Group Testing using left-and-right-regular sparse-graph codes

We consider the problem of non-adaptive group testing of N items out of which K or less items are known to be defective. We propose a testing scheme based on left-and-right-regular sparse-graph codes and a simple iterative decoder. We show that for any arbitrarily small ϵ>0 our scheme requires only m=c_ϵ Klogc_1N/K tests to recover (1-ϵ) fraction of the defective items with high probability (w.h.p) i.e., with probability approaching 1 asymptotically in N and K, where the value of constants c_ϵ and ℓ are a function of the desired error floor ϵ and constant c_1=ℓ/c_ϵ (observed to be approximately equal to 1 for various values of ϵ). More importantly the iterative decoding algorithm has a sub-linear computational complexity of 𝒪(KlogN/K) which is known to be optimal. Also for m=c_2 Klog KlogN/K tests our scheme recovers the whole set of defective items w.h.p. These results are valid for both noiseless and noisy versions of the problem as long as the number of defective items scale sub-linearly with the total number of items, i.e., K=o(N). The simulation results validate the theoretical results by showing a substantial improvement in the number of tests required when compared to the testing scheme based on left-regular sparse-graphs.