An Improved Sketching Algorithm for Edit Distance

TL;DR

This paper improves the efficiency of a sketching protocol for estimating edit distance between strings, reducing message length from roughly k^8 to k^3, enabling more efficient communication.

Contribution

The authors refine the analysis of an existing sketching protocol for edit distance, achieving a significantly lower message length bound of  O(k^3) compared to previous  O(k^8).

Findings

Achieved  O(k^3) message length bound for edit distance sketching.

Built upon and improved the analysis of prior protocols.

Demonstrated a more efficient protocol for distributed edit distance computation.

Abstract

We provide improved upper bounds for the simultaneous sketching complexity of edit distance. Consider two parties, Alice with input $x\in\Sigma^n$ and Bob with input $y\in\Sigma^n$, that share public randomness and are given a promise that the edit distance $\mathsf{ed}(x,y)$ between their two strings is at most some given value $k$. Alice must send a message $sx$ and Bob must send $sy$ to a third party Charlie, who does not know the inputs but shares the same public randomness and also knows $k$. Charlie must output $\mathsf{ed}(x,y)$ precisely as well as a sequence of $\mathsf{ed}(x,y)$ edits required to transform $x$ into $y$. The goal is to minimize the lengths $|sx|, |sy|$ of the messages sent. The protocol of Belazzougui and Zhang (FOCS 2016), building upon the random walk method of Chakraborty, Goldenberg, and Kouck\'y (STOC 2016), achieves a maximum message length of $\tilde O(k^8)$ bits, where $\tilde O(\cdot)$ hides $\mathrm{poly}(\log n)$ factors. In this work we build upon Belazzougui and Zhang's protocol and provide an improved analysis demonstrating that a slight modification of their construction achieves a bound of $\tilde O(k^3)$.