Overlaps help: Improved bounds for group testing with interval queries Original Research Article

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 image. 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 image or image. Exploiting new ideas, we are now able to provide improved lower bounds for any image and image 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 image where optimal strategies were implemented by disjoint queries. The main open problem is whether overlaps help already in the case of small image. Anyway, the remaining gaps between the current upper and lower bounds for any fixed image 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.