree matching that tolerates substitutions, improving over previous methods and expanding the applicability of Cartesian tree analysis in time series and pattern matching.

Contribution

The authors develop a novel algorithm for approximate Cartesian tree matching using Hamming distance, with improved time complexity for certain parameter regimes.

Findings

Algorithm runs in O(n√m·k^{2.5}) for small k

Algorithm runs in O(nk^5) for larger k

Introduces new periodicity notions in Cartesian trees

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.