从字典构造文本问题的经典和量子算法
Classical and Quantum Algorithms for Constructing Text from Dictionary Problem
摘要：
我们研究解决从字典（小字符串序列）构建文本（长字符串）问题的算法。这个问题在生物信息学中有应用，并且与从小片段重建长DNA序列的序列组装方法有关。问题是从可能有交集的字符串$s^1,\dots, s^m$构造一个长度为$n$的字符串$t$。我们提供了一个经典的算法与运行时间$O\left(n+L +m(\log n)^2\right)=\tilde{O}(n+L)$其中$L$是长度的总和$s^1,\dots,s^m$。我们提供了一个量子算法的运行时间$O\left(n +\log n\cdot(\log m+\log\log n)\cdot \sqrt{m\cdot L}\right)=\tilde{O}\left(n +\sqrt{m\cdot L}\right)$。此外，我们证明了经典算法的下界是$Ω(n+L)$。因此，我们的经典算法是最佳的对数因子，和我们的量子算法相比，在字典中的字符串的非恒定长度的情况下，任何经典算法的速度。
英文摘要：
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 $Ω(n+L)$. 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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