Near-Linear Time Insertion-Deletion Codes and (1+$\varepsilon$)-Approximating Edit Distance via Indexing

TL;DR

This paper presents a method to create indexing schemes that enable near-linear time approximation of edit distance, significantly improving decoding efficiency for insertion-deletion error-correcting codes.

Contribution

It introduces fast-decodable indexing schemes for edit distance that facilitate near-linear time approximate computations and enhance decoding algorithms for insertion-deletion codes.

Findings

Constructed indexing schemes with O(1) alphabet size for near-linear time edit distance approximation.

Achieved near-linear time decoding algorithms for insertion-deletion error-correcting codes.

Improved decoding speed for list-decodable insertion-deletion codes.

Abstract

We introduce fast-decodable indexing schemes for edit distance which can be used to speed up edit distance computations to near-linear time if one of the strings is indexed by an indexing string $I$. In particular, for every length $n$ and every $\varepsilon >0$, one can in near linear time construct a string $I \in \Sigma'^n$ with $|\Sigma'| = O_{\varepsilon}(1)$, such that, indexing any string $S \in \Sigma^n$, symbol-by-symbol, with $I$ results in a string $S' \in \Sigma''^n$ where $\Sigma'' = \Sigma \times \Sigma'$ for which edit distance computations are easy, i.e., one can compute a $(1+\varepsilon)$-approximation of the edit distance between $S'$ and any other string in $O(n \text{poly}(\log n))$ time. Our indexing schemes can be used to improve the decoding complexity of state-of-the-art error correcting codes for insertions and deletions. In particular, they lead to near-linear time decoding algorithms for the insertion-deletion codes of [Haeupler, Shahrasbi; STOC `17] and faster decoding algorithms for list-decodable insertion-deletion codes of [Haeupler, Shahrasbi, Sudan; ICALP `18]. Interestingly, the latter codes are a crucial ingredient in the construction of fast-decodable indexing schemes.