SAFFRON: A Fast, Efficient, and Robust Framework for Group Testing based on Sparse-Graph Codes

TL;DR

SAFFRON is a fast, robust, and efficient non-adaptive group testing framework that reliably identifies most defective items with near-optimal tests and decoding complexity, even in noisy conditions.

Contribution

SAFFRON introduces a novel sparse-graph coding based framework for group testing that achieves near-optimal test counts and decoding efficiency, with robustness to noise.

Findings

Recovers at least (1-ε) fraction of defectives with ~6C(ε)K log n tests.

Decoding complexity is order-optimal at O(K log n).

Robust variant recovers all defectives with high probability using a different test scheme.

Abstract

Group testing tackles the problem of identifying a population of $K$ defective items from a set of $n$ items by pooling groups of items efficiently in order to cut down the number of tests needed. The result of a test for a group of items is positive if any of the items in the group is defective and negative otherwise. The goal is to judiciously group subsets of items such that defective items can be reliably recovered using the minimum number of tests, while also having a low-complexity decoding procedure. We describe SAFFRON (Sparse-grAph codes Framework For gROup testiNg), a non-adaptive group testing paradigm that recovers at least a $(1-\epsilon)$-fraction (for any arbitrarily small $\epsilon > 0$) of $K$ defective items with high probability with $m=6C(\epsilon)K\log_2{n}$ tests, where $C(\epsilon)$ is a precisely characterized constant that depends only on $\epsilon$. For instance, it can provably recover at least $(1-10^{-6})K$ defective items with $m \simeq 68 K \log_2{n}$ tests. The computational complexity of the decoding algorithm of SAFFRON is $\mathcal{O}(K\log n)$, which is order-optimal. Further, we robustify SAFFRON such that it can reliably recover the set of $K$ defective items even in the presence of erroneous or noisy test results. We also propose Singleton-Only-SAFFRON, a variant of SAFFRON, that recovers all the $K$ defective items with $m=2e(1+\alpha)K\log K \log_2 n$ tests with probability $1-\mathcal{O}{\left(\frac{1}{K^\alpha}\right)}$, where $\alpha>0$ is a constant. By leveraging powerful design and analysis tools from modern sparse-graph coding theory, SAFFRON is the first approach to reliable, large-scale probabilistic group testing that offers both precisely characterizable number of tests needed (down to the constants) together with order-optimal decoding complexity.