Overlaps help: Improved bounds for group testing with interval queries

Abstract

Given a finite ordered set of items and an unknown distinguished subset P of up to p positive elements, identify the items in P by asking the least number of queries of the type ‘‘does the subset Q intersect P?”, where Q is a subset of consecutive elements of $\{1,2,\dots ,n\}$. This problem arises, e.g., in computational biology, in a particular method for determining splice sites in genes. We consider time-efficient algorithms where queries are arranged in a fixed number s of stages: In each stage, queries are performed in parallel. In a recent bioinformatics paper, we proved optimality (subject to lower-order terms) with respect to the number of queries, of some strategies for the special cases $p=1$ or $s=2$. Exploiting new ideas, we are now able to provide improved lower bounds for any $p⩾2$ and $s⩾3$ and improved upper bounds for larger s. Most notably, our new bounds converge as s grows. Our new query scheme uses overlapping query intervals within a stage, which is effective for large enough s. This contrasts with our previous results for $s⩽2$ where optimal strategies were implemented by disjoint queries. The main open problem is whether overlaps help already in the case of small $s⩾3$. Anyway, the remaining gaps between the current upper and lower bounds for any fixed $s⩾3$ amount to small constant factors in the main term. The paper ends with a discussion of practical implications in the case that the positive elements are well separated.