Optimal trade-offs for pattern matching with $k$ mismatches

TL;DR

This paper introduces a new algorithm for k-mismatch pattern matching that offers a smooth trade-off between existing bounds and proves a matching lower bound under certain conjectures.

Contribution

It presents a novel algorithm that balances the time complexity between previous bounds and establishes a conditional lower bound for combinatorial algorithms.

Findings

New algorithm achieves a smooth time trade-off.

Matching conditional lower bound under the matrix multiplication conjecture.

Improves understanding of the computational limits for k-mismatch pattern matching.

Abstract

Given a pattern of length $m$ and a text of length $n$, the goal in $k$-mismatch pattern matching is to compute, for every $m$-substring of the text, the exact Hamming distance to the pattern or report that it exceeds $k$. This can be solved in either $\widetilde{O}(n \sqrt{k})$ time as shown by Amir et al. [J. Algorithms 2004] or $\widetilde{O}((m + k^2) \cdot n/m)$ time due to a result of Clifford et al. [SODA 2016]. We provide a smooth time trade-off between these two bounds by designing an algorithm working in time $\widetilde{O}( (m + k \sqrt{m}) \cdot n/m)$. We complement this with a matching conditional lower bound, showing that a significantly faster combinatorial algorithm is not possible, unless the combinatorial matrix multiplication conjecture fails.