Does Preprocessing help in Fast Sequence Comparisons?

TL;DR

This paper investigates how preprocessing can significantly improve the efficiency of computing exact and approximate edit distances between strings, especially in scenarios involving many comparisons, leading to new faster algorithms.

Contribution

It introduces novel preprocessing-based algorithms for exact and approximate edit distance computation, outperforming previous methods and enabling faster comparisons in large string pools.

Findings

Exact permutation-LCS computation with $O(n \,\log n)$ preprocessing

Exact edit distance for small $k$ with $O(n \,\log n)$ preprocessing

Approximate edit distance within factor $(7+o(1))$ with subquadratic time

Abstract

We study edit distance computation with preprocessing: the preprocessing algorithm acts on each string separately, and then the query algorithm takes as input the two preprocessed strings. This model is inspired by scenarios where we would like to compute edit distance between many pairs in the same pool of strings. Our results include: Permutation-LCS: If the LCS between two permutations has length $n-k$, we can compute it \textit{ exactly} with $O(n \log(n))$ preprocessing and $O(k \log(n))$ query time. Small edit distance: For general strings, if their edit distance is at most $k$, we can compute it \textit{ exactly} with $O(n\log(n))$ preprocessing and $O(k^2 \log(n))$ query time. Approximate edit distance: For the most general input, we can approximate the edit distance to within factor $(7+o(1))$ with preprocessing time $\tilde{O}(n^2)$ and query time $\tilde{O}(n^{1.5+o(1)})$. All of these results significantly improve over the state of the art in edit distance computation without preprocessing. Interestingly, by combining ideas from our algorithms with preprocessing, we provide new improved results for approximating edit distance without preprocessing in subquadratic time.