We present a near-linear time algorithm that approximates the edit distance between two strings within a polylogarithmic factor; specifically, for strings of length n and every fixed epsilon>0, it can compute a (log n)^O(1/epsilon) approximation in n^(1+epsilon) time. This is an exponential improvement over the previously known factor, 2^(O (sqrt(log n))), with a comparable running time (Ostrovsky and Rabani J.ACM 2007; Andoni and Onak STOC 2009). Previously, no efficient polylogarithmic approximation algorithm was known for any computational task involving edit distance (e.g., nearest neighbor search or sketching).
This result arises naturally in the study of a new asymmetric query model. In this model, the input consists of two strings x and y, and an algorithm can access y in an unrestricted manner, while being charged for querying every symbol of x. Indeed, we obtain our main result by designing an algorithm that makes a small number of queries in this model. We then provide a nearly-matching lower bound on the number of queries.
Our lower bound is the first to expose hardness of edit distance stemming from the input strings being "repetitive", which means that many of their substrings are approximately identical. Consequently, our lower bound provides the first rigorous separation between edit distance and Ulam distance, which is edit distance on non-repetitive strings, such as permutations.
Distributed Exact Weighted All-Pairs Shortest Paths in $\tilde O(n^{5/4})$ Rounds
我们研究计算{ \em所有对最短路径}（APSP）的分布式网络（CONGEST模型）。目标是让（加权）网络中的每个节点都知道使用通信的每个其他节点的距离。该问题允许$(1+o(1))$ -近似$\tilde O(n)$ -时间算法~ \cite{LenzenP-podc15,Nanongkai-STOC14}，它与$\tilde Ω(n)$ -时间下界~ \cite{Nanongkai-STOC14,LenzenP_stoc13,FrischknechtHW12} \footnote{$\tilde Θ$, $\tilde O$ and $\tilde Ω$ hide polylogarithmic factors. Note that the lower bounds also hold even in the unweighted case and in the weighted case with polynomial approximation ratios.}相匹配.没有$ω(n)$下限或$o(m)$上限已知的精确计算。在本文中，我们提出了一个$\tilde O(n^{5/4})$ -时间随机（拉斯维加斯）算法精确加权APSP，这提供了第一个改进的天真$O(m)$ -时间算法时，网络不是那么稀疏。我们的结果也适用于边权重是{PDMATHAMM不对称}（a.k.a.通信是双…
