Approximate Cartesian Tree Matching with Substitutions

TL;DR

This work addresses approximate Cartesian tree matching under substitutions, quantified by the Hamming distance, by introducing a CT-aware periodicity toolbox. The authors design a two-branch algorithm that, depending on the pattern's structure, either marks many candidate starts or trims periodic fragments to enable fast verification, ultimately achieving O(n √m · k^{2.5}) time for k ≤ m^{1/5} and O(n k^5) for larger k, improving over the previous O(n m k) bound in a broad range of regimes. The approach hinges on a novel CT-block-periodicity concept that yields strong locality guarantees and transfers key periodicity ideas from strings to Cartesian trees. The results offer a practical and scalable framework for robust CT-matching in time-series and related applications, supported by a toolbox with potential broader use in Cartesian-tree-based pattern analysis.

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.

Figures (3)

• Figure 1: The running time $t(n,k)$ of algorithms for Approximate CT-Matching with Substitutions as a function of $k$ for the special case when $m= \Theta(n)$, shown on a doubly logarithmic scale.
• Figure 2: The Cartesian tree $\textsf{CT}(X)$ of $X = [4, 5, 6, 1, 2, 7, 7, 8, 3, 9]$ (top left), the tree induced by $\textnormal{psv}_X$ (top right), and the values of $\textnormal{psv}_X$, $\textnormal{nsv}_X$, $\textnormal{PD}(X)$, and $\textnormal{ND}(X)$ (bottom right). For $Y = [14, 15, 16, 11, 12, 17, 17, 18, 13, 19]$, we have $X \approx Y$, illustrating that different sequences can yield the same Cartesian tree. In contrast, the Cartesian tree of $Z = [14, 15, 16, 16, 12, 17, 17, 18, 8, 19]$ (bottom left) is different, i.e., we have $Z \not\approx X$. However, we have $\textnormal{CHd}(X \leadsto Z) = 2$ because we can substitute the values of $X$ at positions 4 and 9 to obtain $X' = [4, 5, 6, \mathbf{6}, 2, 7, 7, 8, \mathbf{1}, 9]\approx Z$.
• Figure 3: An illustration of the settings in \ref{['lem:no_sub_in_run', 'lem:trim']} with $k=3$. The $\textnormal{psv}_Y$ array is shown with grey arrows below string $Y$. \ref{['lem:no_sub_in_run']} establishes that any sequence of $\textnormal{CHd}(X \leadsto Y)$ substitutions that transforms $X$ into $X'$ such that ${X' \approx Y}$ does not modify the fragment highlighted in red; the condition "$\forall i \in [c_r + p, \lvert Y\rvert] : \textnormal{psv}_Y(i) \notin [c_1, c_r)$" ensures that none of these substitutions "interact" with this fragment. \ref{['lem:trim']} establishes that we can trim a pair of fragments (one in $X$ and one in $Y$) without altering the value of $\textnormal{CHd}(X \leadsto Y)$.

Theorems & Definitions (25)

• Theorem 1
• Lemma 2: Periodicity Lemma (weak version) fine1965uniqueness
• Definition 3
• Definition 4
• Lemma 6: exactCTmatching
• Lemma 7: exactCTmatching
• Lemma 9: exactCTmatching
• Definition 10: DBLP:journals/tcs/MatsuokaAIBT16
• Lemma 11
• Definition 12