Cascaded Group Testing

TL;DR

This paper introduces cascaded group testing, a new variant where tests return the first defective item in a specified order, and develops optimal strategies for both adaptive and non-adaptive regimes, significantly reducing the number of tests needed.

Contribution

It defines cascaded group testing, establishes necessary and sufficient conditions for non-adaptive strategies, and provides explicit designs that greatly improve test efficiency for small K.

Findings

Adaptive testing can identify defectives in at most K tests, which is optimal.

Non-adaptive strategies require at least Ω(K^2) tests, but can be achieved with O(K^2 log(N/K)) tests.

For constant K ≥ 3, the proposed iterative design uses only poly(log log N) tests, much fewer than standard methods.

Abstract

In this paper, we introduce a variation of the group testing problem where each test is specified by an ordered subset of items and returns the first defective item in the specified order or returns null if there are no defectives. We refer to this as cascaded group testing and the goal is to identify a small set of $K$ defective items amongst a collection of size $N$, using as few tests as possible for perfect recovery. For the adaptive testing regime, we show that a simple scheme can find all defective items in at most $K$ tests, which is optimal. For the non-adaptive setting, we first come up with a necessary and sufficient condition for any collection of tests to be feasible for recovering all the defectives. Using this, we show that any feasible non-adaptive strategy requires at least $\Omega(K^2)$ tests. In terms of achievability, it is easy to show the existence of a feasible collection of $O(K^2 \log (N/K))$ tests. We show via carefully constructed explicit designs that one can do significantly better for constant $K$. While the cases $K = 1, 2$ are straightforward, the case $K=3$ is already non-trivial and we come up with an iterative design that is asymptotically optimal and requires $\Theta(\log \log N)$ tests. Note that this is in contrast to standard binary group testing, where at least $\Omega(\log N)$ tests are required. For constant $K \ge 3$, our iterative design requires only poly$(\log \log N)$ tests.