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 + k 16 ) 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.