An Improved Algorithm for The k-Dyck Edit Distance Problem

• Published in: ACM transactions on algorithms Vol. 20; no. 3; pp. 1 - 25
• Main Authors: Fried, Dvir, Golan, Shay, Kociumaka, Tomasz, Kopelowitz, Tsvi, Porat, Ely, Starikovskaya, Tatiana
• Format: Journal Article
• Language: English
• Published: New York, NY ACM 21.06.2024
• Subjects: Design and analysis of algorithms; Theory of computation; edit distance; fine-grained complexity; Dyck language
• ISSN: 1549-6325; 1549-6333; 1549-6333
• DOI: 10.1145/3627539

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+k4.544184) 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.