Bille and Gørtz (2011) recently introduced the problem of substring range counting, for which we are asked to store compactly a string $S$ of $n$ characters with integer labels in ([0, u]), such that later, given an interval ([a, b]) and a pattern $P$ of length $m$, we can quickly count the occurrences of $P$ whose first characters' labels are in ([a, b]). They showed how to store $S$ in $\Oh{n \log n / \log \log n}$ space and answer queries in $\Oh{m + \log \log u}$ time. We show that, if $S$ is over an alphabet of size (\polylog (n)), then we can achieve optimal linear space. Moreover, if (u = n \polylog (n)), then we can also reduce the time to $\Oh{m}$. Our results give linear space and time bounds for position-restricted substring counting and the counting versions of indexing substrings with intervals, indexing substrings with gaps and aligned pattern matching.
A palindrome is a string that reads the same forward and backward. A palindromic substring $w$ of a string $T$ is called a minimal unique palindromic substring (MUPS) of $T$ if $w$ occurs only once in $T$ and any proper palindromic substring of $w$ occurs at least twice in $T$. MUPSs are utilized for answering the shortest unique palindromic substring problem, which is motivated by molecular biology [Inoue et al., 2018]. Given a string $T$ of length $n$, all MUPSs of $T$ can be computed in $O(n)$ time. In this paper, we study the problem of updating the set of MUPSs when a character in the inp…
Given a text and a pattern over an alphabet, the pattern matching problem searches for all occurrences of the pattern in the text. An equivalence relation $\approx$ is called a substring consistent equivalence relation (SCER), if for two strings $X$ and $Y$, $X \approx Y$ implies $|X| = |Y|$ and $X[i:j] \approx Y[i:j]$ for all $1 \le i \le j \le |X|$. In this paper, we propose an efficient parallel algorithm for pattern matching under any SCER using the"duel-and-sweep" paradigm. For a pattern of length $m$ and a text of length $n$, our algorithm runs in $O(ξ_m^\mathrm{t} \log^2 m)$ time and $O…
Suppose an oracle knows a string $S$ that is unknown to us and that we want to determine. The oracle can answer queries of the form "Is $s$ a substring of $S$?". In 1995, Skiena and Sundaram showed that, in the worst case, any algorithm needs to ask the oracle $σn/4 -O(n)$ queries in order to be able to reconstruct the hidden string, where $σ$ is the size of the alphabet of $S$ and $n$ its length, and gave an algorithm that spends $(σ-1)n+O(σ\sqrt{n})$ queries to reconstruct $S$. The main contribution of our paper is to improve the above upper-bound in the context where the string is compressi…
A run in a string is a maximal periodic substring. For example, the string $\texttt{bananatree}$ contains the runs $\texttt{anana} = (\texttt{an})^{3/2}$ and $\texttt{ee} = \texttt{e}^2$. There are less than $n$ runs in any length-$n$ string, and computing all runs for a string over a linearly-sortable alphabet takes $\mathcal{O}(n)$ time (Bannai et al., SODA 2015). Kosolobov conjectured that there also exists a linear time runs algorithm for general ordered alphabets (Inf. Process. Lett. 2016). The conjecture was almost proven by Crochemore et al., who presented an $\mathcal{O}(nα(n))$ time a…
In this paper we are interested in indexing texts for substring matching queries with one edit error. That is, given a text $T$ of $n$ characters over an alphabet of size $σ$, we are asked to build a data structure that answers the following query: find all the $occ$ substrings of the text that are at edit distance at most $1$ from a given string $q$ of length $m$. In this paper we show two new results for this problem. The first result, suitable for an unbounded alphabet, uses $O(n\log^εn)$ (where $ε$ is any constant such that $0<ε<1$) words of space and answers to queries in time $O(m+occ)$.…
We present a near-linear time algorithm that approximates the edit distance between two strings within a polylogarithmic factor; specifically, for strings of length n and every fixed epsilon>0, it can compute a (log n)^O(1/epsilon) approximation in n^(1+epsilon) time. This is an exponential improvement over the previously known factor, 2^(O (sqrt(log n))), with a comparable running time (Ostrovsky and Rabani J.ACM 2007; Andoni and Onak STOC 2009). Previously, no efficient polylogarithmic approximation algorithm was known for any computational task involving edit distance (e.g., nearest neighbo…
Given $m$ documents of total length $n$, we consider the problem of finding a longest string common to at least $d \geq 2$ of the documents. This problem is known as the \emph{longest common substring (LCS) problem} and has a classic $O(n)$ space and $O(n)$ time solution (Weiner [FOCS'73], Hui [CPM'92]). However, the use of linear space is impractical in many applications. In this paper we show that for any trade-off parameter $1 \leq τ\leq n$, the LCS problem can be solved in $O(τ)$ space and $O(n^2/τ)$ time, thus providing the first smooth deterministic time-space trade-off from constant to …
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…
Given a string $T$ on an alphabet of size $σ$, we describe a bidirectional Burrows-Wheeler index that takes $O(|T|\logσ)$ bits of space, and that supports the addition \emph{and removal} of one character, on the left or right side of any substring of $T$, in constant time. Previously known data structures that used the same space allowed constant-time addition to any substring of $T$, but they could support removal only from specific substrings of $T$. We also describe an index that supports bidirectional addition and removal in $O(\log{\log{|T|}})$ time, and that occupies a number of words pr…
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…
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…
Let s = s_1 .. s_n be a text (or sequence) on a finite alphabet Σof size σ. A fingerprint in s is the set of distinct characters appearing in one of its substrings. The problem considered here is to compute the set {\cal F} of all fingerprints of all substrings of s in order to answer efficiently certain questions on this set. A substring s_i .. s_j is a maximal location for a fingerprint f in F (denoted by <i,j>) if the alphabet of s_i .. s_j is f and s_{i-1}, s_{j+1}, if defined, are not in f. The set of maximal locations ins is {\cal L} (it is easy to see that |{\cal L}| \leq n σ). Two maxi…
Given a pattern string $P$ of length $n$ consisting of $δ$ distinct characters and a query string $T$ of length $m$, where the characters of $P$ and $T$ are drawn from an alphabet $Σ$ of size $Δ$, the {\em exact string matching} problem consists of finding all occurrences of $P$ in $T$. For this problem, we present a randomized heuristic that in $O(nδ)$ time preprocesses $P$ to identify $sparse(P)$, a rarely occurring substring of $P$, and then use it to find all occurrences of $P$ in $T$ efficiently. This heuristic has an expected search time of $O( \frac{m}{min(|sparse(P)|, Δ)})$, where $|sp…
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…
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 …
We consider the problem of storing a dynamic string $S$ over an alphabet $Σ=\{\,1,\ldots,σ\,\}$ in compressed form. Our representation supports insertions and deletions of symbols and answers three fundamental queries: $\mathrm{access}(i,S)$ returns the $i$-th symbol in $S$, $\mathrm{rank}_a(i,S)$ counts how many times a symbol $a$ occurs among the first $i$ positions in $S$, and $\mathrm{select}_a(i,S)$ finds the position where a symbol $a$ occurs for the $i$-th time. We present the first fully-dynamic data structure for arbitrarily large alphabets that achieves optimal query times for all th…
The palindromic tree (a.k.a. eertree) for a string $S$ of length $n$ is a tree-like data structure that represents the set of all distinct palindromic substrings of $S$, using $O(n)$ space [Rubinchik and Shur, 2018]. It is known that, when $S$ is over an alphabet of size $σ$ and is given in an online manner, then the palindromic tree of $S$ can be constructed in $O(n\logσ)$ time with $O(n)$ space. In this paper, we consider the sliding window version of the problem: For a sliding window of length at most $d$, we present two versions of an algorithm which maintains the palindromic tree of size …
Given a pattern string $P$ of length $n$ and a query string $T$ of length $m$, where the characters of $P$ and $T$ are drawn from an alphabet of size $Δ$, the {\em exact string matching} problem consists of finding all occurrences of $P$ in $T$. For this problem, we present algorithms that in $O(nΔ^2)$ time pre-process $P$ to essentially identify $sparse(P)$, a rarely occurring substring of $P$, and then use it to find occurrences of $P$ in $T$ efficiently. Our algorithms require a worst case search time of $O(m)$, and expected search time of $O(m/min(|sparse(P)|, Δ))$, where $|sparse(P)|$ is …
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…
Given a string $S$, the \emph{compressed indexing problem} is to preprocess $S$ into a compressed representation that supports fast \emph{substring queries}. The goal is to use little space relative to the compressed size of $S$ while supporting fast queries. We present a compressed index based on the Lempel--Ziv 1977 compression scheme. We obtain the following time-space trade-offs: For constant-sized alphabets; (i) $O(m + occ \lg\lg n)$ time using $O(z\lg(n/z)\lg\lg z)$ space, or (ii) $O(m(1 + \frac{\lg^εz}{\lg(n/z)}) + occ(\lg\lg n + \lg^εz))$ time using $O(z\lg(n/z))$ space. For integer al…
Wheeler DFAs (WDFAs) are a sub-class of finite-state automata which is playing an important role in the emerging field of compressed data structures: as opposed to general automata, WDFAs can be stored in just $\logσ+ O(1)$ bits per edge, $σ$ being the alphabet's size, and support optimal-time pattern matching queries on the substring closure of the language they recognize. An important step to achieve further compression is minimization. When the input $\mathcal A$ is a general deterministic finite-state automaton (DFA), the state-of-the-art is represented by the classic Hopcroft's algorithm,…
We consider the approximate pattern matching problem under edit distance. In this problem we are given a pattern $P$ of length $w$ and a text $T$ of length $n$ over some alphabet $Σ$, and a positive integer $k$. The goal is to find all the positions $j$ in $T$ such that there is a substring of $T$ ending at $j$ which has edit distance at most $k$ from the pattern $P$. Recall, the edit distance between two strings is the minimum number of character insertions, deletions, and substitutions required to transform one string into the other. For a position $t$ in $\{1,...,n\}$, let $k_t$ be the smal…
We study the problem of supporting queries on a string $S$ of length $n$ within a space bounded by the size $γ$ of a string attractor for $S$. Recent works showed that random access on $S$ can be supported in optimal $O(\log(n/γ)/\log\log n)$ time within $O\left (γ \rm{polylog}\ n \right)$ space. In this paper, we extend this result to \emph{rank} and \emph{select} queries and provide lower bounds matching our upper bounds on alphabets of polylogarithmic size. Our solutions are given in the form of a space-time trade-off that is more general than the one previously known for grammars and that …
Institute of Computer Science, University of Wrocław（弗罗茨瓦夫大学计算机科学研究所）；Department of Computer Science, Ariel University（阿里埃尔大学计算机科学系）；Max Planck Institute for Informatics（马克斯·普朗克信息学研究所）
Let $S$ be a string of length $n$ over an alphabet $Σ$ and let $Q$ be a subset of $Σ$ of size $q \geq 2$. The 'co-occurrence problem' is to construct a compact data structure that supports the following query: given an integer $w$ return the number of length-$w$ substrings of $S$ that contain each character of $Q$ at least once. This is a natural string problem with applications to, e.g., data mining, natural language processing, and DNA analysis. The state of the art is an $O(\sqrt{nq})$ space data structure that -- with some minor additions -- supports queries in $O(\log\log n)$ time [CPM 20…
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…
Given strings $P$ and $Q$ the (exact) string matching problem is to find all positions of substrings in $Q$ matching $P$. The classical Knuth-Morris-Pratt algorithm [SIAM J. Comput., 1977] solves the string matching problem in linear time which is optimal if we can only read one character at the time. However, most strings are stored in a computer in a packed representation with several characters in a single word, giving us the opportunity to read multiple characters simultaneously. In this paper we study the worst-case complexity of string matching on strings given in packed representation. …
The Cartesian-tree pattern matching is a recently introduced scheme of pattern matching that detects fragments in a sequential data stream which have a similar structure as a query pattern. Formally, Cartesian-tree pattern matching seeks all substrings $S'$ of the text string $S$ such that the Cartesian tree of $S'$ and that of a query pattern $P$ coincide. In this paper, we present a new indexing structure for this problem called the Cartesian-tree Position Heap (CPH). Let $n$ be the length of the input text string $S$, $m$ the length of a query pattern $P$, and $σ$ the alphabet size. We show…
Given strings $P$ of length $m$ and $T$ of length $n$ over an alphabet of size $σ$, the string matching with $k$-mismatches problem is to find the positions of all the substrings in $T$ that are at Hamming distance at most $k$ from $P$. If $T$ can be read only one character at the time the best known bounds are $O(n\sqrt{k\log k})$ and $O(n + n\sqrt{k/w}\log k)$ in the word-RAM model with word length $w$. In the RAM models (including $AC^0$ and word-RAM) it is possible to read up to $\floor{w / \log σ}$ characters in constant time if the characters of $T$ are encoded using $\ceil{\log σ}$ bits…
We consider time-space tradeoffs for exactly computing frequency moments and order statistics over sliding windows. Given an input of length 2n-1, the task is to output the function of each window of length n, giving n outputs in total. Computations over sliding windows are related to direct sum problems except that inputs to instances almost completely overlap. We show an average case and randomized time-space tradeoff lower bound of TS in Omega(n^2) for multi-way branching programs, and hence standard RAM and word-RAM models, to compute the number of distinct elements, F_0, in sliding window…
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…
Indexing labeled graphs for pattern matching is a central challenge of pangenomics. Equi et al. (Algorithmica, 2022) developed the Elastic Founder Graph ($\mathsf{EFG}$) representing an alignment of $m$ sequences of length $n$, drawn from alphabet $Σ$ plus the special gap character: the paths spell the original sequences or their recombination. By enforcing the semi-repeat-free property, the $\mathsf{EFG}$ admits a polynomial-space index for linear-time pattern matching, breaking through the conditional lower bounds on indexing labeled graphs (Equi et al., SOFSEM 2021). In this work we improve…
