In this paper, we present and study the Hamming distance oracle problem. In this problem, the task is to preprocess two strings S and T of lengths n and m, respectively, to obtain a data structure that is able to return the Hamming distance between a substring of S and a substring of T. For strings over a constant-size alphabet, we show that for every x ≤ min{n,m} there is a data structure with Õ(nm/x) preprocessing time and O(x) query time. We also provide a conditional lower bound, showing that for every ε > 0 there is no combinatorial data structure with query time O(x) and preprocessing time O((nm/x)^{1-ε}) unless combinatorial fast matrix multiplication is possible. For strings over a general alphabet, we present a data structure with Õ(nm/√x) pre-processing time and O(x) query time for every x ≤ min {n,m}. Moreover, for every ε > 0 we provide a data structure with a preprocessing time of Õ((n+m)/ε³) that returns with high probability a (1±ε) approximation of the Hamming distance of two input substrings. The query time of the approximation data structure is Õ(1/ε²).

The Longest Common Subsequence (LCS) problem and the Longest Common Substring (LCStr) problem are classical string problems with broad theoretical and practical significance. The former has a quadratic conditional lower bound [FOCS, 2015], while the latter admits a linear-time solution. In this paper, we study a natural variation of these problems, the Longest Common Subsequence-Substring (LCSS) problem. The LCSS problem seeks the longest string that is simultaneously a subsequence of one input string and a substring of the other. This variant bridges LCS and LCStr, raising intriguing algorithmic questions: Does the complexity of computing LCSS interpolate between the linear time of LCStr and the quadratic time of LCS? What about approximability? We also examine a natural extension of LCSS to multiple strings, parameterizing the balance between subsequence and substring requirements. Our results reveal several insights. First, under the SETH conjecture, the inherent complexity of LCSS is quadratic, similar to LCS. In contrast, we provide a linear-time approximation for LCSS. Finally, for the multi-string variant, unlike both problems, we design a quadratic-time algorithm, uncovering deeper structural properties of the problem. By studying the complexity of the LCSS problem, we aim to gain some understanding of what influences whether a variant of the LCS problem behaves more like the standard LCS or like LCStr. Our findings suggest that hybrid constraints can create computational "sweet spots," where problems become more tractable than their pure counterparts. This opens a broader research direction in constraint-mediated algorithm design. Beyond LCSS itself, our work highlights unexpected connections between subsequence and substring constraints, advancing the theoretical understanding of string problems and laying the foundation for new algorithmic techniques and complexity-theoretic insights in the rich space between classical string comparison paradigms.

In this paper we present algorithms for several string problems in the Congested Clique model. In the Congested Clique model, n nodes (computers) are used to solve some problem. The input to the problem is distributed among the nodes, and the communication between the nodes is conducted in rounds. In each round, every node is allowed to send an O(log n)-bit message to every other node in the network. We consider three fundamental string problems in the Congested Clique model. First, we present an O(1) rounds algorithm for string sorting that supports strings of arbitrary length. Second, we present an O(1) rounds combinatorial pattern matching algorithm. Finally, we present an O(log log n) rounds algorithm for the computation of the suffix array and the corresponding Longest Common Prefix array of a given string.

MOTIVATION: Minimizers are the most popular k-mer selection scheme in algorithms and data structures analyzing high-throughput sequencing (HTS) data. In a minimizer scheme, the smallest k-mer by some predefined order is selected as the representative of a sequence window containing w consecutive k-mers, which results in overlapping windows often selecting the same k-mer. Minimizers that achieve the lowest frequency of selected k-mers over a random DNA sequence, termed the expected density, are desired for improved performance of HTS analyses. Yet, no method to date exists to generate minimizers that achieve minimum expected density. Moreover, for k and w values used by common HTS algorithms and data structures, there is a gap between densities achieved by existing selection schemes and the theoretical lower bound. RESULTS: We developed GreedyMini, a toolkit of methods to generate minimizers with low expected or particular density, to improve minimizers, to extend minimizers to larger alphabets, k, and w, and to measure the expected density of a given minimizer efficiently. We demonstrate over various combinations of k and w values, including those of popular HTS methods, that GreedyMini can generate DNA minimizers that achieve expected densities very close to the lower bound, and both expected and particular densities much lower compared to existing selection schemes. Moreover, we show that GreedyMini's k-mer rank-retrieval time is comparable to common k-mer hash functions. We expect GreedyMini to improve the performance of many HTS algorithms and data structures and advance the research of k-mer selection schemes. AVAILABILITY AND IMPLEMENTATION: The toolkit, its source code, and precomputed minimizers for a variety of (k,w) pairs are available via https://github.com/OrensteinLab/GreedyMini.

Abstract. We consider two closely related problems of text indexing in a sublinear working space. The first problem is the sparse suffix tree construction, where a text [Formula: see text] is given in read-only memory, along with a set of suffixes [Formula: see text], and the goal is to construct the compressed trie of all these suffixes ordered lexicographically, using only [Formula: see text] words of space. The second problem is the longest common extension problem, where again a text [Formula: see text] of length [Formula: see text] is given in read-only memory with some trade-off parameter [Formula: see text], and the goal is to construct a data structure that uses [Formula: see text] words of space and can compute for any pair of suffixes their longest common prefix length as fast as possible as a function of [Formula: see text] ([Formula: see text] time for a randomized Las Vegas data structure or [Formula: see text] time for a deterministic data structure). We show how to use ideas based on the locally consistent parsing technique, that were introduced by Sahinalp and Vishkin [ Proceedings of the 26 th Annual ACM Symposium on Theory of Computing, 1994, pp. 300–309 ], in some nontrivial ways in order to improve the known results for the above problems under the space constraints. We introduce the first almost-linear, [Formula: see text], deterministic construction for both problems, where all previous algorithms take at least [Formula: see text] time. We also introduce the first linear-time Las Vegas algorithms for both problems, achieving [Formula: see text] construction time with high probability. This is an improvement over the last result of Gawrychowski and Kociumaka [ Proceedings of the 28 th Annual ACM-SIAM Symposium on Discrete Algorithms, 2017, pp. 425–439 ], which obtained [Formula: see text] time for the Monte Carlo algorithm and [Formula: see text] time with high probability for the Las Vegas algorithm.

Hairpin completion, derived from the hairpin formation observed in DNA biochemistry, is an operation applied to strings, particularly useful in DNA computing. Conceptually, a right hairpin completion operation transforms a string $S$ into $S\cdot S'$ where $S'$ is the reverse complement of a prefix of $S$. Similarly, a left hairpin completion operation transforms a string $S$ into $S'\cdot S$ where $S'$ is the reverse complement of a suffix of $S$. The hairpin completion distance from $S$ to $T$ is the minimum number of hairpin completion operations needed to transform $S$ into $T$. Recently Boneh et al. showed an $O(n^2)$ time algorithm for finding the hairpin completion distance between two strings of length at most $n$. In this paper we show that for any $\varepsilon>0$ there is no $O(n^{2-\varepsilon})$-time algorithm for the hairpin completion distance problem unless the Strong Exponential Time Hypothesis (SETH) is false. Thus, under SETH, the time complexity of the hairpin completion distance problem is quadratic, up to sub-polynomial factors.

We study a natural type of repetitions in 2-dimensional strings. Such a repetition, called a matching frame, is a rectangular substring of size at least 2× 2 with equal marginal rows and equal marginal columns. Matching frames first appeared in literature in the context of Wang tiles. We present two algorithms finding a matching frame with the maximum perimeter in a given n× m input string. The first algorithm solves the problem exactly in Õ(n^{2.5}) time (assuming n ≥ m). The second algorithm finds a (1-ε)-approximate solution in Õ((nm)/ε⁴) time, which is near linear in the size of the input for constant ε. In particular, by setting ε = O(1) the second algorithm decides the existence of a matching frame in a given string in Õ(nm) time. Some technical elements and structural properties used in these algorithms can be of independent interest.

A $λ$-cover of a string $S$ is a set of strings $\{C_i\}_1^λ$ such that every index in $S$ is contained in an occurrence of at least one string $C_i$. The existence of a $1$-cover defines a well-known class of quasi-periodic strings. Quasi-periodicity can be decided in linear time, and all $1$-covers of a string can be reported in linear time plus the size of the output. Since in general it is NP-complete to decide whether a string has a $λ$-cover, the natural next step is the development of efficient algorithms for $2$-covers. Radoszewski and Straszyński [ESA 2020] analysed the particular case where the strings in a $2$-cover must be of the same length. They provided an algorithm that reports all such $2$-covers of $S$ in time near-linear in $|S|$ and in the size of the output. In this work, we consider $2$-covers in full generality. Since every length-$n$ string has $Ω(n^2)$ trivial $2$-covers (every prefix and suffix of total length at least $n$ constitute such a $2$-cover), we state the reporting problem as follows: given a string $S$ and a number $m$, report all $2$-covers $\{C_1,C_2\}$ of $S$ with length $|C_1|+|C_2|$ upper bounded by $m$. We present an $\tilde{O}(n + Output)$ time algorithm solving this problem, with Output being the size of the output. This algorithm admits a simpler modification that finds a $2$-cover of minimum length. We also provide an $\tilde{O}(n)$ time construction of a $2$-cover oracle which, given two substrings $C_1,C_2$ of $S$, reports in poly-logarithmic time whether $\{C_1,C_2\}$ is a $2$-cover of $S$.

Dynamic Time Warping (DTW) distance is the optimal cost of matching two strings when extending runs of letters is for free. Therefore, it is natural to measure the time complexity of DTW in terms of the number of runs n (rather than the string lengths N). In this paper, we give an Õ(n²) time algorithm for computing the DTW distance. This matches (up to log factors) the known (conditional) lower bound, and should be compared with the previous fastest O(n³) time exact algorithm and the Õ(n²) time approximation algorithm. Our method also immediately implies an Õ(nk) time algorithm when the distance is bounded by k. This should be compared with the previous fastest O(n²k) and O(Nk) time exact algorithms and the Õ(nk) time approximation algorithm.

A Dyck sequence is a sequence of opening and closing parentheses (of various types) that is balanced. The Dyck edit distance of a given sequence of parentheses S is the smallest number of edit operations (insertions, deletions, and substitutions) needed to transform S into a Dyck sequence. We consider the threshold Dyck edit distance problem, where the input is a sequence of parentheses S and a positive integer k , and the goal is to compute the Dyck edit distance of S only if the distance is at most k , and otherwise report that the distance is larger than k . Backurs and Onak [PODS’16] showed that the threshold Dyck edit distance problem can be solved in O ( n + k 16 ) time. In this work, we design new algorithms for the threshold Dyck edit distance problem which costs O ( n + k 4.544184 ) time with high probability or O ( n + k 4.853059 ) deterministically. Our algorithms combine several new structural properties of the Dyck edit distance problem, a refined algorithm for fast (min, +) matrix product, and a careful modification of ideas used in Valiant’s parsing algorithm.

A Dyck sequence is a sequence of opening and closing parentheses (of various types) that is balanced. The Dyck edit distance of a given sequence of parentheses S is the smallest number of edit operations (insertions, deletions, and substitutions) needed to transform S into a Dyck sequence. We consider the threshold Dyck edit distance problem, where the input is a sequence of parentheses S and a positive integer k, and the goal is to compute the Dyck edit distance of S only if the distance is at most k, and otherwise report that the distance is larger than k. Backurs and Onak [PODS'16] showed that the threshold Dyck edit distance problem can be solved in O(n + k16) time. In this work, we design new algorithms for the threshold Dyck edit distance problem which costs O(n + k4.782036) time with high probability or O(n + k4.853059) deterministically. Our algorithms combine several new structural properties of the Dyck edit distance problem, a refined algorithm for fast (min, +) matrix product, and a careful modification of ideas used in Valiant's parsing algorithm.

The shift distance $\mathsf{sh}(S_1,S_2)$ between two strings $S_1$ and $S_2$ of the same length is defined as the minimum Hamming distance between $S_1$ and any rotation (cyclic shift) of $S_2$. We study the problem of sketching the shift distance, which is the following communication complexity problem: Strings $S_1$ and $S_2$ of length $n$ are given to two identical players (encoders), who independently compute sketches (summaries) $\mathtt{sk}(S_1)$ and $\mathtt{sk}(S_2)$, respectively, so that upon receiving the two sketches, a third player (decoder) is able to compute (or approximate) $\mathsf{sh}(S_1,S_2)$ with high probability. This paper primarily focuses on the more general $k$-mismatch version of the problem, where the decoder is allowed to declare a failure if $\mathsf{sh}(S_1,S_2)>k$, where $k$ is a parameter known to all parties. Andoni et al. (STOC'13) introduced exact circular $k$-mismatch sketches of size $\widetilde{O}(k+D(n))$, where $D(n)$ is the number of divisors of $n$. Andoni et al. also showed that their sketch size is optimal in the class of linear homomorphic sketches. We circumvent this lower bound by designing a (non-linear) exact circular $k$-mismatch sketch of size $\widetilde{O}(k)$; this size matches communication-complexity lower bounds. We also design $(1\pm \varepsilon)$-approximate circular $k$-mismatch sketch of size $\widetilde{O}(\min(\varepsilon^{-2}\sqrt{k}, \varepsilon^{-1.5}\sqrt{n}))$, which improves upon an $\widetilde{O}(\varepsilon^{-2}\sqrt{n})$-size sketch of Crouch and McGregor (APPROX'11).

We revisit the k-mismatch problem in the streaming model on a pattern of length m and a streaming text of length n, both over a size-σ alphabet. The current state-of-the-art algorithm for the streaming k-mismatch problem, by Clifford et al. [SODA 2019], uses Õ(k) space and Õ(√k) worst-case time per character. The space complexity is known to be (unconditionally) optimal, and the worst-case time per character matches a conditional lower bound. However, there is a gap between the total time cost of the algorithm, which is Õ(n√k), and the fastest known offline algorithm, which costs Õ(n + min(nk/√m, σn)) time. Moreover, it is not known whether improvements over the Õ(n√k) total time are possible when using more than O(k) space. We address these gaps by designing a randomized streaming algorithm for the k-mismatch problem that, given an integer parameter k≤s≤m, uses Õ(s) space and costs Õ(n+min(nk²/m, nk/√s, σnm/s)) total time. For s=m, the total runtime becomes Õ(n + min(nk/√m, σn)), which matches the time cost of the fastest offline algorithm. Moreover, the worst-case time cost per character is still Õ(√k).

We consider the problem of finding, given two documents of total length n, a longest string occurring as a substring of both documents. This problem, known as the Longest Common Substring (LCS) problem, has a classic 𝒪(n)-time solution dating back to the discovery of suffix trees (Weiner, 1973) and their efficient construction for integer alphabets (Farach-Colton, 1997). However, these solutions require Θ(n) space, which is prohibitive in many applications. To address this issue, Starikovskaya and Vildhøj (CPM 2013) showed that for n^{2/3} ≤ s ≤ n, the LCS problem can be solved in 𝒪(s) space and 𝒪̃(n²/s) time. Kociumaka et al. (ESA 2014) generalized this tradeoff to 1 ≤ s ≤ n, thus providing a smooth time-space tradeoff from constant to linear space. In this paper, we obtain a significant speed-up for instances where the length L of the sought LCS is large. For 1 ≤ s ≤ n, we show that the LCS problem can be solved in 𝒪(s) space and 𝒪̃(n²/(L⋅s) +n) time. The result is based on techniques originating from the LCS with Mismatches problem (Flouri et al., 2015; Charalampopoulos et al., CPM 2018), on space-efficient locally consistent parsing (Birenzwige et al., SODA 2020), and on the structure of maximal repetitions (runs) in the input documents.

We consider two closely related problems of text indexing in a sub-linear working space. The first problem is the Sparse Suffix Tree (SST) construction, where a text S is given in read-only memory, along with a set of suffixes B, and the goal is to construct the compressed trie of all these suffixes ordered lexicographically, using only (|B|) words of space. The second problem is the Longest Common Extension (LCE) problem, where again a text S of length n is given in read-only memory with some parameter 1 ≤ τ n, and the goal is to construct a data structure that uses words of space and can compute for any pair of suffixes their longest common prefix length. We show how to use ideas based on the Locally Consistent Parsing technique, that were introduced by Sahinalp and Vishkin [44], in some nontrivial ways in order to improve the known results for the above problems. We introduce new Las-Vegas and deterministic algorithms for both problems. For the randomized algorithms, we introduce the first Las-Vegas SST construction algorithm that takes (n) time. This is an improvement over the last result of Gawrychowski and Kociumaka [22] who obtained (n) time for Monte Carlo algorithm, and time with hight probability for Las-Vegas algorithm. In addition, we introduce a randomized Las-Vegas construction for a data structure that uses words of space, can be constructed in linear time with high probability and answers LCE queries in (τ) time. For the deterministic algorithms, we introduce an SST construction algorithm that takes time (for |B| = Ω(log n)). This is the first almost linear time, (n · polylog n), deterministic SST construction algorithm, where all previous algorithms take at least time. For the LCE problem, we introduce a data structure that uses words of space and answers LCE queries in time, with (n log τ) construction time (for ). This data structure improves both query time and construction time upon the results of Tanimura et al. [47].

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 (i.e., the number of mismatches) between the pattern and the text at every location. Several randomized (1+ε)-approximation algorithms have been proposed in the literature (e.g., by Karloff (Inf. Proc. Lett., 1993), Indyk (FOCS 1998), and Kopelowitz and Porat (SOSA 2018)), with running time of the form O(ε−O(1) nlognlogm), all using fast Fourier transform (FFT). We describe a simple randomized (1+ε)-approximation algorithm that is faster and does not need FFT. Combining our approach with additional ideas leads to numerous new results (all Monte-Carlo randomized) in different settings:

Recently, there has been a growing focus in solving approximate pattern matching problems in the streaming model. Of particular interest are the pattern matching with k-mismatches (KMM) problem and the pattern matching with w-wildcards (PMWC) problem. Motivated by reductions from these problems in the streaming model to the dictionary matching problem, this paper focuses on designing algorithms for the dictionary matching problem in the multi-stream model where there are several independent streams of data (as opposed to just one in the streaming model), and the memory complexity of an algorithm is expressed using two quantities: (1) a read-only shared memory storage area which is shared among all the streams, and (2) local stream memory that each stream stores separately. In the dictionary matching problem in the multi-stream model the goal is to preprocess a dictionary D={P_1,P_2,...,P_d} of d=|D| patterns (strings with maximum length m over alphabet Sigma) into a data structure stored in shared memory, so that given multiple independent streaming texts (where characters arrive one at a time) the algorithm reports occurrences of patterns from D in each one of the texts as soon as they appear. We design two efficient algorithms for the dictionary matching problem in the multi-stream model. The first algorithm works when all the patterns in D have the same length m and costs O(d log m) words in shared memory, O(log m log d) words in stream memory, and O(log m) time per character. The second algorithm works for general D, but the time cost per character becomes O(log m+log d log log d). We also demonstrate the usefulness of our first algorithm in solving both the KMM problem and PMWC problem in the streaming model. In particular, we obtain the first almost optimal (up to poly-log factors) algorithm for the PMWC problem in the streaming model. We also design a new algorithm for the KMM problem in the streaming model that, up to poly-log factors, has the same bounds as the most recent results that use different techniques. Moreover, for most inputs, our algorithm for KMM is significantly faster on average.

In the streaming multi-pattern search problem, which is also known as the streaming dictionary matching problem, a set D={P_1,P_2, . . . ,P_d} of d patterns (strings over an alphabet Sigma), called the dictionary, is given to be preprocessed. Then, a text T arrives one character at a time and the goal is to report, before the next character arrives, the longest pattern in the dictionary that is a current suffix of T. We prove that for a constant size alphabet, there exists a randomized Monte-Carlo algorithm for the streaming dictionary matching problem that takes constant time per character and uses O(d log m) words of space, where m is the length of the longest pattern in the dictionary. In the case where the alphabet size is not constant, we introduce two new randomized Monte-Carlo algorithms with the following complexities: * O(log log |Sigma|) time per character in the worst case and O(d log m) words of space. * O(1/epsilon) time per character in the worst case and O(d |\Sigma|^epsilon log m/epsilon) words of space for any 0<epsilon<= 1. These results improve upon the algorithm of [Clifford et al., ESA'15] which uses O(d log m) words of space and takes O(log log (m+d)) time per character.

In the pattern matching with d wildcards problem we are given a text T of length n and a pattern P of length m that contains d wildcard characters, each denoted by a special symbol '?'. A wildcard character matches any other character. The goal is to establish for each m-length substring of T whether it matches P. In the streaming model variant of the pattern matching with d wildcards problem the text T arrives one character at a time and the goal is to report, before the next character arrives, if the last m characters match P while using only o(m) words of space. In this paper we introduce two new algorithms for the d wildcard pattern matching problem in the streaming model. The first is a randomized Monte Carlo algorithm that is parameterized by a constant 0<=delta<=1. This algorithm uses ~O(d^{1-delta}) amortized time per character and ~O(d^{1+delta}) words of space. The second algorithm, which is used as a black box in the first algorithm, is a randomized Monte Carlo algorithm which uses O(d+log m) worst-case time per character and O(d log m) words of space.