For a text given in advance, the substring minimal suffix queries ask to determine the lexicographically minimal non-empty suffix of a substring specified by the location of its occurrence in the text. We develop a data structure answering such queries optimally: in constant time after linear-time preprocessing. This improves upon the results of Babenko et al. (CPM 2014), whose trade-off solution is characterized by $Θ(n\log n)$ product of these time complexities. Next, we extend our queries to support concatenations of $O(1)$ substrings, for which the construction and query time is preserved. We apply these generalized queries to compute lexicographically minimal and maximal rotations of a given substring in constant time after linear-time preprocessing.
Our data structures mainly rely on properties of Lyndon words and Lyndon factorizations. We combine them with further algorithmic and combinatorial tools, such as fusion trees and the notion of order isomorphism of strings.
We study the following substring suffix selection problem: given a substring of a string T of length n, compute its k-th lexicographically smallest suffix. This a natural generalization of the well-known question of computing the maximal suffix of a string, which is a basic ingredient in many other problems. We first revisit two special cases of the problem, introduced by Babenko, Kolesnichenko and Starikovskaya [CPM'13], in which we are asked to compute the minimal non-empty and the maximal suffixes of a substring. For the maximal suffixes problem, we give a linear-space structure with O(1) q…
Given two strings $T$ and $S$ and a set of strings $P$, for each string $p \in P$, consider the unique substrings of $T$ that have $p$ as their prefix and $S$ as their suffix. Two problems then come to mind; the first problem being the counting of such substrings, and the second problem being the problem of listing all such substrings. In this paper, we describe linear-time, linear-space suffix tree-based algorithms for both problems. More specifically, we describe an $O(|T| + |P|)$ time algorithm for the counting problem, and an $O(|T| + |P| + \#(ans))$ time algorithm for the listing problem,…
We present an improved wavelet tree construction algorithm and discuss its applications to a number of rank/select problems for integer keys and strings. Given a string of length n over an alphabet of size $σ\leq n$, our method builds the wavelet tree in $O(n \log σ/ \sqrt{\log{n}})$ time, improving upon the state-of-the-art algorithm by a factor of $\sqrt{\log n}$. As a consequence, given an array of n integers we can construct in $O(n \sqrt{\log n})$ time a data structure consisting of $O(n)$ machine words and capable of answering rank/select queries for the subranges of the array in $O(\log…
Let two static sequences of strings $P$ and $S$, representing prefix and suffix conditions respectively, be given as input for preprocessing. For the query, let two positive integers $k_1$ and $k_2$ be given, as well as a string $T$ given in an online manner, such that $T_i$ represents the length-$i$ prefix of $T$ for $1 \leq i \leq |T|$. In this paper we are interested in computing the set $\mathit{ans_i}$ of distinct substrings $w$ of $T_i$ such that $k_1 \leq |w| \leq k_2$, and $w$ contains some $p \in P$ as a prefix and some $s \in S$ as a suffix. More specifically, the counting problem is…
We consider the problem of encoding a string of length $n$ from an integer alphabet of size $σ$ so that access and substring equality queries (that is, determining the equality of any two substrings) can be answered efficiently. Any uniquely-decodable encoding supporting access must take $n\logσ+ Θ(\log (n\logσ))$ bits. We describe a new data structure matching this lower bound when $σ\leq n^{O(1)}$ while supporting both queries in optimal $O(1)$ time. Furthermore, we show that the string can be overwritten in-place with this structure. The redundancy of $Θ(\log n)$ bits and the constant query…
Joint Graduate School of Mathematics for Innovation, Kyushu University（九州大学数学创新联合研究生院）；Department of Information Science and Technology, Kyushu University（九州大学信息科学技术系）；Graduate School of Informatics and Engineering, University of Electro-Communications（电波大学信息与工程学院）；Department of Informatics, Kyushu University（九州大学信息学系）
We consider a sliding window $W$ over a stream of characters from some alphabet of constant size. The user wants to perform deterministic substring matching on the current sliding window content and obtain positions of the matches. We present an indexed version of the sliding window using the suffix tree, the link tree and the lowest common ancestor. The data structure of size $Θ(|W|)$ has optimal time queries $Θ(m+occ)$ and amortized constant time updates, where $m$ is the length of the query string and $occ$ is the number of its occurrences.
We revisit the problem of finding shortest unique substring (SUS) proposed recently by [6]. We propose an optimal $O(n)$ time and space algorithm that can find an SUS for every location of a string of size $n$. Our algorithm significantly improves the $O(n^2)$ time complexity needed by [6]. We also support finding all the SUSes covering every location, whereas the solution in [6] can find only one SUS for every location. Further, our solution is simpler and easier to implement and can also be more space efficient in practice, since we only use the inverse suffix array and longest common prefix…
Recently Kubica et al. (Inf. Process. Let., 2013) and Kim et al. (submitted to Theor. Comp. Sci.) introduced order-preserving pattern matching. In this problem we are looking for consecutive substrings of the text that have the same "shape" as a given pattern. These results include a linear-time order-preserving pattern matching algorithm for polynomially-bounded alphabet and an extension of this result to pattern matching with multiple patterns. We make one step forward in the analysis and give an $O(\frac{n\log{n}}{\log\log{n}})$ time randomized algorithm constructing suffix trees in the ord…
We revisit various string indexing problems with range reporting features, namely, position-restricted substring searching, indexing substrings with gaps, and indexing substrings with intervals. We obtain the following main results. {itemize} We give efficient reductions for each of the above problems to a new problem, which we call \emph{substring range reporting}. Hence, we unify the previous work by showing that we may restrict our attention to a single problem rather than studying each of the above problems individually. We show how to solve substring range reporting with optimal query tim…
The weighted ancestor problem is a well-known generalization of the predecessor problem to trees. It is known to require $Ω(\log\log n)$ time for queries provided $O(n\mathop{\mathrm{polylog}} n)$ space is available and weights are from $[0..n]$, where $n$ is the number of tree nodes. However, when applied to suffix trees, the problem, surprisingly, admits an $O(n)$-space solution with constant query time, as was shown by Gawrychowski, Lewenstein, and Nicholson (Proc. ESA 2014). This variant of the problem can be reformulated as follows: given the suffix tree of a string $s$, we need a data st…
We consider an index data structure for similar strings. The generalized suffix tree can be a solution for this. The generalized suffix tree of two strings $A$ and $B$ is a compacted trie representing all suffixes in $A$ and $B$. It has $|A|+|B|$ leaves and can be constructed in $O(|A|+|B|)$ time. However, if the two strings are similar, the generalized suffix tree is not efficient because it does not exploit the similarity which is usually represented as an alignment of $A$ and $B$. In this paper we propose a space/time-efficient suffix tree of alignment which wisely exploits the similarity i…
A grammar compression algorithm, called GCIS, is introduced in this work. GCIS is based on the induced suffix sorting algorithm SAIS, presented by Nong et al. in 2009. The proposed solution builds on the factorization performed by SAIS during suffix sorting. A context-free grammar is used to replace factors by non-terminals. The algorithm is then recursively applied on the shorter sequence of non-terminals. The resulting grammar is encoded by exploiting some redundancies, such as common prefixes between right-hands of rules, sorted according to SAIS. GCIS excels for its low space and time requ…
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 $n^{2/3} \le s \le n^{1-o(1)}$, the LCS problem can be s…
The suffix array $SA[1..n]$ of a text $T$ of length $n$ is a permutation of $\{1,\ldots,n\}$ describing the lexicographical ordering of suffixes of $T$, and it is considered to be among of the most important data structures in string algorithms, with dozens of applications in data compression, bioinformatics, and information retrieval. One of the biggest drawbacks of the suffix array is that it is very difficult to maintain under text updates: even a single character substitution can completely change the contents of the suffix array. Thus, the suffix array of a dynamic text is modelled using …
Suffix tree (and the closely related suffix array) are fundamental structures capturing all substrings of a given text essentially by storing all its suffixes in the lexicographical order. In some applications, we work with a subset of $b$ interesting suffixes, which are stored in the so-called sparse suffix tree. Because the size of this structure is $Θ(b)$, it is natural to seek a construction algorithm using only $O(b)$ words of space assuming read-only random access to the text. We design a linear-time Monte Carlo algorithm for this problem, hence resolving an open question explicitly stat…
Two strings $x$ and $y$ over $Σ\cup Π$ of equal length are said to \emph{parameterized match} (\emph{p-match}) if there is a renaming bijection $f:Σ\cup Π\rightarrow Σ\cup Π$ that is identity on $Σ$ and transforms $x$ to $y$ (or vice versa). The \emph{p-matching} problem is to look for substrings in a text that p-match a given pattern. In this paper, we propose \emph{parameterized suffix automata} (\emph{p-suffix automata}) and \emph{parameterized directed acyclic word graphs} (\emph{PDAWGs}) which are the p-matching versions of suffix automata and DAWGs. While suffix automata and DAWGs are eq…
Indexing highly repetitive texts - such as genomic databases, software repositories and versioned text collections - has become an important problem since the turn of the millennium. A relevant compressibility measure for repetitive texts is r, the number of runs in their Burrows-Wheeler Transforms (BWTs). One of the earliest indexes for repetitive collections, the Run-Length FM-index, used O(r) space and was able to efficiently count the number of occurrences of a pattern of length m in the text (in loglogarithmic time per pattern symbol, with current techniques). However, it was unable to lo…
We study quantum algorithms for several fundamental string problems, including Longest Common Substring, Lexicographically Minimal String Rotation, and Longest Square Substring. These problems have been widely studied in the stringology literature since the 1970s, and are known to be solvable by near-linear time classical algorithms. In this work, we give quantum algorithms for these problems with near-optimal query complexities and time complexities. Specifically, we show that: - Longest Common Substring can be solved by a quantum algorithm in $\tilde O(n^{2/3})$ time, improving upon the rece…
String attractors [STOC 2018] are combinatorial objects recently introduced to unify all known dictionary compression techniques in a single theory. A set $Γ\subseteq [1..n]$ is a $k$-attractor for a string $S\in[1..σ]^n$ if and only if every distinct substring of $S$ of length at most $k$ has an occurrence straddling at least one of the positions in $Γ$. Finding the smallest $k$-attractor is NP-hard for $k\geq3$, but polylogarithmic approximations can be found using reductions from dictionary compressors. It is easy to reduce the $k$-attractor problem to a set-cover instance where string's po…
CWI（荷兰数学与计算机科学研究学会）；Vrije Universiteit（阿姆斯特丹自由大学）；King’s College London（伦敦国王学院）；Birkbeck, University of London（伦敦大学伯贝克学院）；The Cyprus Institute（塞浦路斯研究所）
We propose algorithms that, given the input string of length $n$ over integer alphabet of size $σ$, construct the Burrows-Wheeler transform (BWT), the permuted longest-common-prefix (PLCP) array, and the LZ77 parsing in $O(n/\log_σn+r\,{\rm polylog}\,n)$ time and working space, where $r$ is the number of runs in the BWT of the input. These are the essential components of many compressed indexes such as compressed suffix tree, FM-index, and grammar and LZ77-based indexes, but also find numerous applications in sequence analysis and data compression. The value of $r$ is a common measure of repet…
Genome assembly is a fundamental problem in Bioinformatics, where for a given set of overlapping substrings of a genome, the aim is to reconstruct the source genome. The classical approaches to solving this problem use assembly graphs, such as de Bruijn graphs or overlap graphs, which maintain partial information about such overlaps. For genome assembly algorithms, these graphs present a trade-off between overlap information stored and scalability. Thus, Hierarchical Overlap Graph (HOG) was proposed to overcome the limitations of both these approaches. For a given set $P$ of $n$ strings, the f…
