In this paper, we study edit distance (ED) and longest common subsequence (LCS) in the asymmetric streaming model, introduced by Saks and Seshadhri [SS13]. As an intermediate model between the random access model and the streaming model, this model allows one to have streaming access to one string and random access to the other string.
Our first main contribution is a systematic study of space lower bounds for ED and LCS in the asymmetric streaming model. Previously, there are no explicitly stated results in this context, although some lower bounds about LCS can be inferred from the lower bounds for longest increasing subsequence (LIS) in [SW07][GG10][EJ08]. Yet these bounds only work for large alphabet size. In this paper, we develop several new techniques to handle ED in general and LCS for small alphabet size, thus establishing strong lower bounds for both problems. In particular, our lower bound for ED provides an exponential separation between edit distance and Hamming distance in the asymmetric streaming model. Our lower bounds also extend to LIS and longest non-decreasing sequence (LNS) in the standard streaming model. Together with previous results, our bounds provide an almost complete picture for these two problems.
As our second main contribution, we give improved algorithms for ED and LCS in the asymmetric streaming model. For ED, we improve the space complexity of the constant factor approximation algorithms in [FHRS20][CJLZ20] from $\tilde{O}(\frac{n^δ}δ)$ to $O(\frac{d^δ}δ\;\mathsf{polylog}(n))$, where $n$ is the length of each string and $d$ is the edit distance between the two strings. For LCS, we give the first $1/2+ε$ approximation algorithm with space $n^δ$ for any constant $δ>0$, over a binary alphabet.
New algorithm for finding longest increasing subsequence is discussed. This algorithm is based on the ideas of idempotent mathematics and uses Max-Plus idempotent semiring. Problem of finding longest increasing sub- sequence is reformulated in a matrix form and solved with linear algebra.
Given a string $T$ with length $n$ whose characters are drawn from an ordered alphabet of size $σ$, its longest Lyndon subsequence is a longest subsequence of $T$ that is a Lyndon word. We propose algorithms for finding such a subsequence in $O(n^3)$ time with $O(n)$ space, or online in $O(n^3 σ)$ space and time. Our first result can be extended to find the longest common Lyndon subsequence of two strings of length $n$ in $O(n^4 σ)$ time using $O(n^3)$ space.
This paper reformulates the problem of finding a longest common increasing subsequence of the two given input sequences in a very succinct way. An extremely simple linear space algorithm based on the new formula can find a longest common increasing subsequence of sizes $n$ and $m$ respectively, in time $O(nm)$ using additional $\min\{n,m\}+1$ space.
We present an algorithm computing the longest periodic subsequence of a string of length $n$ in $O(n^7)$ time with $O(n^4)$ words of space. We obtain improvements when restricting the exponents or extending the search allowing the reported subsequence to be subperiodic down to $O(n^3)$ time and $O(n^2)$ words of space.
We present the first $\mathrm{o}(n)$-space polynomial-time algorithm for computing the length of a longest common subsequence. Given two strings of length $n$, the algorithm runs in $\mathrm{O}(n^{3})$ time with $\mathrm{O}\left(\frac{n \log^{1.5} n}{2^{\sqrt{\log n}}}\right)$ bits of space.
The Longest Common Increasing Subsequence (LCIS) is a variant of the classical Longest Common Subsequence (LCS), in which we additionally require the common subsequence to be strictly increasing. While the well-known "Four Russians" technique can be used to find LCS in subquadratic time, it does not seem applicable to LCIS. Recently, Duraj [STACS 2020] used a completely different method based on the combinatorial properties of LCIS to design an $\mathcal{O}(n^2(\log\log n)^2/\log^{1/6}n)$ time algorithm. We show that an approach based on exploiting tabulation can be used to construct an asympt…
The longest square subsequence (LSS) problem consists of computing a longest subsequence of a given string $S$ that is a square, i.e., a longest subsequence of form $XX$ appearing in $S$. It is known that an LSS of a string $S$ of length $n$ can be computed using $O(n^2)$ time [Kosowski 2004], or with (model-dependent) polylogarithmic speed-ups using $O(n^2 (\log \log n)^2 / \log^2 n)$ time [Tiskin 2013]. We present the first algorithm for LSS whose running time depends on other parameters, i.e., we show that an LSS of $S$ can be computed in $O(r \min\{n, M\}\log \frac{n}{r} + n + M \log n)$ t…
Given a sequence of integers, we want to find a longest increasing subsequence of the sequence. It is known that this problem can be solved in $O(n \log n)$ time and space. Our goal in this paper is to reduce the space consumption while keeping the time complexity small. For $\sqrt{n} \le s \le n$, we present algorithms that use $O(s \log n)$ bits and $O(\frac{1}{s} \cdot n^{2} \cdot \log n)$ time for computing the length of a longest increasing subsequence, and $O(\frac{1}{s} \cdot n^{2} \cdot \log^{2} n)$ time for finding an actual subsequence. We also show that the time complexity of our al…
The Longest Common Subsequence Problem (LCS) deals with finding the longest subsequence among a given set of strings. The LCS problem is an NP-hard problem which makes it a target for lots of effort to find a better solution with heuristics methods. The baseline for most famous heuristics functions is a tabular random, probabilistic approach. This approach approximates the length of the LCS in each iteration. The combination of beam search and tabular probabilistic-based heuristics has led to a large number of proposals and achievements in algorithms for solving the LCS problem. In this work, …
We study the problem of computing a longest increasing subsequence in a sequence $S$ of $n$ distinct elements in the presence of persistent comparison errors. In this model, every comparison between two elements can return the wrong result with some fixed (small) probability $ p $, and comparisons cannot be repeated. Computing the longest increasing subsequence exactly is impossible in this model, therefore, the objective is to identify a subsequence that (i) is indeed increasing and (ii) has a length that approximates the length of the longest increasing subsequence. We present asymptotically…
The Longest Common Increasing Subsequence problem (LCIS) is a natural variant of the celebrated Longest Common Subsequence (LCS) problem. For LCIS, as well as for LCS, there is an $O(n^2)$-time algorithm and a SETH-based conditional lower bound of $O(n^{2-\varepsilon})$. For LCS, there is also the Masek-Paterson $O(n^2 / \log{n})$-time algorithm, which does not seem to adapt to LCIS in any obvious way. Hence, a natural question arises: does any (slightly) sub-quadratic algorithm exist for the Longest Common Increasing Subsequence problem? We answer this question positively, presenting a $O(n^2…
At CPM 2017, Castelli et al. define and study a new variant of the Longest Common Subsequence Problem, termed the Longest Filled Common Subsequence Problem (LFCS). For the LFCS problem, the input consists of two strings $A$ and $B$ and a multiset of characters $\mathcal{M}$. The goal is to insert the characters from $\mathcal{M}$ into the string $B$, thus obtaining a new string $B^*$, such that the Longest Common Subsequence (LCS) between $A$ and $B^*$ is maximized. Casteli et al. show that the problem is NP-hard and provide a 3/5-approximation algorithm for the problem. In this paper we study…
We consider the problem of identifying tandem scattered subsequences within a string. Our algorithm identifies a longest subsequence which occurs twice without overlap in a string. This algorithm is based on the Hunt-Szymanski algorithm, therefore its performance improves if the string is not self similar. This occurs naturally on strings over large alphabets. Our algorithm relies on new results for data structures that support dynamic longest increasing sub-sequences. In the process we also obtain improved algorithms for the decremental string comparison problem.
The 2-LCPS problem, first introduced by Chowdhury et al. [Fundam. Inform., 129(4):329-340, 2014], asks one to compute (the length of) a longest palindromic common subsequence between two given strings $A$ and $B$. We show that the 2-LCPS problem is at least as hard as the well-studied longest common subsequence problem for four strings (the 4-LCS problem). Then, we present a new algorithm which solves the 2-LCPS problem in $O(σM^2 + n)$ time, where $n$ denotes the length of $A$ and $B$, $M$ denotes the number of matching positions between $A$ and $B$, and $σ$ denotes the number of distinct cha…
We consider the longest common subsequence (LCS) problem with the restriction that the common subsequence is required to consist of at least $k$ length substrings. First, we show an $O(mn)$ time algorithm for the problem which gives a better worst-case running time than existing algorithms, where $m$ and $n$ are lengths of the input strings. Furthermore, we mainly consider the LCS in at least $k$ length order-isomorphic substrings problem. We show that the problem can also be solved in $O(mn)$ worst-case time by an easy-to-implement algorithm.
Given a pair of strings, the problems of computing their Longest Common Subsequence and Edit Distance have been extensively studied for decades. For exact algorithms, LCS and Edit Distance (with character insertions and deletions) are equivalent; the state of the art running time is (almost) quadratic and this is tight under plausible fine-grained complexity assumptions. But for approximation algorithms the picture is different: there is a long line of works with improved approximation factors for Edit Distance, but for LCS (with binary strings) only a trivial $1/2$-approximation was known. In…
In this paper, we propose a data structure, a quadruple neighbor list (QN-list, for short), to support real time queries of all longest increasing subsequence (LIS) and LIS with constraints over sequential data streams. The QN-List built by our algorithm requires $O(w)$ space, where $w$ is the time window size. The running time for building the initial QN-List takes $O(w\log w)$ time. Applying the QN-List, insertion of the new item takes $O(\log w)$ time and deletion of the first item takes $O(w)$ time. To the best of our knowledge, this is the first work to support both LIS enumeration and LI…
In this paper we address the constrained longest common subsequence problem. Given two sequences $X$, $Y$ and a constrained sequence $P$, a sequence $Z$ is a constrained longest common subsequence for $X$ and $Y$ with respect to $P$ if $Z$ is the longest subsequence of $X$ and $Y$ such that $P$ is a subsequence of $Z$. Recently, Tsai \cite{Tsai} proposed an $O(n^2 \cdot m^2 \cdot r)$ time algorithm to solve this problem using dynamic programming technique, where $n$, $m$ and $r$ are the lengths of $X$, $Y$ and $P$, respectively. In this paper, we present a simple algorithm to solve the constra…
In this paper, we provide new approximation algorithms for dynamic variations of the longest increasing subsequence (\textsf{LIS}) problem, and the complementary distance to monotonicity (\textsf{DTM}) problem. In this setting, operations of the following form arrive sequentially: (i) add an element, (ii) remove an element, or (iii) substitute an element for another. At every point in time, the algorithm has an approximation to the longest increasing subsequence (or distance to monotonicity). We present a $(1+ε)$-approximation algorithm for \textsf{DTM} with polylogarithmic worst-case update t…
We study dynamic algorithms for the longest increasing subsequence (\textsf{LIS}) problem. A dynamic \textsf{LIS} algorithm maintains a sequence subject to operations of the following form arriving one by one: (i) insert an element, (ii) delete an element, or (iii) substitute an element for another. After performing each operation, the algorithm must report the length of the longest increasing subsequence of the current sequence. Our main contribution is the first exact dynamic \textsf{LIS} algorithm with sublinear update time. More precisely, we present a randomized algorithm that performs ea…
The Longest Common Subsequence (LCS) of two strings is a fundamental string similarity measure with a classical dynamic programming solution taking quadratic time. Despite significant efforts, little progress was made in improving the runtime. Even in the realm of approximation, not much was known for linear time algorithms beyond the trivial $\sqrt{n}$-approximation. Recent breakthrough result provided a $n^{0.497}$-factor approximation algorithm [HSSS19], which was more recently improved to a $n^{0.4}$-factor one [BCD21]. The latter paper also showed a $n^{2-2.5α}$ time algorithm which outpu…
Given a set of $k$ strings $I$, their longest common subsequence (LCS) is the string with the maximum length that is a subset of all the strings in $I$. A data-structure for this problem preprocesses $I$ into a data-structure such that the LCS of a set of query strings $Q$ with the strings of $I$ can be computed faster. Since the problem is NP-hard for arbitrary $k$, we allow an error that allows some characters to be replaced by other characters. We define the approximation version of the problem with an extra input $m$, which is the length of the regular expression (regex) that describes the…
In this paper, we revisit the much studied LCS problem for two given sequences. Based on the algorithm of Iliopoulos and Rahman for solving the LCS problem, we have suggested 3 new improved algorithms. We first reformulate the problem in a very succinct form. The problem LCS is abstracted to an abstract data type DS on an ordered positive integer set with a special operation Update(S,x). For the two input sequences X and Y of equal length n, the first improved algorithm uses a van Emde Boas tree for DS and its time and space complexities are O(R\log\log n+n) and O(R), where R is the number of …
A heapable sequence is a sequence of numbers that can be arranged in a "min-heap data structure". Finding a longest heapable subsequence of a given sequence was proposed by Byers, Heeringa, Mitzenmacher, and Zervas (ANALCO 2011) as a generalization of the well-studied longest increasing subsequence problem and its complexity still remains open. An equivalent formulation of the longest heapable subsequence problem is that of finding a maximum-sized binary tree in a given permutation directed acyclic graph (permutation DAG). In this work, we study parameterized algorithms for both longest heapab…
Longest Run Subsequence is a problem introduced recently in the context of the scaffolding phase of genome assembly (Schrinner et al., WABI 2020). The problem asks for a maximum length subsequence of a given string that contains at most one run for each symbol (a run is a maximum substring of consecutive identical symbols). The problem has been shown to be NP-hard and to be fixed-parameter tractable when the parameter is the size of the alphabet on which the input string is defined. In this paper we further investigate the complexity of the problem and we show that it is fixed-parameter tracta…
Polynomial-Time Solutions for Longest Common Subsequence Related Problems Between a Sequence and a Pangenome Graph
College of Big Data Statistics, Guizhou University of Finance and Economics（贵州财经大学大数据统计学院）；Algorithm and Computational Complexity Laboratory of School of Information, Guizhou University of Finance and Economics（贵州财经大学信息学院算法与计算复杂性实验室）
String similarity, longest common subsequence and shortest edit scripts are the triplets of problem that related to each other. There are different algorithms exist to generate edit script by solving longest common subsequence problem. This paper proposes an algorithm that uses string similarity problem to generate shortest edit script. For this we use the famous Levenshtein distance algorithm, which computes a numerical value that represents similarity between the strings from 0 to n, where n is the length of longest input string, and produce the shortest edit script which contains instructio…
We consider the general problem of the Longest Common Subsequence (LCS) on weighted sequences. Weighted sequences are an extension of classical strings, where in each position every letter of the alphabet may occur with some probability. Previous results presented a PTAS and noticed that no FPTAS is possible unless P=NP. In this paper we essentially close the gap between upper and lower bounds by improving both. First of all, we provide an EPTAS for bounded alphabets (which is the most natural case), and prove that there does not exist any EPTAS for unbounded alphabets unless FPT=W[1]. Further…
The similarity between a pair of time series, i.e., sequences of indexed values in time order, is often estimated by the dynamic time warping (DTW) distance, instead of any in the well-studied family of measures including the longest common subsequence (LCS) length and the edit distance. Although it may seem as if the DTW and the LCS(-like) measures are essentially different, we reveal that the DTW distance can be represented by the longest increasing subsequence (LIS) length of a sequence of integers, which is the LCS length between the integer sequence and itself sorted. For a given pair of …
Longest Increasing Subsequence (LIS) is a fundamental statistic of a sequence, and has been studied for decades. While the LIS of a sequence of length $n$ can be computed exactly in time $O(n\log n)$, the complexity of estimating the (length of the) LIS in sublinear time, especially when LIS $\ll n$, is still open. We show that for any integer $n$ and any $λ= o(1)$, there exists a (randomized) non-adaptive algorithm that, given a sequence of length $n$ with LIS $\ge λn$, approximates the LIS up to a factor of $1/λ^{o(1)}$ in $n^{o(1)} / λ$ time. Our algorithm improves upon prior work substanti…
We consider the classic problem of computing the Longest Common Subsequence (LCS) of two strings of length $n$. While a simple quadratic algorithm has been known for the problem for more than 40 years, no faster algorithm has been found despite an extensive effort. The lack of progress on the problem has recently been explained by Abboud, Backurs, and Vassilevska Williams [FOCS'15] and Bringmann and Künnemann [FOCS'15] who proved that there is no subquadratic algorithm unless the Strong Exponential Time Hypothesis fails. This has led the community to look for subquadratic approximation algorit…
A word $u$ is a subsequence of another word $w$ if $u$ can be obtained from $w$ by deleting some of its letters. The word $w$ with alph$(w)=Σ$ is called $k$-subsequence universal if the set of subsequences of length $k$ of $w$ contains all possible words of length $k$ over $Σ$. We propose a series of efficient algorithms computing the minimal number of edit operations (insertion, deletion, substitution) one needs to apply to a given word in order to reach the set of $k$-subsequence universal words.
In this paper, we consider a generalized longest common subsequence problem, in which a constraining sequence of length $s$ must be included as a substring and the other constraining sequence of length $t$ must be excluded as a subsequence of two main sequences and the length of the result must be maximal. For the two input sequences $X$ and $Y$ of lengths $n$ and $m$, and the given two constraining sequences of length $s$ and $t$, we present an $O(nmst)$ time dynamic programming algorithm for solving the new generalized longest common subsequence problem. The time complexity can be reduced fu…
