Quantum algorithms for SCS and text assembling problems

Abstract

In this paper, we consider two versions of the Text Assembling problem. We are given a sequence of strings $s^1,\dots,s^n$ of total length $L$ that is a dictionary, and a string $t$ of length $m$ that is a text. The first version of the problem is assembling $t$ from the dictionary. The second version is the ``Shortest Superstring Problem''(SSP) or the ``Shortest Common Superstring Problem''(SCS). In this case, $t$ is not given, and we should construct the shortest string (we call it superstring) that contains each string from the given sequence as a substring.These problems are connected with the sequence assembly method for reconstructing a long DNA sequence from small fragments. For both problems, we suggest new quantum algorithms that work better than their classical counterparts. In the first case, we present a quantum algorithm with $O(m+\log m\sqrt{nL})$ query complexity. In the case of SSP, we present a quantum algorithm with $\tilde{O}(n^3 1.728^n +L +n^{1.5}\sqrt{L})$ query complexity. Here $\tilde{O}$ hides not only constants but logarithms of $L$ and $n$ also.