An Improved Algorithm for The k-Dyck Edit Distance Problem

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 + k4.782036) time with high probability or O(n + k4.853059) ) 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.