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Sparse Combinatorial Group Testing

机译：稀疏组合组测试

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In combinatorial group testing, the primary objective is to fully identify the set of at most d defective items from a pool of n items using as few tests as possible. The celebrated result for the combinatorial group testing problem is that the number of tests, denoted by t , can be made logarithmic in n when {d} = {O}(ext {poly}(log {n})) . However, state-of-the-art group testing codes require the items to be tested {w} = Omega left ({rac {{d} log {n}}{log {d} + log log {n}} }ight) times and tests to include ho = Omega left ({rac {{n}}{{d} log _{{d}} {n}}}ight) items. In many emerging applications, items can only participate in a limited number of tests and tests are constrained to include a limited number of items. In this paper, we study the "sparse" regime for the group testing problem where we restrict the number of tests each item can participate in by {w}_{max } or the number of items each test can include by ho _{max } in both noiseless and noisy settings. These constraints lead to a largely unexplored regime where t is a fractional power of n, rather than logarithmic in n as in the classical setting. Our results characterize the number of tests t needed in this regime as a function of {w}_{max } or ho _{max } and show, for example, that t decreases drastically when {w}_{max } is increased beyond a bare minimum. In particular, in the noiseless case it can be shown that if {w}_{max } leq {d} , then we must have {t}={n} , i.e., testing every item individually is optimal. We show that if {w}_{max }={d}+1 , the number of tests decreases suddenly from {t}={n} to {t} = Theta ({d} sqrt {{n}}) . The order-optimal construction is obtained via a modification of the classical Kautz-Singleton construction, which is known to be suboptimal for the classical group testing problem. For the more general case, when {w}_{max }={{ ld}}+1 for integer {l}>1 , the modified Kautz-Singleton construction requires {t} = Theta left ({{d} {n}<^>{rac {1}{{l}+1}}}ight) tests, which we prove to be near order-optimal. We also show that our constructions have a favorable encoding and decoding complexity, i.e. they can be decoded in ({poly}({d}) + {O}({t})) -time and each entry in any codeword can be computed in {poly}(log {n}) memory space. We finally discuss an application of our results to the construction of energy-limited random access schemes for Internet of Things networks, which provided the initial motivation for our work.
机译：在组合组测试中，主要目标是通过尽可能少的测试完全识别来自N个项目的池中的最多D缺陷物品。组合组测试问题的庆祝结果是，由T表示的测试数量，当{d} = {o}（ text {poly}（ log {n}））时，可以在n中进行对数。但是，最先进的组测试代码要求测试的项目{w} = oomega left（{ frac {{d} log {n}} { log {d} + log 对{n}}} 右）次数和测试包括 rho = ω left（{ frac {{n}} {{d} log _ {{d}} {n}} 右）项目。在许多新兴应用程序中，物品只能参与有限数量的测试，并且测试被限制为包括有限数量的项目。在本文中，我们研究了组测试问题的“稀疏”制度，其中我们限制了每个项目的测试数量可以参与{w} _ { max}或每个测试可以包括 rho _的项目数{ max}在无噪声和嘈杂的设置中。这些约束导致主要是未探测的制度，其中T是n的分数力，而不是在经典设置中的n中的对数。我们的结果表征了该制度中所需的测试T的数量，作为{w} _ { max}或 rho _ { max}的函数，并且例如，当{w} _ { max时，t急剧下降}超出了最小的最小值。特别地，在无噪声的情况下，可以示出，如果{w} _ { max} leq {d}，则必须具有{t} = {n}，即，单独测试每个项目是最佳的。我们表明，如果{w} _ { max} = {d} +1，则测试的数量突然从{t} = {n}突然减小到{t} = theta（{d} sqrt {{n} }）。通过修改经典的Kautz-Singleton结构来获得订单最优结构，这已知是经典群体测试问题的次优。对于更常规的情况，当整数{l}> 1的{w} _ { max} = {{ld}} + 1时，修改后的kautz-singleton构造需要{t} = theta left（{{d {n} <^> { frac {1} {{l} +1}} 右）测试，我们证明是靠近订单的最佳状态。我们还表明，我们的结构具有良好的编码和解码复杂性，即它们可以在（{poly}（{d}）+ {o}（{t}））中解码 - 可以计算任何码字中的每个条目在{poly}（ log {n}）内存空间。我们终于讨论了我们的结果，以建造了用于网络网络网络的能量有限的随机接入方案，为我们的工作提供了初始动机。

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