Maxime Crochemore, Lucian Ilie
The cornerstone of any algorithm computing all repetitions in a string of length n in O(n) time is the fact that the number of runs (or maximal repetitions) is O(n). We give a simple proof of this result. As a consequence of our approach, the stronger result concerning the linearity of the sum of exponents of all runs follows easily.
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In this paper we present two algorithms for the following problem: given a string and a rational $e > 1$, detect in the online fashion the earliest occurrence of a repetition of exponent $\ge e$ in the string. 1. The first algorithm supports the backtrack operation removing the last letter of the input string. This solution runs in $O(n\log m)$ time and $O(m)$ space, where $m$ is the maximal length of a string generated during the execution of a given sequence of $n$ read and backtrack operations. 2. The second algorithm works in $O(n\logσ)$ time and $O(n)$ space, where $n$ is the length of th…
Highly-repetitive collections of strings are increasingly being amassed by genome sequencing and genetic variation experiments, as well as by storing all versions of human-generated files, like webpages and source code. Existing indexes for locating all the exact occurrences of a pattern in a highly-repetitive string take advantage of a single measure of repetition. However, multiple, distinct measures of repetition all grow sublinearly in the length of a highly-repetitive string. In this paper we explore the practical advantages of combining data structures whose size depends on distinct meas…
Simple Linear-time Repetition Factorization
本文研究字符串重复因子分解问题，提出一种不依赖 Union-Find 和区间穿刺数据结构的简单 O(n) 时间算法，改进了此前依赖复杂数据结构的线性时间解法。
The notions of periodicity and repetitions in strings, and hence these of runs and squares, naturally extend to two-dimensional strings. We consider two types of repetitions in 2D-strings: 2D-runs and quartics (quartics are a 2D-version of squares in standard strings). Amir et al. introduced 2D-runs, showed that there are $O(n^3)$ of them in an $n \times n$ 2D-string and presented a simple construction giving a lower bound of $Ω(n^2)$ for their number (TCS 2020). We make a significant step towards closing the gap between these bounds by showing that the number of 2D-runs in an $n \times n$ 2D-…
Optimal Bounds for Distinct Quartics
本研究解决了二维字符串中不同四次方的渐近紧界问题，证明其数量为$\u00a0(n^2 \u00a0 n)$，并给出了最坏情况下最优的$\u00a0(n^2 \u00a0 n)$时间计算算法，完善了高维字符串重复结构的研究。
This paper provides an upper bound for several subsets of maximal repeats and maximal pairs in compressed strings and also presents a formerly unknown relationship between maximal pairs and the run-length Burrows-Wheeler transform. This relationship is used to obtain a different proof for the Burrows-Wheeler conjecture which has recently been proven by Kempa and Kociumaka in "Resolution of the Burrows-Wheeler Transform Conjecture". More formally, this paper proves that a string $S$ with $z$ LZ77-factors and without $q$-th powers has at most $73(\log_2 |S|)(z+2)^2$ runs in the run-length Burrow…
Compact Enumeration of Maximal Closed Substrings in Run-Length Encoded Strings
研究从字符串游程编码直接枚举最大封闭子串，引入紧凑族表示，证明\(O(m^2)\)个族有时必要且总是充分，借助后缀树等处理不同情况，能在特定时间和空间内列出所有最大封闭子串的紧凑表示。
A run is a maximal occurrence of a repetition $v$ with a period $p$ such that $2p \le |v|$. The maximal number of runs in a string of length $n$ was studied by several authors and it is known to be between $0.944 n$ and $1.029 n$. We investigate highly periodic runs, in which the shortest period $p$ satisfies $3p \le |v|$. We show the upper bound $0.5n$ on the maximal number of such runs in a string of length $n$ and construct a sequence of words for which we obtain the lower bound $0.406 n$.
A border of a string is a non-empty prefix of the string that is also a suffix of the string, and a string is unbordered if it has no border other than itself. Loptev, Kucherov, and Starikovskaya [CPM 2015] conjectured the following: If we pick a string of length $n$ from a fixed non-unary alphabet uniformly at random, then the expected maximum length of its unbordered factors is $n - O(1)$. We confirm this conjecture by proving that the expected value is, in fact, ${n - Θ(σ^{-1})}$, where $σ$ is the size of the alphabet. This immediately implies that we can find such a maximal unbordered fact…
Given a string $S$ of length $n$, its maximal unbordered factor is the longest factor which does not have a border. In this work we investigate the relationship between $n$ and the length of the maximal unbordered factor of $S$. We prove that for the alphabet of size $σ\ge 5$ the expected length of the maximal unbordered factor of a string of length~$n$ is at least $0.99 n$ (for sufficiently large values of $n$). As an application of this result, we propose a new algorithm for computing the maximal unbordered factor of a string.
Efficient algorithms for enumerating maximal common subsequences of two strings
本研究针对两字符串最大公共子序列枚举问题，提出三种不同时空复杂度的高效算法，分别适配不同的资源与输出效率需求，部分结构还支持条件式MCS检索。
There has been surprisingly little work on algorithms for sorting strings on distributed-memory parallel machines. We develop efficient algorithms for this problem based on the multi-way merging principle. These algorithms inspect only characters that are needed to determine the sorting order. Moreover, communication volume is reduced by also communicating (roughly) only those characters and by communicating repetitions of the same prefixes only once. Experiments on up to 1280 cores reveal that these algorithm are often more than five times faster than previous algorithms.
Computing Maximal Repeating Subsequences in a String
本文提出在单个字符串中高效计算最大重复子序列的方法，改进了传统算法的时间复杂度。
Almost succinct representation of maximal palindromes
本文提出了一种新的紧凑表示方法，用于存储字符串中的所有极大回文，能够在O(1)时间内检索任意位置的回文长度，同时支持在O(log n)时间内计算任意子串的最长回文。
Squares (fragments of the form $xx$, for some string $x$) are arguably the most natural type of repetition in strings. The basic algorithmic question concerning squares is to check if a given string of length $n$ is square-free, that is, does not contain a fragment of such form. Main and Lorentz [J. Algorithms 1984] designed an $\mathcal{O}(n\log n)$ time algorithm for this problem, and proved a matching lower bound assuming the so-called general alphabet, meaning that the algorithm is only allowed to check if two characters are equal. However, their lower bound also assumes that there are $Ω(…
We study a new generalization of palindromes and gapped palindromes called block palindromes. A block palindrome is a string that becomes a palindrome when identical substrings are replaced with a distinct character. We investigate several properties of block palindromes and in particular, study substrings of a string which are block palindromes. In so doing, we introduce the notion of a \emph{maximal block palindrome}, which leads to a compact representation of all block palindromes that occur in a string. We also propose an algorithm which enumerates all maximal block palindromes that appear…
Maximal Closed Substrings
该研究提出极大闭子串（MCS）的定义，明确其与游程、极大带间隙重复的关联，并给出一种能在O(n log n)时间内定位长度为n的字符串中所有MCS的算法。
Fast Computation of $k$-Runs, Parameterized Squares, and Other Generalised Squares
本文证明 $k$-错配重复输出规模为 $O(nk\log k)$，并据此给出 $O(nσ\log σ)$ 枚举与 $O(n\log n)$ 计数不等价参数化平方的算法。
In this paper we compare the difference in performance of two of the Prediction by Partial Matching (PPM) family of compressors (PPM$^*$ and the original Bounded PPM algorithm) and the Lempel-Ziv 78 (LZ) algorithm. We construct an infinite binary sequence whose worst-case compression ratio for PPM$^*$ is $0$, while Bounded PPM's and LZ's best-case compression ratios are at least $1/2$ and $1$ respectively. This sequence is an enumeration of all binary strings in order of length, i.e. all strings of length $1$ followed by all strings of length $2$ and so on. It is therefore normal, and is built…
We describe a RAM algorithm computing all runs (maximal repetitions) of a given string of length $n$ over a general ordered alphabet in $O(n\log^{\frac{2}3} n)$ time and linear space. Our algorithm outperforms all known solutions working in $Θ(n\logσ)$ time provided $σ= n^{Ω(1)}$, where $σ$ is the alphabet size. We conjecture that there exists a linear time RAM algorithm finding all runs.
Indexing highly repetitive strings (i.e., strings with many repetitions) for fast queries has become a central research topic in string processing, because it has a wide variety of applications in bioinformatics and natural language processing. Although a substantial number of indexes for highly repetitive strings have been proposed thus far, developing compressed indexes that support various queries remains a challenge. The run-length Burrows-Wheeler transform (RLBWT) is a lossless data compression by a reversible permutation of an input string and run-length encoding, and it has received int…
Online computation of maximal closed substrings
提出在线后缀树与link-cut树结合的数据结构LCST，实现在线计算字符串中所有最大封闭子串，总时间O(n log n)，空间O(n)。
The compression of highly repetitive strings (i.e., strings with many repetitions) has been a central research topic in string processing, and quite a few compression methods for these strings have been proposed thus far. Among them, an efficient compression format gathering increasing attention is the run-length Burrows--Wheeler transform (RLBWT), which is a run-length encoded BWT as a reversible permutation of an input string on the lexicographical order of suffixes. State-of-the-art construction algorithms of RLBWT have a serious issue with respect to (i) non-optimal computation time or (ii…
A Compact DAG for Storing and Searching Maximal Common Subsequences
本研究针对最大公共子序列（MCSs）紧凑表示的复杂度难题，提出多项式时间构造多项式规模DAG的方法，支持MCSs的高效枚举、计数与随机访问，填补了MCSs与LCSs间的复杂度差距，为新型序列分析奠定基础。
Regularities in strings are often related to periods and covers, which have extensively been studied, and algorithms for their efficient computation have broad application. In this paper we concentrate on computing cyclic regularities of strings, in particular, we propose several efficient algorithms for computing: (i) cyclic periodicity; (ii) all cyclic periodicity; (iii) maximal local cyclic periodicity; (iv) cyclic covers.
Cadences are structurally maximal arithmetic progressions of indices corresponding to equal characters in an underlying string. This paper provides a polynomial time detection algorithm for 3-cadences in grammar-compressed binary strings. This algorithm also translates to a linear time detection algorithm for 3-cadences in uncompressed binary strings. Furthermore, this paper proves that several variants of the cadence detection problem are NP-complete for grammar-compressed strings. As a consequence, the equidistant subsequence matching problem with patterns of length three is NP-complete for …
The Complexity of Maximal Common Subsequence Enumeration
本文研究了极大公共子序列枚举问题的复杂性，证明了在多字符串间枚举的困难性并排除了输出多项式时间算法的可能性，同时通过参数化复杂性分析提供了规避该难处理性的结果。
Enumerating characteristic substrings (e.g., maximal repeats, minimal unique substrings, and minimal absent words) in a given string has been an important research topic because there are a wide variety of applications in various areas such as string processing and computational biology. Although several enumeration algorithms for characteristic substrings have been proposed, they are not space-efficient in that their space-usage is proportional to the length of an input string. Recently, the run-length encoded Burrows-Wheeler transform (RLBWT) has attracted increased attention in string proce…
Finding the common subsequences of $L$ multiple strings has many applications in the area of bioinformatics, computational linguistics, and information retrieval. A well-known result states that finding a Longest Common Subsequence (LCS) for $L$ strings is NP-hard, e.g., the computational complexity is exponential in $L$. In this paper, we develop a randomized algorithm, referred to as {\em Random-MCS}, for finding a random instance of Maximal Common Subsequence ($MCS$) of multiple strings. A common subsequence is {\em maximal} if inserting any character into the subsequence no longer yields a…
Relating Left and Right Extensions of Maximal Repeats
本文研究字符串极大重复左右扩展数量之比，证明其最坏情况下为 Θ(√n)，并给出依赖字母表大小的渐近紧界。
Iterated hash functions process strings recursively, one character at a time. At each iteration, they compute a new hash value from the preceding hash value and the next character. We prove that iterated hashing can be pairwise independent, but never 3-wise independent. We show that it can be almost universal over strings much longer than the number of hash values; we bound the maximal string length given the collision probability.
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