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 $O(\frac {m^k}{k!} + occ$)($O(m + \frac {log_Σ^kn}{k!} + occ$) in the average case) of query time for any online/offline pattern, where $occ$ is the number of outputs. As well, a tree structure of $O(2^{k}n\frac{log_Σ^{k}n}{k!})$ words and $O(\frac {m^k}{k!} + 3^{k}occ$)($O(m + \frac {log_Σ^kn}{k!} + 3^{k}occ$) in the average case) query time for edit distance for any online/offline pattern.
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…
In this paper, we design new sublinear-time algorithms for solving the gap edit distance problem and for embedding edit distance to Hamming distance. For the gap edit distance problem, we give an $\tilde{O}(\frac{n}{k}+k^2)$-time greedy algorithm that distinguishes between length-$n$ input strings with edit distance at most $k$ and those with edit distance exceeding $(3k+5)k$. This is an improvement and a simplification upon the result of Goldenberg, Krauthgamer, and Saha [FOCS 2019], where the $k$ vs $Θ(k^2)$ gap edit distance problem is solved in $\tilde{O}(\frac{n}{k}+k^3)$ time. We further…
Edit distance is a fundamental measure of distance between strings and has been widely studied in computer science. While the problem of estimating edit distance has been studied extensively, the equally important question of actually producing an alignment (i.e., the sequence of edits) has received far less attention. Somewhat surprisingly, we show that any algorithm to estimate edit distance can be used in a black-box fashion to produce an approximate alignment of strings, with modest loss in approximation factor and small loss in run time. Plugging in the result of Andoni, Krauthgamer, and …
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）
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 …
We give cell-probe bounds for the computation of edit distance, Hamming distance, convolution and longest common subsequence in a stream. In this model, a fixed string of $n$ symbols is given and one $δ$-bit symbol arrives at a time in a stream. After each symbol arrives, the distance between the fixed string and a suffix of most recent symbols of the stream is reported. The cell-probe model is perhaps the strongest model of computation for showing data structure lower bounds, subsuming in particular the popular word-RAM model. * We first give an $Ω((δ\log n)/(w+\log\log n))$ lower bound for t…
Constant Rate Isometric Embeddings of Hamming Metric into Edit Metric
Computer Science Institute of Charles University, Prague（查理大学计算机科学研究所）；UMONS – Université de Mons（蒙斯大学）；The Academic College of Tel Aviv-Yaffo（特拉维夫雅法学术学院）；Rutgers University（罗格斯大学）；INSAIT, Sofia University ”St. Kliment Ohridski” & ETH Zürich（索菲亚大学“圣克利门特奥赫里德斯基”研究所（INSAIT）及苏黎世联邦理工学院）
We consider a class of pattern matching problems where a normalising transformation is applied at every alignment. Normalised pattern matching plays a key role in fields as diverse as image processing and musical information processing where application specific transformations are often applied to the input. By considering the class of polynomial transformations of the input, we provide fast algorithms and the first lower bounds for both new and old problems. Given a pattern of length m and a longer text of length n where both are assumed to contain integer values only, we first show O(n log …
We consider a range of simply stated dynamic data structure problems on strings. An update changes one symbol in the input and a query asks us to compute some function of the pattern of length $m$ and a substring of a longer text. We give both conditional and unconditional lower bounds for variants of exact matching with wildcards, inner product, and Hamming distance computation via a sequence of reductions. As an example, we show that there does not exist an $O(m^{1/2-\varepsilon})$ time algorithm for a large range of these problems unless the online Boolean matrix-vector multiplication conje…
We introduce the notion of Levenshtein graphs, an analog to Hamming graphs but using the edit distance instead of the Hamming distance; in particular, Levenshtein graphs allow for underlying strings (nodes) of different lengths. We characterize various properties of these graphs, including a necessary and sufficient condition for their geodesic distance to be identical to the edit distance, their automorphism group and determining number, and an upper bound on their metric dimension. Regarding the latter, we construct a resolving set composed of two-run strings and an algorithm that computes t…
We present space lower bounds for online pattern matching under a number of different distance measures. Given a pattern of length m and a text that arrives one character at a time, the online pattern matching problem is to report the distance between the pattern and a sliding window of the text as soon as the new character arrives. We require that the correct answer is given at each position with constant probability. We give Omega(m) bit space lower bounds for L_1, L_2, L_\infty, Hamming, edit and swap distances as well as for any algorithm that computes the cross-correlation/convolution. We…
A pattern $α$ is a string of variables and terminal letters. We say that $α$ matches a word $w$, consisting only of terminal letters, if $w$ can be obtained by replacing the variables of $α$ by terminal words. The matching problem, i.e., deciding whether a given pattern matches a given word, was heavily investigated: it is NP-complete in general, but can be solved efficiently for classes of patterns with restricted structure. If we are interested in what is the minimum Hamming distance between $w$ and any word $u$ obtained by replacing the variables of $α$ by terminal words (so matching under …
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}}(\…
We consider the question of interactive communication, in which two remote parties perform a computation while their communication channel is (adversarially) noisy. We extend here the discussion into a more general and stronger class of noise, namely, we allow the channel to perform insertions and deletions of symbols. These types of errors may bring the parties "out of sync", so that there is no consensus regarding the current round of the protocol. In this more general noise model, we obtain the first interactive coding scheme that has a constant rate and resists noise rates of up to $1/18-\…
In this work, we revisit the fundamental and well-studied problem of approximate pattern matching under edit distance. Given an integer $k$, a pattern $P$ of length $m$, and a text $T$ of length $n \ge m$, the task is to find substrings of $T$ that are within edit distance $k$ from $P$. Our main result is a streaming algorithm that solves the problem in $\tilde{O}(k^5)$ space and $\tilde{O}(k^8)$ amortised time per character of the text, providing answers correct with high probability. (Hereafter, $\tilde{O}(\cdot)$ hides a $\mathrm{poly}(\log n)$ factor.) This answers a decade-old question: s…
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…
In this paper, we study edit distance (ED) and longest common subsequence (LCS) in the asymmetric streaming model, introduced by Saks and Seshadhri [SS13]. As an intermediate model between the random access model and the streaming model, this model allows one to have streaming access to one string and random access to the other string. Our first main contribution is a systematic study of space lower bounds for ED and LCS in the asymmetric streaming model. Previously, there are no explicitly stated results in this context, although some lower bounds about LCS can be inferred from the lower boun…
We revisit the task of computing the edit distance in sublinear time. In the $(k,K)$-gap edit distance problem the task is to distinguish whether the edit distance of two strings is at most $k$ or at least $K$. It has been established by Goldenberg, Krauthgamer and Saha (FOCS '19), with improvements by Kociumaka and Saha (FOCS '20), that the $(k,k^2)$-gap problem can be solved in time $\widetilde O(n/k+\operatorname{poly}(k))$. One of the most natural questions in this line of research is whether the $(k,k^2)$-gap is best-possible for the running time $\widetilde O(n/k+\operatorname{poly}(k))$…
Real-world data often comes in compressed form. Analyzing compressed data directly (without decompressing it) can save space and time by orders of magnitude. In this work, we focus on fundamental sequence comparison problems and try to quantify the gain in time complexity when the underlying data is highly compressible. We consider grammar compression, which unifies many practically relevant compression schemes. For two strings of total length $N$ and total compressed size $n$, it is known that the edit distance and a longest common subsequence (LCS) can be computed exactly in time $\tilde{O}(…
Given an LZW/LZ78 compressed text, we want to find an approximate occurrence of a given pattern of length m. The goal is to achieve time complexity depending on the size n of the compressed representation of the text instead of its length. We consider two specific definitions of approximate matching, namely the Hamming distance and the edit distance, and show how to achieve O(nm^0.5k^2) and O(nm^0.5k^3) running time, respectively, where k is the bound on the distance. Both algorithms use just linear space. Even for very small values of k, the best previously known solutions required O(nm) time…
