Non-adaptive algorithms for threshold group testing with consecutive positives

Given up to $d$ positive items in a large population of $n$ items ($d \ll n$), the goal of threshold group testing is to efficiently identify the positives via tests, where a test on a subset of items is positive if the subset contains at least $u$ positive items, negative if it contains up to $\ell $ positive items and arbitrary (either positive or negative) otherwise. The parameter $g = u - \ell - 1$ is called the gap. In non-adaptive strategies, all tests are fixed in advance and can be represented as a measurement matrix, in which each row and column represent a test and an item, respectively. In this paper, we consider non-adaptive threshold group testing with consecutive positives in which the items are linearly ordered and the positives are consecutive in that order. We show that by designing deterministic and strongly explicit measurement matrices, $\lceil \log _{2}{\lceil \frac {n}{d} \rceil } \rceil + 2d + 3$ (respectively, $\lceil \log _{2}{\lceil \frac {n}{d} \rceil } \rceil + 3d$) tests suffice to identify the positives in $O \left ( \log _{2}{\frac {n}{d}} + d \right )$ time when $g = 0$ (respectively, $g> 0$). The results significantly improve the state-of-the-art scheme that needs $15 \lceil \log _{2}{\lceil \frac {n}{d} \rceil } \rceil + 4d + 71$ tests to identify the positives in $O \left ( \frac {n}{d} \log _{2}{\frac {n}{d}} + ud^{2} \right )$ time, and whose associated measurement matrices are random and (non-strongly) explicit.