We revisit a fundamental problem in string matching: given a pattern of length m and a text of length n, both over an alphabet of size $σ$, compute the Hamming distance between the pattern and the text at every location. Several $(1+ε)$-approximation algorithms have been proposed in the literature, with running time of the form $O(ε^{-O(1)}n\log n\log m)$, all using fast Fourier transform (FFT). We describe a simple $(1+ε)$-approximation algorithm that is faster and does not need FFT. Combining our approach with additional ideas leads to numerous new results:
- We obtain the first linear-time approximation algorithm; the running time is $O(ε^{-2}n)$.
- We obtain a faster exact algorithm computing all Hamming distances up to a given threshold k; its running time improves previous results by logarithmic factors and is linear if $k\le\sqrt m$.
- We obtain approximation algorithms with better $ε$-dependence using rectangular matrix multiplication. The time-bound is $Õ(n)$ when the pattern is sufficiently long: $m\ge ε^{-28}$. Previous algorithms require $Õ(ε^{-1}n)$ time.
- When k is not too small, we obtain a truly sublinear-time algorithm to find all locations with Hamming distance approximately (up to a constant factor) less than k, in $O((n/k^{Ω(1)}+occ)n^{o(1)})$ time, where occ is the output size. The algorithm leads to a property tester, returning true if an exact match exists and false if the Hamming distance is more than $δm$ at every location, running in $Õ(δ^{-1/3}n^{2/3}+δ^{-1}n/m)$ time.
- We obtain a streaming algorithm to report all locations with Hamming distance approximately less than k, using $Õ(ε^{-2}\sqrt k)$ space. Previously, streaming algorithms were known for the exact problem with Õ(k) space or for the approximate problem with $Õ(ε^{-O(1)}\sqrt m)$ space.
Text-to-pattern distance is a fundamental problem in string matching, where given a pattern of length $m$ and a text of length $n$, over an integer alphabet, we are asked to compute the distance between pattern and the text at every location. The distance function can be e.g. Hamming distance or $\ell_p$ distance for some parameter $p > 0$. Almost all state-of-the-art exact and approximate algorithms developed in the past $\sim 40$ years were using FFT as a black-box. In this work we present $\widetilde{O}(n/\varepsilon^2)$ time algorithms for $(1\pm\varepsilon)$-approximation of $\ell_2$ dist…
The algorithmic tasks of computing the Hamming distance between a given pattern of length $m$ and each location in a text of length $n$ is one of the most fundamental algorithmic tasks in string algorithms. Unfortunately, there is evidence that for a text $T$ of size $n$ and a pattern $P$ of size $m$, one cannot compute the exact Hamming distance for all locations in $T$ in time which is less than $\tilde O(n\sqrt m)$. However, Karloff~\cite{karloff} showed that if one is willing to suffer a $1\pmε$ approximation, then it is possible to solve the problem with high probability, in $\tilde O(\fr…
Given a text $T$ of length $n$ and a pattern $P$ of length $m$, the approximate pattern matching problem asks for computation of a particular \emph{distance} function between $P$ and every $m$-substring of $T$. We consider a $(1\pm\varepsilon)$ multiplicative approximation variant of this problem, for $\ell_p$ distance function. In this paper, we describe two $(1+\varepsilon)$-approximate algorithms with a runtime of $\widetilde{O}(\frac{n}{\varepsilon})$ for all (constant) non-negative values of $p$. For constant $p \ge 1$ we show a deterministic $(1+\varepsilon)$-approximation algorithm. Pre…
Error Tree is a novel tree structure that is mainly oriented to solve the approximate pattern matching problems, Hamming and edit distances, as well as the wildcards matching problem. The input is a text of length $n$ over a fixed alphabet of length $Σ$, a pattern of length $m$, and $k$. The output is to find all positions that have $\leq$ $k$ Hamming distance, edit distance, or wildcards matching with $P$. The algorithm proposes for Hamming distance and wildcards matching a tree structure that needs $O(n\frac{log_Σ^{k}n}{k!})$ words and takes $O(\frac {m^k}{k!} + occ$)($O(m + \frac {log_Σ^kn}…
Quasiperiodicity in strings was introduced almost 30 years ago as an extension of string periodicity. The basic notions of quasiperiodicity are cover and seed. A cover of a text $T$ is a string whose occurrences in $T$ cover all positions of $T$. A seed of text $T$ is a cover of a superstring of $T$. In various applications exact quasiperiodicity is still not sufficient due to the presence of errors. We consider approximate notions of quasiperiodicity, for which we allow approximate occurrences in $T$ with a small Hamming, Levenshtein or weighted edit distance. In previous work Sip et al. (200…
We consider the streaming complexity of a fundamental task in approximate pattern matching: the $k$-mismatch problem. It asks to compute Hamming distances between a pattern of length $n$ and all length-$n$ substrings of a text for which the Hamming distance does not exceed a given threshold $k$. In our problem formulation, we report not only the Hamming distance but also, on demand, the full \emph{mismatch information}, that is the list of mismatched pairs of symbols and their indices. The twin challenges of streaming pattern matching derive from the need both to achieve small working space an…
We consider the problem of approximating the arboricity of a graph $G= (V,E)$, which we denote by $\mathsf{arb}(G)$, in sublinear time, where the arboricity of a graph is the minimal number of forests required to cover its edges. An algorithm for this problem may perform degree and neighbor queries, and is allowed a small error probability. We design an algorithm that outputs an estimate $\hatα$, such that with probability $1-1/\textrm{poly}(n)$, $\mathsf{arb}(G)/c\log^2 n \leq \hatα \leq \mathsf{arb}(G)$, where $n=|V|$ and $c$ is a constant. The expected query complexity and running time of t…
Let $G$ be a strongly connected directed graph. We consider the following three problems, where we wish to compute the smallest strongly connected spanning subgraph of $G$ that maintains respectively: the $2$-edge-connected blocks of $G$ (\textsf{2EC-B}); the $2$-edge-connected components of $G$ (\textsf{2EC-C}); both the $2$-edge-connected blocks and the $2$-edge-connected components of $G$ (\textsf{2EC-B-C}). All three problems are NP-hard, and thus we are interested in efficient approximation algorithms. For \textsf{2EC-C} we can obtain a $3/2$-approximation by combining previously known re…
We consider the problem of computing distance between a pattern of length $n$ and all $n$-length subwords of a text in the streaming model. In the streaming setting, only the Hamming distance ($L_0$) has been studied. It is known that computing the exact Hamming distance between a pattern and a streaming text requires $Ω(n)$ space (folklore). Therefore, to develop sublinear-space solutions, one must relax their requirements. One possibility to do so is to compute only the distances bounded by a threshold $k$, see~[SODA'19, Clifford, Kociumaka, Porat] and references therein. The motivation for …
We revisit the complexity of one of the most basic problems in pattern matching. In the k-mismatch problem we must compute the Hamming distance between a pattern of length m and every m-length substring of a text of length n, as long as that Hamming distance is at most k. Where the Hamming distance is greater than k at some alignment of the pattern and text, we simply output "No". We study this problem in both the standard offline setting and also as a streaming problem. In the streaming k-mismatch problem the text arrives one symbol at a time and we must give an output before processing any f…
An Algorithmic Bridge Between Hamming and Levenshtein Distances
Academic College of Tel Aviv-Yafo（特拉维夫雅法学术学院）；Max Planck Institute for Informatics（马克斯·普朗克计算机科学研究所）；Weizmann Institute of Science（魏茨曼科学研究所）；University of California, San Diego（加利福尼亚大学圣地亚哥分校）
In the $k$-mismatch problem we are given a pattern of length $n$ and a text and must find all locations where the Hamming distance between the pattern and the text is at most $k$. A series of recent breakthroughs have resulted in an ultra-efficient streaming algorithm for this problem that requires only $O(k \log \frac{n}{k})$ space and $O(\log \frac{n}{k} (\sqrt{k \log k} + \log^3 n))$ time per letter [Clifford, Kociumaka, Porat, SODA 2019]. In this work, we consider a strictly harder problem called dictionary matching with $k$ mismatches. In this problem, we are given a dictionary of $d$ pat…
We generalize Karp-Rabin string matching to handle multiple patterns in $\mathcal{O}(n \log n + m)$ time and $\mathcal{O}(s)$ space, where $n$ is the length of the text and $m$ is the total length of the $s$ patterns, returning correct answers with high probability. As a prime application of our algorithm, we show how to approximate the LZ77 parse of a string of length $n$. If the optimal parse consists of $z$ phrases, using only $\mathcal{O}(z)$ working space we can return a parse consisting of at most $(1+\varepsilon)z$ phrases in $\mathcal{O}(\varepsilon^{-1}n\log n)$ time, for any $\vareps…
The text-to-pattern Hamming distances problem asks to compute the Hamming distances between a given pattern of length $m$ and all length-$m$ substrings of a given text of length $n\ge m$. We focus on the $k$-mismatch version of the problem, where a distance needs to be returned only if it does not exceed a threshold $k$. We assume $n\le 2m$ (in general, one can partition the text into overlapping blocks). In this work, we show data structures for the dynamic version of this problem supporting two operations: An update performs a single-letter substitution in the pattern or the text, and a quer…
Edit distance is a measure of similarity of two strings based on the minimum number of character insertions, deletions, and substitutions required to transform one string into the other. The edit distance can be computed exactly using a dynamic programming algorithm that runs in quadratic time. Andoni, Krauthgamer, and Onak (2010) gave a nearly linear time algorithm that approximates edit distance within an approximation factor $\text{poly}(\log n)$. In this paper, we provide an algorithm with running time $\tilde{O}(n^{2-2/7})$ that approximates the edit distance within a constant factor.
We consider computing the longest palindrome in a text of length $n$ in the streaming model, where the characters arrive one-by-one, and we do not have random access to the input. While computing the answer exactly using sublinear memory is not possible in such a setting, one can still hope for a good approximation guarantee. We focus on the two most natural variants, where we aim for either additive or multiplicative approximation of the length of the longest palindrome. We first show that there is no point in considering Las Vegas algorithms in such a setting, as they cannot achieve sublinea…
The problem of finding \emph{distance} between \emph{pattern} of length $m$ and \emph{text} of length $n$ is a typical way of generalizing pattern matching to incorporate dissimilarity score. For both Hamming and $L_1$ distances only a super linear upper bound $\widetilde{O}(n\sqrt{m})$ are known, which prompts the question of relaxing the problem: either by asking for $(1 \pm \varepsilon)$ approximate distance (every distance is reported up to a multiplicative factor), or $k$-approximated distance (distances exceeding $k$ are reported as $\infty$). We focus on $L_1$ distance, for which we sho…
Although real-world text datasets, such as DNA sequences, are far from being uniformly random, average-case string searching algorithms perform significantly better than worst-case ones in most applications of interest. In this paper, we study the problem of computing the longest prefix of each suffix of a given string of length $n$ over a constant-sized alphabet that occurs elsewhere in the string with $k$-errors. This problem has already been studied under the Hamming distance model. Our first result is an improvement upon the state-of-the-art average-case time complexity for non-constant $k…
An elastic-degenerate string is a sequence of $n$ finite sets of strings of total length $N$, introduced to represent a set of related DNA sequences, also known as a pangenome. The ED string matching (EDSM) problem consists in reporting all occurrences of a pattern of length $m$ in an ED text. This problem has recently received some attention by the combinatorial pattern matching community, culminating in an $\tilde{\mathcal{O}}(nm^{ω-1})+\mathcal{O}(N)$-time algorithm [Bernardini et al., SIAM J. Comput. 2022], where $ω$ denotes the matrix multiplication exponent and the $\tilde{\mathcal{O}}(\…
The noise sensitivity of a Boolean function $f: \{0,1\}^n \rightarrow \{0,1\}$ is one of its fundamental properties. A function of a positive noise parameter $δ$, it is denoted as $NS_δ[f]$. Here we study the algorithmic problem of approximating it for monotone $f$, such that $NS_δ[f] \geq 1/n^{C}$ for constant $C$, and where $δ$ satisfies $1/n \leq δ\leq 1/2$. For such $f$ and $δ$, we give a randomized algorithm performing $O\left(\frac{\min(1,\sqrt{n} δ\log^{1.5} n) }{NS_δ[f]} \text{poly}\left(\frac{1}ε\right)\right)$ queries and approximating $NS_δ[f]$ to within a multiplicative factor of $…
University of California, Berkeley（加州大学伯克利分校）；University of Texas, Austin（德克萨斯大学奥斯汀分校）；ENSAE（法国国立统计与经济管理学院）；Google Brain（谷歌大脑）
We study the complexity of approximating the multimarginal optimal transport (MOT) distance, a generalization of the classical optimal transport distance, considered here between $m$ discrete probability distributions supported each on $n$ support points. First, we show that the standard linear programming (LP) representation of the MOT problem is not a minimum-cost flow problem when $m \geq 3$. This negative result implies that some combinatorial algorithms, e.g., network simplex method, are not suitable for approximating the MOT problem, while the worst-case complexity bound for the determin…
In this paper, we present a novel iterative Monte Carlo method for approximating the stationary probability of a single state of a positive recurrent Markov chain. We utilize the characterization that the stationary probability of a state $i$ is inversely proportional to the expected return time of a random walk beginning at $i$. Our method obtains an $ε$-multiplicative close estimate with probability greater than $1 - α$ using at most $\tilde{O}\left(t_{\text{mix}} \ln(1/α) / π_i ε^2 \right)$ simulated random walk steps on the Markov chain across all iterations, where $t_{\text{mix}}$ is the …
Given a pattern of length $m$ and a text of length $n$, the goal in $k$-mismatch pattern matching is to compute, for every $m$-substring of the text, the exact Hamming distance to the pattern or report that it exceeds $k$. This can be solved in either $\widetilde{O}(n \sqrt{k})$ time as shown by Amir et al. [J. Algorithms 2004] or $\widetilde{O}((m + k^2) \cdot n/m)$ time due to a result of Clifford et al. [SODA 2016]. We provide a smooth time trade-off between these two bounds by designing an algorithm working in time $\widetilde{O}( (m + k \sqrt{m}) \cdot n/m)$. We complement this with a mat…
Approximate pattern matching is a natural and well-studied problem on strings: Given a text $T$, a pattern $P$, and a threshold $k$, find (the starting positions of) all substrings of $T$ that are at distance at most $k$ from $P$. We consider the two most fundamental string metrics: the Hamming distance and the edit distance. Under the Hamming distance, we search for substrings of $T$ that have at most $k$ mismatches with $P$, while under the edit distance, we search for substrings of $T$ that can be transformed to $P$ with at most $k$ edits. Exact occurrences of $P$ in $T$ have a very simple …
Zwick's $(1+\varepsilon)$-approximation algorithm for the All Pairs Shortest Path (APSP) problem runs in time $\widetilde{O}(\frac{n^ω}{\varepsilon} \log{W})$, where $ω\le 2.373$ is the exponent of matrix multiplication and $W$ denotes the largest weight. This can be used to approximate several graph characteristics including the diameter, radius, median, minimum-weight triangle, and minimum-weight cycle in the same time bound. Since Zwick's algorithm uses the scaling technique, it has a factor $\log W$ in the running time. In this paper, we study whether APSP and related problems admit approx…
In this paper we study lower bounds for the fundamental problem of text indexing with mismatches and differences. In this problem we are given a long string of length $n$, the "text", and the task is to preprocess it into a data structure such that given a query string $Q$, one can quickly identify substrings that are within Hamming or edit distance at most $k$ from $Q$. This problem is at the core of various problems arising in biology and text processing. While exact text indexing allows linear-size data structures with linear query time, text indexing with $k$ mismatches (or $k$ differences…
We present a new streaming algorithm for the $k$-Mismatch problem, one of the most basic problems in pattern matching. Given a pattern and a text, the task is to find all substrings of the text that are at the Hamming distance at most $k$ from the pattern. Our algorithm is enhanced with an important new feature called Error Correcting, and its complexities for $k=1$ and for a general $k$ are comparable to those of the solutions for the $k$-Mismatch problem by Porat and Porat (FOCS 2009) and Clifford et al. (SODA 2016). In parallel to our research, a yet more efficient algorithm for the $k$-Mis…
The edit distance between two strings is defined as the smallest number of insertions, deletions, and substitutions that need to be made to transform one of the strings to another one. Approximating edit distance in subquadratic time is "one of the biggest unsolved problems in the field of combinatorial pattern matching". Our main result is a quantum constant approximation algorithm for computing the edit distance in truly subquadratic time. More precisely, we give an $O(n^{1.858})$ quantum algorithm that approximates the edit distance within a factor of $7$. We further extend this result to a…
We study variants of the mean problem under the $p$-Dynamic Time Warping ($p$-DTW) distance, a popular and robust distance measure for sequential data. In our setting we are given a set of finite point sequences over an arbitrary metric space and we want to compute a mean point sequence of given length that minimizes the sum of $p$-DTW distances, each raised to the $q$\textsuperscript{th} power, between the input sequences and the mean sequence. In general, the problem is $\mathrm{NP}$-hard and known not to be fixed-parameter tractable in the number of sequences. On the positive side, we show …
Directed Steiner Tree (DST) is a central problem in combinatorial optimization and theoretical computer science: Given a directed graph $G=(V, E)$ with edge costs $c \in \mathbb{R}_{\geq 0}^E$, a root $r \in V$ and $k$ terminals $K\subseteq V$, we need to output the minimum-cost arborescence in $G$ that contains an $r$\textrightarrow $t$ path for every $t \in K$. Recently, Grandoni, Laekhanukit and Li, and independently Ghuge and Nagarajan, gave quasi-polynomial time $O(\log^2k/\log \log k)$-approximation algorithms for the problem, which are tight under popular complexity assumptions. In this…
We revisit the deadline version of the discrete time-cost tradeoff problem for the special case of bounded depth. Such instances occur for example in VLSI design. The depth of an instance is the number of jobs in a longest chain and is denoted by $d$. We prove new upper and lower bounds on the approximability. First we observe that the problem can be regarded as a special case of finding a minimum-weight vertex cover in a $d$-partite hypergraph. Next, we study the natural LP relaxation, which can be solved in polynomial time for fixed $d$ and -- for time-cost tradeoff instances -- up to an arb…
We introduce fast-decodable indexing schemes for edit distance which can be used to speed up edit distance computations to near-linear time if one of the strings is indexed by an indexing string $I$. In particular, for every length $n$ and every $\varepsilon >0$, one can in near linear time construct a string $I \in Σ'^n$ with $|Σ'| = O_{\varepsilon}(1)$, such that, indexing any string $S \in Σ^n$, symbol-by-symbol, with $I$ results in a string $S' \in Σ''^n$ where $Σ'' = Σ\times Σ'$ for which edit distance computations are easy, i.e., one can compute a $(1+\varepsilon)$-approximation of the e…
The Longest Common Subsequence (LCS) is a fundamental string similarity measure, and computing the LCS of two strings is a classic algorithms question. A textbook dynamic programming algorithm gives an exact algorithm in quadratic time, and this is essentially best possible under plausible fine-grained complexity assumptions, so a natural problem is to find faster approximation algorithms. When the inputs are two binary strings, there is a simple $\frac{1}{2}$-approximation in linear time: compute the longest common all-0s or all-1s subsequence. It has been open whether a better approximation …
