arXiv subjects
Publications and source records attributed to Solon P. Pissis.
Given a string $T$ of length $n$ over an alphabet $Σ\subset \{1,2,\ldots,n^{O(1)}\}$ of size $σ$, we are to preprocess $T$ so that given a range $[i,j]$, we can return a representation of a shortest string over $Σ$ that is absent in the fragment $T[i]\cdots T[j]$ of $T$. We present an $O(n)$-space data structure that answers such queries in constant time and can be constructed in $O(n\log_σn)$ time.
An elastic-degenerate (ED) string is a sequence of $n$ sets of strings of total length $N$, which was recently proposed to model a set of similar sequences. The ED string matching (EDSM) problem is to find all occurrences of a pattern of length $m$ in an ED text. An $O(nm^{1.5}\sqrt{\log m}+N)$-time algorithm for EDSM is known [Aoyama et al., CPM 2018]. The standard assumption in the prior work on this question is that $N$ is substantially larger than both $n$ and $m$, and thus we would like to have a linear dependency on the former. Under this assumption, the natural open problem is whether we can decrease the 1.5 exponent in the time complexity, similarly as in the related (but, to the best of our knowledge, not equivalent) word break problem [Backurs and Indyk, FOCS 2016]. Our starting point is a conditional lower bound for EDSM. We use the popular combinatorial Boolean Matrix Multiplication (BMM) conjecture stating that there is no truly subcubic combinatorial algorithm for BMM [Abboud and Williams, FOCS 2014]. By designing an appropriate reduction we show that a combinatorial algorithm solving the EDSM problem in $O(nm^{1.5-e}+N)$ time, for any $e>0$, refutes this conjecture. Our reduction should be understood as an indication that decreasing the exponent requires fast matrix multiplication. String periodicity and fast Fourier transform are two standard tools in string algorithms. Our main technical contribution is that we successfully combine these tools with fast matrix multiplication to design a non-combinatorial $\tilde{O}(nm^{ω-1}+N)$-time algorithm for EDSM, where $ω$ denotes the matrix multiplication exponent. To the best of our knowledge, we are the first to combine these tools. In particular, using the fact that $ω<2.373$ [Le Gall, ISSAC 2014; Williams, STOC 2012], we obtain an $O(nm^{1.373}+N)$-time algorithm for EDSM.
The $k$-mismatch problem consists in computing the Hamming distance between a pattern $P$ of length $m$ and every length-$m$ substring of a text $T$ of length $n$, if this distance is no more than $k$. In many real-world applications, any cyclic rotation of $P$ is a relevant pattern, and thus one is interested in computing the minimal distance of every length-$m$ substring of $T$ and any cyclic rotation of $P$. This is the circular pattern matching with $k$ mismatches ($k$-CPM) problem. A multitude of papers have been devoted to solving this problem but, to the best of our knowledge, only average-case upper bounds are known. In this paper, we present the first non-trivial worst-case upper bounds for the $k$-CPM problem. Specifically, we show an $O(nk)$-time algorithm and an $O(n+\frac{n}{m}\,k^4)$-time algorithm. The latter algorithm applies in an extended way a technique that was very recently developed for the $k$-mismatch problem [Bringmann et al., SODA 2019]. A preliminary version of this work appeared at FCT 2019. In this version we improve the time complexity of the main algorithm from $O(n+\frac{n}{m}\,k^5)$ to $O(n+\frac{n}{m}\,k^4)$.
String data are often disseminated to support applications such as location-based service provision or DNA sequence analysis. This dissemination, however, may expose sensitive patterns that model confidential knowledge. In this paper, we consider the problem of sanitizing a string by concealing the occurrences of sensitive patterns, while maintaining data utility, in two settings that are relevant to many common string processing tasks. In the first setting, we aim to generate the minimal-length string that preserves the order of appearance and frequency of all non-sensitive patterns. Such a string allows accurately performing tasks based on the sequential nature and pattern frequencies of the string. To construct such a string, we propose a time-optimal algorithm, TFS-ALGO. We also propose another time-optimal algorithm, PFS-ALGO, which preserves a partial order of appearance of non-sensitive patterns but produces a much shorter string that can be analyzed more efficiently. The strings produced by either of these algorithms are constructed by concatenating non-sensitive parts of the input string. However, it is possible to detect the sensitive patterns by ``reversing'' the concatenation operations. In response, we propose a heuristic, MCSR-ALGO, which replaces letters in the strings output by the algorithms with carefully selected letters, so that sensitive patterns are not reinstated, implausible patterns are not introduced, and occurrences of spurious patterns are prevented. In the second setting, we aim to generate a string that is at minimal edit distance from the original string, in addition to preserving the order of appearance and frequency of all non-sensitive patterns. To construct such a string, we propose an algorithm, ETFS-ALGO, based on solving specific instances of approximate regular expression matching.
A weighted string, also known as a position weight matrix, is a sequence of probability distributions over some alphabet. We revisit the Weighted Shortest Common Supersequence (WSCS) problem, introduced by Amir et al. [SPIRE 2011], that is, the SCS problem on weighted strings. In the WSCS problem, we are given two weighted strings $W_1$ and $W_2$ and a threshold $\mathit{Freq}$ on probability, and we are asked to compute the shortest (standard) string $S$ such that both $W_1$ and $W_2$ match subsequences of $S$ (not necessarily the same) with probability at least $\mathit{Freq}$. Amir et al. showed that this problem is NP-complete if the probabilities, including the threshold $\mathit{Freq}$, are represented by their logarithms (encoded in binary). We present an algorithm that solves the WSCS problem for two weighted strings of length $n$ over a constant-sized alphabet in $\mathcal{O}(n^2\sqrt{z} \log{z})$ time. Notably, our upper bound matches known conditional lower bounds stating that the WSCS problem cannot be solved in $\mathcal{O}(n^{2-\varepsilon})$ time or in $\mathcal{O}^*(z^{0.5-\varepsilon})$ time unless there is a breakthrough improving upon long-standing upper bounds for fundamental NP-hard problems (CNF-SAT and Subset Sum, respectively). We also discover a fundamental difference between the WSCS problem and the Weighted Longest Common Subsequence (WLCS) problem, introduced by Amir et al. [JDA 2010]. We show that the WLCS problem cannot be solved in $\mathcal{O}(n^{f(z)})$ time, for any function $f(z)$, unless $\mathrm{P}=\mathrm{NP}$.
A word $x$ that is absent from a word $y$ is called minimal if all its proper factors occur in $y$. Given a collection of $k$ words $y_1,y_2,\ldots,y_k$ over an alphabet $Σ$, we are asked to compute the set $\mathrm{M}^{\ell}_{y_{1}\#\ldots\#y_{k}}$ of minimal absent words of length at most $\ell$ of word $y=y_1\#y_2\#\ldots\#y_k$, $\#\notinΣ$. In data compression, this corresponds to computing the antidictionary of $k$ documents. In bioinformatics, it corresponds to computing words that are absent from a genome of $k$ chromosomes. This computation generally requires $Ω(n)$ space for $n=|y|$ using any of the plenty available $\mathcal{O}(n)$-time algorithms. This is because an $Ω(n)$-sized text index is constructed over $y$ which can be impractical for large $n$. We do the identical computation incrementally using output-sensitive space. This goal is reasonable when $||\mathrm{M}^{\ell}_{y_{1}\#\ldots\#y_{N}}||=o(n)$, for all $N\in[1,k]$. For instance, in the human genome, $n \approx 3\times 10^9$ but $||\mathrm{M}^{12}_{y_{1}\#\ldots\#y_{k}}|| \approx 10^6$. We consider a constant-sized alphabet for stating our results. We show that all $\mathrm{M}^{\ell}_{y_{1}},\ldots,\mathrm{M}^{\ell}_{y_{1}\#\ldots\#y_{k}}$ can be computed in $\mathcal{O}(kn+\sum^{k}_{N=1}||\mathrm{M}^{\ell}_{y_{1}\#\ldots\#y_{N}}||)$ total time using $\mathcal{O}(\mathrm{MaxIn}+\mathrm{MaxOut})$ space, where $\mathrm{MaxIn}$ is the length of the longest word in $\{y_1,\ldots,y_{k}\}$ and $\mathrm{MaxOut}=\max\{||\mathrm{M}^{\ell}_{y_{1}\#\ldots\#y_{N}}||:N\in[1,k]\}$. Proof-of-concept experimental results are also provided confirming our theoretical findings and justifying our contribution.
In this paper we introduce a new family of string processing problems. We are given two or more strings and we are asked to compute a factor common to all strings that preserves a specific property and has maximal length. Here we consider three fundamental string properties: square-free factors, periodic factors, and palindromic factors under three different settings, one per property. In the first setting, we are given a string $x$ and we are asked to construct a data structure over $x$ answering the following type of on-line queries: given string $y$, find a longest square-free factor common to $x$ and $y$. In the second setting, we are given $k$ strings and an integer $1 < k'\leq k$ and we are asked to find a longest periodic factor common to at least $k'$ strings. In the third setting, we are given two strings and we are asked to find a longest palindromic factor common to the two strings. We present linear-time solutions for all settings. We anticipate that our paradigm can be extended to other string properties or settings.
In the longest common substring (LCS) problem, we are given two strings $S$ and $T$, each of length at most $n$, and we are asked to find a longest string occurring as a fragment of both $S$ and $T$. This is a classical and well-studied problem in computer science with a known $\mathcal{O}(n)$-time solution. In the fully dynamic version of the problem, edit operations are allowed in either of the two strings, and we are asked to report an LCS after each such operation. We present the first solution to this problem that requires sublinear time per edit operation. In particular, we show how to return an LCS in $\tilde{\mathcal{O}}(n^{2/3})$ time (or $\tilde{\mathcal{O}}(\sqrt{n})$ time if edits are allowed in only one of the two strings) after each operation using $\tilde{\mathcal{O}}(n)$ space. This line of research was recently initiated by the authors [SPIRE 2017] in a somewhat restricted dynamic variant. An $\tilde{\mathcal{O}}(n)$-sized data structure that returns an LCS of the two strings after a single edit operation (that is reverted afterwards) in $\tilde{\mathcal{O}}(1)$ time was presented. At CPM 2018, three papers studied analogously restricted dynamic variants of problems on strings. We show that our techniques can be used to obtain fully dynamic algorithms for several classical problems on strings, namely, computing the longest repeat, the longest palindrome and the longest Lyndon substring of a string. The only previously known sublinear-time dynamic algorithms for problems on strings were obtained for maintaining a dynamic collection of strings for comparison queries and for pattern matching with the most recent advances made by Gawrychowski et al. [SODA 2018] and by Clifford et al. [STACS 2018].
A border u of a word w is a proper factor of w occurring both as a prefix and as a suffix. The maximal unbordered factor of w is the longest factor of w which does not have a border. Here an O(n log n)-time with high probability (or O(n log n log^2 log n)-time deterministic) algorithm to compute the Longest Unbordered Factor Array of w for general alphabets is presented, where n is the length of w. This array specifies the length of the maximal unbordered factor starting at each position of w. This is a major improvement on the running time of the currently best worst-case algorithm working in O(n^{1.5} ) time for integer alphabets [Gawrychowski et al., 2015].
Sequence comparison is a prerequisite to virtually all comparative genomic analyses. It is often realised by sequence alignment techniques, which are computationally expensive. This has led to increased research into alignment-free techniques, which are based on measures referring to the composition of sequences in terms of their constituent patterns. These measures, such as $q$-gram distance, are usually computed in time linear with respect to the length of the sequences. In this paper, we focus on the complementary idea: how two sequences can be efficiently compared based on information that does not occur in the sequences. A word is an {\em absent word} of some sequence if it does not occur in the sequence. An absent word is {\em minimal} if all its proper factors occur in the sequence. Here we present the first linear-time and linear-space algorithm to compare two sequences by considering {\em all} their minimal absent words. In the process, we present results of combinatorial interest, and also extend the proposed techniques to compare circular sequences. We also present an algorithm that, given a word $x$ of length $n$, computes the largest integer for which all factors of $x$ of that length occur in some minimal absent word of $x$ in time and space $\cO(n)$. Finally, we show that the known asymptotic upper bound on the number of minimal absent words of a word is tight.
In the Longest Common Factor with $k$ Mismatches (LCF$_k$) problem, we are given two strings $X$ and $Y$ of total length $n$, and we are asked to find a pair of maximal-length factors, one of $X$ and the other of $Y$, such that their Hamming distance is at most $k$. Thankachan et al. show that this problem can be solved in $\mathcal{O}(n \log^k n)$ time and $\mathcal{O}(n)$ space for constant $k$. We consider the LCF$_k$($\ell$) problem in which we assume that the sought factors have length at least $\ell$, and the LCF$_k$($\ell$) problem for $\ell=Ω(\log^{2k+2} n)$, which we call the Long LCF$_k$ problem. We use difference covers to reduce the Long LCF$_k$ problem to a task involving $m=\mathcal{O}(n/\log^{k+1}n)$ synchronized factors. The latter can be solved in $\mathcal{O}(m \log^{k+1}m)$ time, which results in a linear-time algorithm for Long LCF$_k$. In general, our solution to LCF$_k$($\ell$) for arbitrary $\ell$ takes $\mathcal{O}(n + n \log^{k+1} n/\sqrt{\ell})$ time.
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 $k$ and using only linear space under the Hamming distance model. Notably, we show that our technique can be extended to the edit distance model with the same time and space complexities. Specifically, our algorithms run in $\mathcal{O}(n \log^k n \log \log n)$ time on average using $\mathcal{O}(n)$ space. We show that our technique is applicable to several algorithmic problems in computational biology and elsewhere.
In a \emph{weighted sequence}, for every position of the sequence and every letter of the alphabet a probability of occurrence of this letter at this position is specified. Weighted sequences are commonly used to represent imprecise or uncertain data, for example, in molecular biology where they are known under the name of Position-Weight Matrices. Given a probability threshold $\frac1z$, we say that a string $P$ of length $m$ occurs in a weighted sequence $X$ at position $i$ if the product of probabilities of the letters of $P$ at positions $i,\ldots,i+m-1$ in $X$ is at least $\frac1z$. In this article, we consider an \emph{indexing} variant of the problem, in which we are to preprocess a weighted sequence to answer multiple pattern matching queries. We present an $O(nz)$-time construction of an $O(nz)$-sized index for a weighted sequence of length $n$ over a constant-sized alphabet that answers pattern matching queries in optimal, $O(m+Occ)$ time, where $Occ$ is the number of occurrences reported. The cornerstone of our data structure is a novel construction of a family of $\lfloor z \rfloor$ special strings that carries the information about all the strings that occur in the weighted sequence with a sufficient probability. We obtain a weighted index with the same complexities as in the most efficient previously known index by Barton et al. (CPM 2016), but our construction is significantly simpler. The most complex algorithmic tool required in the basic form of our index is the suffix tree which we use to develop a new, more straightforward index for the so-called property matching problem. We provide an implementation of our data structure. Our construction allows us also to obtain a significant improvement over the complexities of the approximate variant of the weighted index presented by Biswas et al. (EDBT 2016) and an improvement of the space complexity of their general index.
In the Range Minimum Query (RMQ) problem, we are given an array $A$ of $n$ numbers and we are asked to answer queries of the following type: for indices $i$ and $j$ between $0$ and $n-1$, query $\text{RMQ}_A(i,j)$ returns the index of a minimum element in the subarray $A[i..j]$. Answering a small batch of RMQs is a core computational task in many real-world applications, in particular due to the connection with the Lowest Common Ancestor (LCA) problem. With small batch, we mean that the number $q$ of queries is $o(n)$ and we have them all at hand. It is therefore not relevant to build an $Ω(n)$-sized data structure or spend $Ω(n)$ time to build a more succinct one. It is well-known, among practitioners and elsewhere, that these data structures for online querying carry high constants in their pre-processing and querying time. We would thus like to answer this batch efficiently in practice. With efficiently in practice, we mean that we (ultimately) want to spend $n + \mathcal{O}(q)$ time and $\mathcal{O}(q)$ space. We write $n$ to stress that the number of operations per entry of $A$ should be a very small constant. Here we show how existing algorithms can be easily modified to satisfy these conditions. The presented experimental results highlight the practicality of this new scheme. The most significant improvement obtained is for answering a small batch of LCA queries. A library implementation of the presented algorithms is made available.
In the k-mappability problem, we are given a string x of length n and integers m and k, and we are asked to count, for each length-m factor y of x, the number of other factors of length m of x that are at Hamming distance at most k from y. We focus here on the version of the problem where k = 1. The fastest known algorithm for k = 1 requires time O(mn log n/ log log n) and space O(n). We present two algorithms that require worst-case time O(mn) and O(n log^2 n), respectively, and space O(n), thus greatly improving the state of the art. Moreover, we present an algorithm that requires average-case time and space O(n) for integer alphabets if m = Ω(log n/ log σ), where σ is the alphabet size.
The observed frequency of the longest proper prefix, the longest proper suffix, and the longest infix of a word $w$ in a given sequence $x$ can be used for classifying $w$ as avoided or overabundant. The definitions used for the expectation and deviation of $w$ in this statistical model were described and biologically justified by Brendel et al. (J Biomol Struct Dyn 1986). We have very recently introduced a time-optimal algorithm for computing all avoided words of a given sequence over an integer alphabet (Algorithms Mol Biol 2017). In this article, we extend this study by presenting an $\mathcal{O}(n)$-time and $\mathcal{O}(n)$-space algorithm for computing all overabundant words in a sequence $x$ of length $n$ over an integer alphabet. Our main result is based on a new non-trivial combinatorial property of the suffix tree $\mathcal{T}$ of $x$: the number of distinct factors of $x$ whose longest infix is the label of an explicit node of $\mathcal{T}$ is no more than $3n-4$. We further show that the presented algorithm is time-optimal by proving that $\mathcal{O}(n)$ is a tight upper bound for the number of overabundant words. Finally, we present experimental results, using both synthetic and real data, which justify the effectiveness and efficiency of our approach in practical terms.
We study pattern matching problems on two major representations of uncertain sequences used in molecular biology: weighted sequences (also known as position weight matrices, PWM) and profiles (i.e., scoring matrices). In the simple version, in which only the pattern or only the text is uncertain, we obtain efficient algorithms with theoretically-provable running times using a variation of the lookahead scoring technique. We also consider a general variant of the pattern matching problems in which both the pattern and the text are uncertain. Central to our solution is a special case where the sequences have equal length, called the consensus problem. We propose algorithms for the consensus problem parameterized by the number of strings that match one of the sequences. As our basic approach, a careful adaptation of the classic meet-in-the-middle algorithm for the knapsack problem is used. On the lower bound side, we prove that our dependence on the parameter is optimal up to lower-order terms conditioned on the optimality of the original algorithm for the knapsack problem.
Longest common extension queries (LCE queries) and runs are ubiquitous in algorithmic stringology. Linear-time algorithms computing runs and preprocessing for constant-time LCE queries have been known for over a decade. However, these algorithms assume a linearly-sortable integer alphabet. A recent breakthrough paper by Bannai et.\ al.\ (SODA 2015) showed a link between the two notions: all the runs in a string can be computed via a linear number of LCE queries. The first to consider these problems over a general ordered alphabet was Kosolobov (\emph{Inf.\ Process.\ Lett.}, 2016), who presented an $O(n (\log n)^{2/3})$-time algorithm for answering $O(n)$ LCE queries. This result was improved by Gawrychowski et.\ al.\ (accepted to CPM 2016) to $O(n \log \log n)$ time. In this work we note a special \emph{non-crossing} property of LCE queries asked in the runs computation. We show that any $n$ such non-crossing queries can be answered on-line in $O(n α(n))$ time, which yields an $O(n α(n))$-time algorithm for computing runs.