Elastic-degenerate string comparison

IF 1 4区计算机科学 Q3 COMPUTER SCIENCE, THEORY & METHODS Information and Computation Pub Date : 2025-05-01 Epub Date: 2025-03-14 DOI:10.1016/j.ic.2025.105296

Estéban Gabory , Moses Njagi Mwaniki , Nadia Pisanti , Solon P. Pissis , Jakub Radoszewski , Michelle Sweering , Wiktor Zuba

引用次数: 0

Abstract

An elastic-degenerate (ED) string T is a sequence of n sets

T [1], \dots, T [n]

containing m strings in total whose cumulative length is N. We call n, m, and N the length, the cardinality and the size of T, respectively. The language of T is defined as

L (T) = {S_{1} \dots S_{n} : S_{i} \in T [i] for all i \in [1, n]}

. Given two ED strings, how fast can we check whether the two languages they represent have a nonempty intersection? We call this problem the ED String Intersection (EDSI) problem. For two ED strings

T_{1}

and

T_{2}

of lengths

n_{1}

and

n_{2}

, cardinalities

m_{1}

and

m_{2}

, and sizes

N_{1}

and

N_{2}

, respectively, we show the following:

•
There is no $O ({(N_{1} N_{2})}^{1 - ϵ})$ -time algorithm, for any $ϵ > 0$ , for EDSI even if $T_{1}$ and $T_{2}$ are over a binary alphabet, unless the Strong Exponential-Time Hypothesis is false.
•
There is no combinatorial $O ({(N_{1} + N_{2})}^{1.2 - ϵ} f (n_{1}, n_{2}))$ -time algorithm, for any $ϵ > 0$ and any function f, for EDSI even if $T_{1}$ and $T_{2}$ are over a binary alphabet, unless the Boolean Matrix Multiplication conjecture is false.
•
An $O (N_{1} \log N_{1} \log n_{1} + N_{2} \log N_{2} \log n_{2})$ -time algorithm for outputting a compact representation of the intersection language of two unary ED strings. When $T_{1}$ and $T_{2}$ are given in a compact representation, we show that the problem is NP-complete.
•
An $O (N_{1} m_{2} + N_{2} m_{1})$ -time algorithm for EDSI.
•
An $\tilde{O} (N_{1}^{ω - 1} n_{2} + N_{2}^{ω - 1} n_{1})$ -time algorithm for EDSI, where ω is the matrix multiplication exponent; the $\tilde{O}$ notation suppresses factors that are polylogarithmic in the input size.

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弹性简并字符串比较

弹性简并（ED）字符串T是由n个集合T[1]，…，T[n]组成的序列，其中总共包含m个字符串，其累积长度为n。我们将n， m， n分别称为T的长度，基数和大小。T的语言定义为L(T)={S1⋯Sn:Si∈T[i]，对于所有i∈[1，n]}。给定两个ED字符串，我们可以多快地检查它们所代表的两种语言是否有非空交集？我们称这个问题为ED弦相交问题。对于长度n1和n2，基数m1和m2，大小分别为n1和n2的两个ED字符串T1和T2，我们展示了以下内容：•对于任何ϵ>；0，对于EDSI，即使T1和T2在二进制字母表上，也没有O（(N1N2)1−λ）时间算法，除非强指数时间假设是假的。•对于任何ϵ>；0和任何函数f，对于EDSI，除非布尔矩阵乘法猜想为假，否则不存在组合O(（N1+N2)1.2−ϵf(N1, N2)）时间算法，即使T1和T2在二进制字母表上。•用于输出两个一元ED字符串的相交语言的紧凑表示的O(N1log (N1log)) n +N2log (N2log) n时间算法。当T1和T2以紧表示形式给出时，我们证明了问题是np完全的。•EDSI的O（N1m2+N2m1）时间算法。•EDSI的O ~ （N1ω−1n2+N2ω−1n1）时间算法，其中ω为矩阵乘法指数；O ~表示法抑制了输入大小中的多对数因子。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Information and Computation 工程技术-计算机：理论方法

CiteScore

2.30

自引率

0.00%

发文量

119

审稿时长

140 days

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