Minimizers in Semi-Dynamic Strings

TL;DR

This paper introduces a new data structure for efficiently maintaining minimizers in semi-dynamic strings, enabling faster updates and reduced space usage while preserving linear time complexity for computing minimizers.

Contribution

It presents the first semi-dynamic string data structure supporting constant-time minimizer queries and updates, with a novel sublinear space algorithm for minimizer set computation.

Findings

Supports minimizer queries in O(1) amortized time with semi-dynamic updates.

Achieves O(n) time and O(√w) space for computing minimizers of a string.

Demonstrates practical effectiveness through application and experimental evaluation.

Abstract

Minimizers sampling is one of the most widely-used mechanisms for sampling strings. Let $S=S[0]\ldots S[n-1]$ be a string over an alphabet $\Sigma$. In addition, let $w\geq 2$ and $k\geq 1$ be two integers and $\rho=(\Sigma^k,\leq)$ be a total order on $\Sigma^k$. The minimizer of window $X=S[i\mathinner{.\,.} i+w+k-2]$ is the smallest position in $[i,i+w-1]$ where the smallest length-$k$ substring of $S[i\mathinner{.\,.} i+w+k-2]$ based on $\rho$ starts. The set of minimizers for all $i\in[0,n-w-k+1]$ is the set $\mathcal{M}_{w,k,\rho}(S)$ of the minimizers of $S$. The set $\mathcal{M}_{w,k,\rho}(S)$ can be computed in $\mathcal{O}(n)$ time. The folklore algorithm for this computation computes the minimizer of every window in $\mathcal{O}(1)$ amortized time using $\mathcal{O}(w)$ working space. It is thus natural to pose the following two questions: Question 1: Can we efficiently support other dynamic updates on the window? Question 2: Can we improve on the $\mathcal{O}(w)$ working space? We answer both questions in the affirmative: 1. We term a string $X$ semi-dynamic when one is allowed to insert or delete a letter at any of its ends. We show a data structure that maintains a semi-dynamic string $X$ and supports minimizer queries in $X$ in $\mathcal{O}(1)$ time with amortized $\mathcal{O}(1)$ time per update operation. 2. We show that this data structure can be modified to occupy strongly sublinear space without increasing the asymptotic complexity of its operations. To the best of our knowledge, this yields the first algorithm for computing $\mathcal{M}_{w,k,\rho}(S)$ in $\mathcal{O}(n)$ time using $\mathcal{O}(\sqrt{w})$ working space. We complement our theoretical results with a concrete application and an experimental evaluation.