On Minimizers of Minimum Density

TL;DR

This paper addresses the problem of finding minimizers with the lowest possible density in biological sampling schemes, providing algorithms for exact solutions, analyzing their properties via automata theory, and computing optimal minimizers for various parameters.

Contribution

It introduces an efficient algorithm for finding minimum density minimizers, analyzes their properties using regular languages, and computes optimal solutions for specific parameter sets.

Findings

Algorithms for exact minimum density minimizers are developed.

Automata-based analysis provides insights into asymptotic minimizer properties.

Computed minimizers for various parameters outperform average density and theoretical bounds.

Abstract

Minimizers are sampling schemes with numerous applications in computational biology. Assuming a fixed alphabet of size $\sigma$, a minimizer is defined by two integers $k,w\ge2$ and a linear order $\rho$ on strings of length $k$ (also called $k$-mers). A string is processed by a sliding window algorithm that chooses, in each window of length $w+k-1$, its minimal $k$-mer with respect to $\rho$. A key characteristic of the minimizer is its density, which is the expected frequency of chosen $k$-mers among all $k$-mers in a random infinite $\sigma$-ary string. Minimizers of smaller density are preferred as they produce smaller samples with the same guarantee: each window is represented by a $k$-mer. The problem of finding a minimizer of minimum density for given input parameters $(\sigma,k,w)$ has a huge search space of $(\sigma^k)!$ and is representable by an ILP of size $\tilde\Theta(\sigma^{k+w})$, which has worst-case solution time that is doubly-exponential in $(k+w)$ under standard complexity assumptions. We solve this problem in $w\cdot 2^{\sigma^k+O(k)}$ time and provide several additional tricks reducing the practical runtime and search space. As a by-product, we describe an algorithm computing the average density of a minimizer within the same time bound. Then we propose a novel method of studying minimizers via regular languages and show how to find, via the eigenvalue/eigenvector analysis over finite automata, minimizers with the minimal density in the asymptotic case $w\to\infty$. Implementing our algorithms, we compute the minimum density minimizers for $(\sigma,k)\in\{(2,2),(2,3),(2,4),(2,5),(4,2)\}$ and \textbf{all} $w\ge 2$. The obtained densities are compared against the average density and the theoretical lower bounds, including the new bound presented in this paper.