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,…,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 p2 and s3 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 s2 where optimal strategies were implemented by disjoint queries. The main open problem is whether overlaps help already in the case of small s3. Anyway, the remaining gaps between the current upper and lower bounds for any fixed s3 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.

Keywords: Group testing; Interval query; Non-adaptive strategy; Computational molecular biology

Article Outline

1. Introduction

2. Preliminaries

3. Lower bounds for multistage interval group testing

4. An asymptotically optimal strategy for multistage interval group testing

5. Lower bound for three stages further improved

6. Discussion

7. A weaker adversary

References