A Dyck sequence is a sequence of opening and closing parentheses (of various types) that is balanced. The Dyck edit distance of a given sequence of parentheses S is the smallest number of edit operations (insertions, deletions, and substitutions) needed to transform S into a Dyck sequence. We consider the threshold Dyck edit distance problem, where the input is a sequence of parentheses S and a positive integer k, and the goal is to compute the Dyck edit distance of S only if the distance is at most k, and otherwise report that the distance is larger than k. Backurs and Onak [PODS’16] showed that the threshold Dyck edit distance problem can be solved in O(n + k¹⁶) time.

In this work, we design new algorithms for the threshold Dyck edit distance problem which costs O(n + k^4.544184) time with high probability or O(n + k^4.853059) deterministically. Our algorithms combine several new structural properties of the Dyck edit distance problem, a refined algorithm for fast (min , +) matrix product, and a careful modification of ideas used in Valiant’s parsing algorithm.

A戴克序列是平衡的左括号和右括号（各种类型）的序列。这戴克给定括号序列S的编辑距离是将S转换为戴克顺序。我们考虑阈值戴克编辑距离问题，其中输入是括号序列S和正整数k，目标是计算戴克仅当距离至多为k 时才编辑S的距离，否则报告距离大于k。Backurs 和 Onak [PODS'16] 表明阈值戴克编辑距离问题可以在O ( n + k ¹⁶ )时间内解决。

在这项工作中，我们为阈值设计了新的算法戴克编辑距离问题，高概率花费O ( n + k ^4.544184 ) 时间，或者确定性花费O ( n + k ^{4.853059 ) 时间。}我们的算法结合了几个新的结构特性戴克编辑距离问题，快速 (min , +) 矩阵乘积的改进算法，以及对 Valiant 解析算法中使用的思想的仔细修改。