We show how to build several data structures of central importance to string processing, taking as input the Burrows-Wheeler transform (BWT) and using small extra working space. Let $n$ be the text length and $σ$ be the alphabet size. We first provide two algorithms that enumerate all LCP values and suffix tree intervals in $O(n\logσ)$ time using just $o(n\logσ)$ bits of working space on top of the input BWT. Using these algorithms as building blocks, for any parameter $0 < ε\leq 1$ we show how to build the PLCP bitvector and the balanced parentheses representation of the suffix tree topology in $O\left(n(\logσ+ ε^{-1}\cdot \log\log n)\right)$ time using at most $n\logσ\cdot(ε+ o(1))$ bits of working space on top of the input BWT and the output. In particular, this implies that we can build a compressed suffix tree from the BWT using just succinct working space (i.e. $o(n\logσ)$ bits) and any time in $Θ(n\logσ) + ω(n\log\log n)$. This improves the previous most space-efficient algorithms, which worked in $O(n)$ bits and $O(n\log n)$ time. We also consider the problem of merging BWTs of string collections, and provide a solution running in $O(n\logσ)$ time and using just $o(n\logσ)$ bits of working space. An efficient implementation of our LCP construction and BWT merge algorithms use (in RAM) as few as $n$ bits on top of a packed representation of the input/output and process data as fast as $2.92$ megabases per second.
Suffix trees are a fundamental data structure in stringology, but their space usage, though linear, is an important problem for its applications. We design and implement a new compressed suffix tree targeted to highly repetitive texts, such as large genomic collections of the same species. Our suffix tree tree builds on Block Trees, a recent Lempel-Ziv-bounded data structure that captures the repetitiveness of its input. We use Block Trees to compress the topology of the suffix tree, and augment the Block Tree nodes with data that speeds up suffix tree navigation. Our compressed suffix tree is…
Suffix trees are one of the most versatile data structures in stringology, with many applications in bioinformatics. Their main drawback is their size, which can be tens of times larger than the input sequence. Much effort has been put into reducing the space usage, leading ultimately to compressed suffix trees. These compressed data structures can efficiently simulate the suffix tree, while using space proportional to a compressed representation of the sequence. In this work, we take a new approach to compressed suffix trees for repetitive sequence collections, such as collections of individu…
Spaced seeds are important tools for similarity search in bioinformatics, and using several seeds together often significantly improves their performance. With existing approaches, however, for each seed we keep a separate linear-size data structure, either a hash table or a spaced suffix array (SSA). In this paper we show how to compress SSAs relative to normal suffix arrays (SAs) and still support fast random access to them. We first prove a theoretical upper bound on the space needed to store an SSA when we already have the SA. We then present experiments indicating that our approach works …
We show that the compressed suffix array and the compressed suffix tree for a string of length $n$ over an integer alphabet of size $σ\leq n$ can both be built in $O(n)$ (randomized) time using only $O(n\logσ)$ bits of working space. The previously fastest construction algorithms that used $O(n\logσ)$ bits of space took times $O(n\log\logσ)$ and $O(n\log^εn)$ respectively (where $ε$ is any positive constant smaller than $1$). In the passing, we show that the Burrows-Wheeler transform of a string of length $n$ over an alphabet of size $σ$ can be built in deterministic $O(n)$ time and space $O(n…
In this paper we study the structure of suffix trees. Given an unlabeled tree $τ$ on $n$ nodes and suffix links of its internal nodes, we ask the question "Is $τ$ a suffix tree?", i.e., is there a string $S$ whose suffix tree has the same topological structure as $τ$? We place no restrictions on $S$, in particular we do not require that $S$ ends with a unique symbol. This corresponds to considering the more general definition of implicit or extended suffix trees. Such general suffix trees have many applications and are for example needed to allow efficient updates when suffix trees are built o…
Temporal graphs represent binary relationships that change along time. They can model the dynamism of, for example, social and communication networks. Temporal graphs are defined as sets of contacts that are edges tagged with the temporal intervals when they are active. This work explores the use of the Compressed Suffix Array (CSA), a well-known compact and self-indexed data structure in the area of text indexing, to represent large temporal graphs. The new structure, called Temporal Graph CSA (TGCSA), is experimentally compared with the most competitive compact data structures in the state-o…
The precise analysis of the variance of the profile of a suffix tree has been a longstanding open problem. We analyze three regimes of the asymptotic growth of the variance of the profile of a suffix tree built from a randomly generated binary string, in the nonuniform case. We utilize combinatorics on words, singularity analysis, and the Mellin transform.
The suffix tree is a very important data structure in string processing, but it suffers from a huge space consumption. In large-scale applications, compressed suffix trees (CSTs) are therefore used instead. A CST consists of three (compressed) components: the suffix array, the LCP-array, and data structures for simulating navigational operations on the suffix tree. The LCP-array stores the lengths of the longest common prefixes of lexicographically adjacent suffixes, and it can be computed in linear time. In this paper, we present new LCP-array construction algorithms that are fast and very sp…
We consider the problem of constructing a sparse suffix tree (or suffix array) for $b$ suffixes of a given text $T$ of size $n$, using only $O(b)$ words of space during construction time. Breaking the naive bound of $Ω(nb)$ time for this problem has occupied many algorithmic researchers since a different structure, the (evenly spaced) sparse suffix tree, was introduced by K{ä}rkk{ä}inen and Ukkonen in 1996. While in the evenly spaced sparse suffix tree the suffixes considered must be evenly spaced in $T$, here there is no constraint on the locations of the suffixes. We show that the sparse suf…
When augmented with the longest common prefix (LCP) array and some other structures, the suffix array can solve many string processing problems in optimal time and space. A compressed representation of the LCP array is also one of the main building blocks in many compressed suffix tree proposals. In this paper, we describe a new compressed LCP representation: the sampled LCP array. We show that when used with a compressed suffix array (CSA), the sampled LCP array often offers better time/space trade-offs than the existing alternatives. We also show how to construct the compressed representatio…
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…
We study a suffix tree built from a sequence generated by a Markovian source. Such sources are more realistic probabilistic models for text generation, data compression, molecular applications, and so forth. We prove that the average size of such a suffix tree is asymptotically equivalent to the average size of a trie built over $n$ independent sequences from the same Markovian source. This equivalence is only known for memoryless sources. We then derive a formula for the size of a trie under Markovian model to complete the analysis for suffix trees. We accomplish our goal by applying some nov…
Suffix trees have emerged to be very fast for pattern searching yielding O (m) time, where m is the pattern size. Unfortunately their high memory requirements make it impractical to work with huge amounts of data. We present a memory efficient algorithm of a generalized suffix tree which reduces the space size by a factor of 10 when the size of the pattern is known beforehand. Experiments on the chromosomes and Pizza&Chili corpus show significant advantages of our algorithm over standard linear time suffix tree construction in terms of memory usage for pattern searching.
Suffix tree construction algorithms based on suffix links are popular because they are simple to implement, can operate online in linear time, and because the suffix links are often convenient for pattern matching. We present an approach using edge-oriented suffix links, which reduces the number of branch lookup operations (known to be a bottleneck in construction time) with some additional techniques to reduce construction cost. We discuss various effects of our approach and compare it to previous techniques. An experimental evaluation shows that we are able to reduce construction time to aro…
Given the query string of length $m$, we explore a parallel query in a static suffix tree based data structure for $p \ll n$, where $p$ is the number of processors and $n$ is the length of the text. We present three results on CREW PRAM. The parallel query in the suffix trie requires $O(m + p)$ work, $O(m/p + \lg p)$ time and $O(n^2)$ space in the worst case. We extend the same technique to the suffix tree where we show it is, by design, inherently sequential in the worst case. Finally we perform the parallel query using an interleaved approach and achieve $O(m \lg p)$ work, $O(\frac{m}{p} \lg…
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 show that the compressed suffix array and the compressed suffix tree of a string $T$ can be built in $O(n)$ deterministic time using $O(n\logσ)$ bits of space, where $n$ is the string length and $σ$ is the alphabet size. Previously described deterministic algorithms either run in time that depends on the alphabet size or need $ω(n\log σ)$ bits of working space. Our result has immediate applications to other problems, such as yielding the first linear-time LZ77 and LZ78 parsing algorithms that use $O(n \logσ)$ bits.
The classical, ubiquitous, predecessor problem is to construct a data structure for a set of integers that supports fast predecessor queries. Its generalization to weighted trees, a.k.a. the weighted ancestor problem, has been extensively explored and successfully reduced to the predecessor problem. It is known that any solution for both problems with an input set from a polynomially bounded universe that preprocesses a weighted tree in O(n polylog(n)) space requires Ω(loglogn) query time. Perhaps the most important and frequent application of the weighted ancestors problem is for suffix trees…
The run-length compressed Burrows-Wheeler transform (RLBWT) used in conjunction with the backward search introduced in the FM index is the centerpiece of most compressed indexes working on highly-repetitive data sets like biological sequences. Compared to grammar indexes, the size of the RLBWT is often much bigger, but queries like counting the occurrences of long patterns can be done much faster than on any existing grammar index so far. In this paper, we combine the virtues of a grammar with the RLBWT by building the RLBWT on top of a special grammar based on induced suffix sorting. Our expe…
A suffix tree is able to efficiently locate a pattern in an indexed string, but not in general the most recent copy of the pattern in an online stream, which is desirable in some applications. We study the most general version of the problem of locating a most recent match: supporting queries for arbitrary patterns, at each step of processing an online stream. We present augmentations to Ukkonen's suffix tree construction algorithm for optimal-time queries, maintaining indexing time within a logarithmic factor in the size of the indexed string. We show that the algorithm is applicable to slidi…
We solve the problem of finding interspersed maximal repeats using a suffix array construction. As it is well known, all the functionality of suffix trees can be handled by suffix arrays, gaining practicality. Our solution improves the suffix tree based approaches for the repeat finding problem, being particularly well suited for very large inputs. We prove the corrrectness and complexity of the algorithms.
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 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.
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…
Suffix trees have recently become very successful data structures in handling large data sequences such as DNA or Protein sequences. Consequently parallel architectures have become ubiquitous. We present a novel alphabet-dependent parallel algorithm which attempts to take advantage of the perverseness of the multicore architecture. Microsatellites are important for their biological relevance hence our algorithm is based on time efficient construction for identification of such. We experimentally achieved up to 15x speedup over the sequential algorithm on different input sizes of biological seq…
We consider construction of the suffix tree and the directed acyclic word graph (DAWG) indexing data structures for a collection $\mathcal{T}$ of texts, where a new symbol may be appended to any text in $\mathcal{T} = \{T_1, \ldots, T_K\}$, at any time. This fully-online scenario, which arises in dynamically indexing multi-sensor data, is a natural generalization of the long solved semi-online text indexing problem, where texts $T_1, \ldots, T_{k}$ are permanently fixed before the next text $T_{k+1}$ is processed for each $1 \leq k < K$. We present fully-online algorithms that construct the su…
The suffix trees are fundamental data structures for various kinds of string processing. The suffix tree of a string $T$ of length $n$ has $O(n)$ nodes and edges, and the string label of each edge is encoded by a pair of positions in $T$. Thus, even after the tree is built, the input text $T$ needs to be kept stored and random access to $T$ is still needed. The linear-size suffix tries (LSTs), proposed by Crochemore et al. [Linear-size suffix tries, TCS 638:171-178, 2016], are a `stand-alone' alternative to the suffix trees. Namely, the LST of a string $T$ of length $n$ occupies $O(n)$ total s…
Packing several characters into one computer word is a simple and natural way to compress the representation of a string and to speed up its processing. Exploiting this idea, we propose an index for a packed string, based on a {\em sparse suffix tree} \cite{KU-96} with appropriately defined suffix links. Assuming, under the standard unit-cost RAM model, that a word can store up to $\log_σn$ characters ($σ$ the alphabet size), our index takes $O(n/\log_σn)$ space, i.e. the same space as the packed string itself. The resulting pattern matching algorithm runs in time $O(m+r^2+r\cdot occ)$, where …
A suffix tree is a data structure used mainly for pattern matching. It is known that the space complexity of simple suffix trees is quadratic in the length of the string. By a slight modification of the simple suffix trees one gets the compact suffix trees, which have linear space complexity. The motivation of this paper is the question whether the space complexity of simple suffix trees is quadratic not only in the worst case, but also in expectation.
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…
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…
Given a string $T$, it is known that its suffix tree can be represented using the compact directed acyclic word graph (CDAWG) with $e_T$ arcs, taking overall $O(e_T+e_{\overline{T}})$ words of space, where ${\overline{T}}$ is the reverse of $T$, and supporting some key operations in time between $O(1)$ and $O(\log{\log{n}})$ in the worst case. This representation is especially appealing for highly repetitive strings, like collections of similar genomes or of version-controlled documents, in which $e_T$ grows sublinearly in the length of $T$ in practice. In this paper we augment such representa…
We deal with the problem of maintaining the suffix tree indexing structure for a fully-online collection of multiple strings, where a new character can be prepended to any string in the collection at any time. The only previously known algorithm for the problem, recently proposed by Takagi et al. [Algorithmica 82(5): 1346-1377 (2020)], runs in $O(N \log σ)$ time and $O(N)$ space on the word RAM model, where $N$ denotes the total length of the strings and $σ$ denotes the alphabet size. Their algorithm makes heavy use of the nearest marked ancestor (NMA) data structure on semi-dynamic trees, tha…
We prove that longest common prefix (LCP) information can be stored in much less space than previously known. More precisely, we show that in the presence of the text and the suffix array, o(n) additional bits are sufficient to answer LCP-queries asymptotically in the same time that is needed to retrieve an entry from the suffix array. This yields the smallest compressed suffix tree with sub-logarithmic navigation time.
Suffix arrays encode the lexicographical order of all suffixes of a text and are often combined with the Longest Common Prefix array (LCP) to simulate navigational queries on the suffix tree in reduced space. In space-critical applications such as sparse and compressed text indexing, only information regarding the lexicographical order of a size-$b$ subset of all $n$ text suffixes is often needed. Such information can be stored space-efficiently (in $b$ words) in the sparse suffix array (SSA). The SSA and its relative sparse LCP array (SLCP) can be used as a space-efficient substitute of the s…
Suffix trees and suffix arrays are two of the most widely used data structures for text indexing. Each uses linear space and can be constructed in linear time for polynomially sized alphabets. However, when it comes to answering queries with worst-case deterministic time bounds, the prior does so in $O(m\log|Σ|)$ time, where $m$ is the query size, $|Σ|$ is the alphabet size, and the latter does so in $O(m+\log n)$ time, where $n$ is the text size. If one wants to output all appearances of the query, an additive cost of $O(occ)$ time is sufficient, where $occ$ is the size of the output. We prop…
Online Computation of Palindromes and Suffix Trees on Tries
Joint Graduate School of Mathematics for Innovation, Kyushu University（九州大学数学创新联合研究生院）；NTT Communication Science Laboratories（NTT通信科学实验室）；Department of Computer and Network Engineering, University of Electro-Communications（电子通信大学计算机与网络工程系）；Department of Informatics, Kyushu University（九州大学信息学系）；M&D Data Science Center, Institute of Integrated Research, Institute of Science Tokyo（东京科学大学综合研究研究所M&D数据科学中心）
Matching statistics were introduced to solve the approximate string matching problem, which is a recurrent subroutine in bioinformatics applications. In 2010, Ohlebusch et al. [SPIRE 2010] proposed a time and space efficient algorithm for computing matching statistics which relies on some components of a compressed suffix tree - notably, the longest common prefix (LCP) array. In this paper, we show how their algorithm can be generalized from strings to Wheeler deterministic finite automata. Most importantly, we introduce a notion of LCP array for Wheeler automata, thus establishing a first cle…
We present several results about position heaps, a relatively new alternative to suffix trees and suffix arrays. First, we show that, if we limit the maximum length of patterns to be sought, then we can also limit the height of the heap and reduce the worst-case cost of insertions and deletions. Second, we show how to build a position heap in linear time independent of the size of the alphabet. Third, we show how to augment a position heap such that it supports access to the corresponding suffix array, and vice versa. Fourth, we introduce a variant of a position heap that can be simulated effi…
We show that both the Lempel Ziv 77- and the 78-factorization of a text of length $n$ on an integer alphabet of size $σ$ can be computed in $O(n \lg \lg σ)$ time (linear time if we allow randomization) using $O(n \lg σ)$ bits of working space. Given that a compressed representation of the suffix tree is loaded into RAM, we can compute both factorizations in $O(n)$ time using $z \lg n + O(n)$ bits of space, where $z$ is the number of factors.
We present the first worst-case linear time algorithm that directly computes the parameterized suffix and LCP arrays for constant sized alphabets. Previous algorithms either required quadratic time or the parameterized suffix tree to be built first. More formally, for a string over static alphabet $Σ$ and parameterized alphabet $Π$, our algorithm runs in $O(nπ)$ time and $O(n)$ words of space, where $π$ is the number of distinct symbols of $Π$ in the string.
In this paper, a contrastive evaluation of massively parallel implementations of suffix tree and suffix array to accelerate genome sequence matching are proposed based on Intel Core i7 3770K quad-core and NVIDIA GeForce GTX680 GPU. Besides suffix array only held approximately 20%~30% of the space relative to suffix tree, the coalesced binary search and tile optimization make suffix array clearly outperform suffix tree using GPU. Consequently, the experimental results show that multiple genome sequence matching based on suffix array is more than 99 times speedup than that of CPU serial implemen…
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…
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…
Text indexing is a classical algorithmic problem that has been studied for over four decades: given a text $T$, pre-process it off-line so that, later, we can quickly count and locate the occurrences of any string (the query pattern) in $T$ in time proportional to the query's length. The earliest optimal-time solution to the problem, the suffix tree, dates back to 1973 and requires up to two orders of magnitude more space than the plain text just to be stored. In the year 2000, two breakthrough works showed that efficient queries can be achieved without this space overhead: a fast index be sto…
