General Coverage Models: Structure, Monotonicity, and Shotgun Sequencing

TL;DR

This paper introduces a combinatorial approach to analyze coverage times in models inspired by shotgun DNA sequencing, providing exact formulas, asymptotic analysis, and bounds for various sampling schemes.

Contribution

It develops a unifying combinatorial framework for coverage problems, enabling exact calculations and asymptotic analysis for window-based models and uniform regular models.

Findings

Exact expressions for coverage times in window models.

Asymptotic behavior characterized for cyclic and non-cyclic models.

Bounds established for uniform -regular models and batch sampling.

Abstract

We study coverage processes in which each draw reveals a subset of $[n]$, and the goal is to determine the expected number of draws until all items are seen at least once. A classical example is the Coupon Collector's Problem, where each draw reveals exactly one item. Motivated by shotgun DNA sequencing, we introduce a model where each draw is a contiguous window of fixed length, in both cyclic and non-cyclic variants. We develop a unifying combinatorial tool that shifts the task of finding coverage time from probability, to a counting problem over families of subsets of $[n]$ that together contain all items, enabling exact calculation. Using this result, we obtain exact expressions for the window models. We then leverage past results on a continuous analogue of the cyclic window model to analyze the asymptotic behavior of both models. We further study what we call uniform $\ell$-regular models, where every draw has size $\ell$ and every item appears in the same number of admissible draws. We compare these to the batch sampling model, in which all $\ell$-subsets are drawn uniformly at random and present upper and lower bounds, which were also obtained independently by Berend and Sher. We conjecture, and prove for special cases, that this model maximizes the coverage time among all uniform $\ell$-regular models. Finally, we prove a universal upper bound on the entire class of uniform $\ell$-regular models, which illuminates the fact that many sampling models share the same leading asymptotic order, while potentially differing significantly in lower-order terms.