Elastic-degenerate string comparison

Abstract

An elastic-degenerate (ED) string T is a sequence of n sets $T\left[1\right],\dots ,T\left[n\right]$ 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 $L\left(T\right)=\{S_1\cdots S_n:S_i\in T[i]\text{ for all }i\in [1,n\left]\right\}$. 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:

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  There is no $O\left({\left(N_1{N}_2\right)}^{1-ϵ}\right)$-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.
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  There is no combinatorial $O\left({(N_1+N_2)}^{1.2-ϵ}f\right(n_1,n_2\left)\right)$-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.
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  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.
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  An $O(N_1{m}_2+N_2{m}_1)$-time algorithm for EDSI.
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  An $\tilde{O}(N_1^{\omega -1}{n}_2+N_2^{\omega -1}{n}_1)$-time algorithm for EDSI, where ω is the matrix multiplication exponent; the $\tilde{O}$ notation suppresses factors that are polylogarithmic in the input size.