An Improved Algorithm for The $k$-Dyck Edit Distance Problem

TL;DR

This paper introduces faster algorithms for the threshold Dyck edit distance problem, reducing the complexity from previous solutions by leveraging new structural insights and advanced matrix multiplication techniques.

Contribution

The authors develop new algorithms that compute the threshold Dyck edit distance more efficiently, improving the time complexity from $O(n+k^{16})$ to approximately $O(n+k^{4.54})$, using novel structural properties and matrix multiplication methods.

Findings

Achieved $O(n+k^{4.544184})$ time algorithm with high probability.

Developed a deterministic $O(n+k^{4.853059})$ time algorithm.

Enhanced understanding of Dyck edit distance structure.

Abstract

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.