Streaming dictionary matching with mismatches

TL;DR

This paper extends efficient streaming algorithms from the $k$-mismatch problem to the more complex dictionary matching with $k$ mismatches, providing new algorithms with specific space and time bounds and establishing lower bounds.

Contribution

It introduces a novel streaming algorithm for dictionary matching with $k$ mismatches and proves a lower bound on space complexity for this problem.

Findings

Developed a randomized streaming algorithm with $O(k d ext{polylog}(n))$ space.

Achieved $O(k ext{polylog}(n) + | ext{occ}|)$ time per position.

Proved a lower bound of $ ext{Omega}(k d)$ bits of space for any streaming algorithm.

Abstract

In the $k$-mismatch problem we are given a pattern of length $n$ and a text and must find all locations where the Hamming distance between the pattern and the text is at most $k$. A series of recent breakthroughs have resulted in an ultra-efficient streaming algorithm for this problem that requires only $O(k \log \frac{n}{k})$ space and $O(\log \frac{n}{k} (\sqrt{k \log k} + \log^3 n))$ time per letter [Clifford, Kociumaka, Porat, SODA 2019]. In this work, we consider a strictly harder problem called dictionary matching with $k$ mismatches. In this problem, we are given a dictionary of $d$ patterns, where the length of each pattern is at most $n$, and must find all substrings of the text that are within Hamming distance $k$ from one of the patterns. We develop a streaming algorithm for this problem with $O(k d \log^k d \mathrm{polylog}(n))$ space and $O(k \log^{k} d \mathrm{polylog}(n) + |\mathrm{occ}|)$ time per position of the text. The algorithm is randomised and outputs correct answers with high probability. On the lower bound side, we show that any streaming algorithm for dictionary matching with $k$ mismatches requires $\Omega(k d)$ bits of space.