Classical and Quantum Algorithms for Constructing Text from Dictionary Problem

TL;DR

This paper introduces efficient classical and quantum algorithms for reconstructing a text from a dictionary of small strings, with the quantum approach providing a significant speed-up over classical methods.

Contribution

The paper presents the first classical and quantum algorithms for the string reconstruction problem with proven optimality and speed-up results.

Findings

Classical algorithm runs in rac{n+L}{ ext{polylog}(n)} time.

Quantum algorithm achieves a rac{ ext{sqrt}(mL)}{ ext{polylog}(n)} speed-up.

Classical lower bound established as rac{n+L}{ ext{constant}}.

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.