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arXiv · 2005.14335

Classical and Quantum Algorithms for Constructing Text from Dictionary Problem

Kamil Khadiev·Vladislav Remidovskii
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Abstract

We study algorithms for solving the problem of constructing a text (long string) from a dictionary (sequence of small strings). The problem has an application in bioinformatics and has a connection with the Sequence assembly method for reconstructing a long DNA sequence from small fragments. The problem is constructing a string $t$ of length $n$ from strings $s^1,\dots, s^m$ with possible intersections. We provide a classical algorithm with running time $O\left(n+L +m(\log n)^2\right)=\tilde{O}(n+L)$ where $L$ is the sum of lengths of $s^1,\dots,s^m$. We provide a quantum algorithm with running time $O\left(n +\log n\cdot(\log m+\log\log n)\cdot \sqrt{m\cdot L}\right)=\tilde{O}\left(n +\sqrt{m\cdot L}\right)$. Additionally, we show that the lower bound for the classical algorithm is $\Omega(n+L)$. Thus, our classical algorithm is optimal up to a log factor, and our quantum algorithm shows speed-up comparing to any classical algorithm in a case of non-constant length of strings in the dictionary.

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Kamil Khadiev, Vladislav Remidovskii. 2020-05-28. Classical and Quantum Algorithms for Constructing Text from Dictionary Problem. https://arxiv.org/abs/2005.14335

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