The edit distance between two strings is defined as the smallest number of insertions , deletions , and substitutions that need to be made to transform one of the strings to another one. Approximating edit distance in subquadratic time is “one of the biggest unsolved problems in the field of combinatorial pattern matching” [37]. Our main result is a quantum constant approximation algorithm for computing the edit distance in truly subquadratic time. More precisely, we give an quantum algorithm that approximates the edit distance within a factor of 3. We further extend this result to an quantum algorithm that approximates the edit distance within a larger constant factor. Our solutions are based on a framework for approximating edit distance in parallel settings. This framework requires as black box an algorithm that computes the distances of several smaller strings all at once. For a quantum algorithm, we reduce the black box to metric estimation and provide efficient algorithms for approximating it. We further show that this framework enables us to approximate edit distance in distributed settings. To this end, we provide a MapReduce algorithm to approximate edit distance within a factor of , with sublinearly many machines and sublinear memory. Also, our algorithm runs in a logarithmic number of rounds.

中文翻译：

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在真正的次二次时间内逼近编辑距离：Quantum 和 MapReduce

这编辑距离两个字符串之间的最小数目定义为插入,删除， 和换人需要将其中一个字符串转换为另一个字符串。在次二次时间内近似编辑距离是“组合模式匹配领域中最大的未解决问题之一”[37]。我们的主要成果是一种量子常数逼近算法，用于在真正的次二次时间内计算编辑距离。更准确地说，我们给出一个 将编辑距离近似为 3 倍的量子算法。我们进一步将此结果扩展到 在较大的常数因子内逼近编辑距离的量子算法。我们的解决方案基于在并行设置中近似编辑距离的框架。该框架需要一个算法作为黑盒，一次计算几个较小字符串的距离。对于量子算法，我们将黑盒简化为度量估计并提供有效的算法来近似它。我们进一步表明，该框架使我们能够近似分布式设置中的编辑距离。为此，我们提供了一种 MapReduce 算法来将编辑距离近似为一个因子 ，具有亚线性多台机器和亚线性内存。此外，我们的算法以对数轮数运行。