Group Testing with Blocks of Positives

Abstract

The main goal of group testing is to identify a small number of positive items among a large population of n items. In this work, we consider a new model of group testing in which the input items are linearly ordered, and the positives are subsets of small blocks (at unknown locations) of consecutive items over that order. When the number of blocks is at least one and at most k, and the number of items in a block is at most d, we show that there exists a deterministic and explicit design that can identify the positives with O(k <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> d log (n/d)) tests in O(poly(k, n/d)+kd) time. The number of tests in our proposed design is less than that of in standard combinatorial group testing by a factor of at least d/ log (kd). We also show that there exists a randomized design that can identify the positives with O(k(log (n/d)+d log k)) tests in O(k(log <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> (n/d)+k log k+d log k)) time with high probability.