Nonadaptive Noise-Resilient Group Testing with Order-Optimal Tests and Fast-and-Reliable Decoding
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| Format: | Preprint |
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2023
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| _version_ | 1866917630474977280 |
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| author | Guruswami, Venkatesan Wang, Hsin-Po |
| author_facet | Guruswami, Venkatesan Wang, Hsin-Po |
| contents | Group testing (GT) is the Boolean version of spare signal recovery and, due to its simplicity, a marketplace for ideas that can be brought to bear upon related problems, such as heavy hitters, compressed sensing, and multiple access channels. The definition of a "good" GT varies from one buyer to another, but it generally includes (i) usage of nonadaptive tests, (ii) limiting to $O(k \log n)$ tests, (iii) resiliency to test noise, (iv) $O(k \mathrm{poly}(\log n))$ decoding time, and (v) lack of mistakes. In this paper, we propose $Gacha~GT$. Gacha is an elementary and self-contained, versatile and unified scheme that, for the first time, satisfies all criteria for a fairly large region of parameters, namely when $\log k < \log(n)^{1-1/O(1)}$. Outside this parameter region, Gacha can be specialized to outperform the state-of-the-art partial-recovery GTs, exact-recovery GTs, and worst-case GTs.
The new idea Gacha brings to the market is a redesigned Reed--Solomon code for probabilistic list-decoding at diminishing code rates over reasonably-large alphabets. Normally, list-decoding a vanilla Reed--Solomon code is equivalent to the nontrivial task of identifying the subsets of points that fit low-degree polynomials. In this paper, we explicitly tell the decoder which points belong to the same polynomial, thus reducing the complexity and enabling the improvement on GT. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2311_08283 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Nonadaptive Noise-Resilient Group Testing with Order-Optimal Tests and Fast-and-Reliable Decoding Guruswami, Venkatesan Wang, Hsin-Po Information Theory Data Structures and Algorithms Group testing (GT) is the Boolean version of spare signal recovery and, due to its simplicity, a marketplace for ideas that can be brought to bear upon related problems, such as heavy hitters, compressed sensing, and multiple access channels. The definition of a "good" GT varies from one buyer to another, but it generally includes (i) usage of nonadaptive tests, (ii) limiting to $O(k \log n)$ tests, (iii) resiliency to test noise, (iv) $O(k \mathrm{poly}(\log n))$ decoding time, and (v) lack of mistakes. In this paper, we propose $Gacha~GT$. Gacha is an elementary and self-contained, versatile and unified scheme that, for the first time, satisfies all criteria for a fairly large region of parameters, namely when $\log k < \log(n)^{1-1/O(1)}$. Outside this parameter region, Gacha can be specialized to outperform the state-of-the-art partial-recovery GTs, exact-recovery GTs, and worst-case GTs. The new idea Gacha brings to the market is a redesigned Reed--Solomon code for probabilistic list-decoding at diminishing code rates over reasonably-large alphabets. Normally, list-decoding a vanilla Reed--Solomon code is equivalent to the nontrivial task of identifying the subsets of points that fit low-degree polynomials. In this paper, we explicitly tell the decoder which points belong to the same polynomial, thus reducing the complexity and enabling the improvement on GT. |
| title | Nonadaptive Noise-Resilient Group Testing with Order-Optimal Tests and Fast-and-Reliable Decoding |
| topic | Information Theory Data Structures and Algorithms |
| url | https://arxiv.org/abs/2311.08283 |