Faster two-dimensional pattern matching with $k$ mismatches

Jonas Ellert; Paweł Gawrychowski; Adam Górkiewicz; Tatiana Starikovskaya

Home/Archive/arXiv

arXiv · 2410.22109

Faster two-dimensional pattern matching with $k$ mismatches

Jonas Ellert·Paweł Gawrychowski·Adam Górkiewicz·Tatiana Starikovskaya

View original source ↗Read source document ↗Export metadata ↓

Abstract

The classical pattern matching asks for locating all occurrences of one string, called the pattern, in another, called the text, where a string is simply a sequence of characters. Due to the potential practical applications, it is desirable to seek approximate occurrences, for example by bounding the number of mismatches. This problem has been extensively studied, and by now we have a good understanding of the best possible time complexity as a function of $n$ (length of the text), $m$ (length of the pattern), and $k$ (number of mismatches). In particular, we know that for $k=\mathcal{O}(\sqrt{m})$, we can achieve quasi-linear time complexity [Gawrychowski and Uzna\'nski, ICALP 2018]. We consider a natural generalisation of the approximate pattern matching problem to two-dimensional strings, which are simply square arrays of characters. The exact version of this problem has been extensively studied in the early 90s. While periodicity, which is the basic tool for one-dimensional pattern matching, admits a natural extension to two dimensions, it turns out to become significantly more challenging to work with, and it took some time until an alphabet-independent linear-time algorithm has been obtained by Galil and Park [SICOMP 1996]. In the approximate two-dimensional pattern matching, we are given a pattern of size $m\times m$ and a text of size $n\times n$, and ask for all locations in the text where the pattern matches with at most $k$ mismatches. The asymptotically fastest algorithm for this algorithm works in $\mathcal{O}(kn^{2})$ time [Amir and Landau, TCS 1991]. We provide a new insight into two-dimensional periodicity to improve on these 30-years old bounds. Our algorithm works in $\tilde{\mathcal{O}}((m^{2}+mk^{5/4})n^{2}/m^{2})$ time, which is $\tilde{\mathcal{O}}(n^{2})$ for $k=\mathcal{O}(m^{4/5})$.

Explore related subjects

cs.DS

Keep this discovery

BibTeX RIS

Jonas Ellert, Paweł Gawrychowski, Adam Górkiewicz, Tatiana Starikovskaya. 2024-10-29. Faster two-dimensional pattern matching with $k$ mismatches. https://arxiv.org/abs/2410.22109

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Quasi-Monte Carlo Beyond Hardy-Krause II: $(1 + \varepsilon)n$ Samples Suffice

Numerical integration studies how well one can estimate the integral of a function $f$ over $[0,1)^d$ using $n$ sample points. The two classical methods, Monte Carlo (MC) and quasi-Monte Carlo (QMC), have complementary strengths and weaknesses, and a fundamental question is to design an approach that combines the benefits of both. Recently, building on the transference principle in discrepancy theory, Bansal and Jiang~\cite{BJ25a} gave a randomized QMC method that bridges MC and QMC guarantees using only i.i.d.\ samples. Their method also goes beyond the classical Koksma--Hlawka inequality: it achieves integration error $\widetilde{O}_d(\sigma_{\mathsf{SO}}(f)/n)$, where the smoothed-out variation $\sigma_{\mathsf{SO}}(f)$ can be substantially smaller than the Hardy--Krause variation that governs the classical bound. However, their algorithm requires $n^2$ i.i.d.\ samples as input, and this quadratic blowup is inherent to any method based on the transference principle. In this work, we bypass the quadratic blowup: for any constant $\varepsilon > 0$, we show that $(1+\varepsilon)n$ i.i.d.\ samples suffice to both obtain the beyond-Hardy--Krause guarantee of~\cite{BJ25a}, resolving an open problem posed there, and to produce low-discrepancy point sequences. Our algorithms are variants of the online Haar-thinning method of Dwivedi, Feldheim, Gurel-Gurevich, and Ramdas~\cite{DFG+19}.

cs.DS↗

arXiv2026

A deterministic $(1+\varepsilon)^n$ approximation for the permanent of a nonnegative matrix

For every fixed $0<\varepsilon\le1$, we give a deterministic strongly polynomial algorithm that, given a nonnegative matrix $A\in\mathbb{R}_{\ge0}^{n\times n}$, returns $Q$ satisfying $\operatorname{per} A\le Q\le(1+\varepsilon)^n\operatorname{per} A$.

cs.DS↗

arXiv2026

Single-Exponential Algorithms and a Polynomial Kernel for Strong Connectivity Augmentation

Strong Connectivity Augmentation (SCA) asks whether a directed acyclic graph can be made strongly connected by adding at most $k$ prescribed links whose total weight is within a given budget. Klinkby, Misra, and Saurabh (SODA 2021) gave an $O^*(2^{O(k\log k)})$-time algorithm and asked whether the problem admits a single-exponential parameterized algorithm and a polynomial kernel. We answer both questions affirmatively: SCA can be solved in $O^*(9^k)$ time and admits a polynomial kernel with $O(k^4)$ vertices and $O(k^{16})$ bits. For unweighted SCA, we obtain $O^*(4^k)$ time and a kernel with $O(k^3)$ vertices. Our algorithms are based on a particularly simple reduction to Strongly Connected Spanning Subgraph with two edge costs.

cs.DS↗