非自适应分组检测中部分检测的紧尺度-局部性界
A Tight Scale-Locality Bound for Partial Detection in Non-Adaptive Group Testing
AI总结：
该研究针对缺陷项总数未知的随机非自适应分组检测部分检测问题，证明了其测试次数的紧下界与匹配上界，确定了该场景下的复杂度为Θ(ℓ log²(n/ℓ))。
AI中文摘要：
针对目标为找到任意ℓ个缺陷项但缺陷项总数d未知的随机非自适应分组检测问题，我们给出了一个下界。Bshouty与Haddad-Zaknoon此前证明了测试次数的上界为O(ℓ log² n)，下界为Ω(ℓ log² n / (log ℓ + log log n))；我们证明了与之匹配的下界。更一般地，我们表明，对于每个缺陷集都以恒定概率成功的每一种随机非自适应算法，必须使用Ω(ℓ log²(n/ℓ))次测试。证明如下：在固定的d值下，找到ℓ个缺陷项需要约ℓ log(n/d)比特的信息；另一方面，一个固定的分组测试仅在其大小适配d的尺度时才具有信息性，在d的所有对数尺度上，单个测试仅贡献O(1)比特，对所有尺度求和即可得到该下界。我们还记录了匹配的上界O(ℓ log²(n/ℓ))，该上界通过对d的二分式猜测并行运行已知d的算法得到。因此，对于恒定成功概率，未知d的部分检测的随机非自适应复杂度为Θ(ℓ log²(n/ℓ))。
英文摘要：
We give a lower bound for randomized non-adaptive group testing when the goal is to find any $\ell$ defective items but the total number $d$ of defectives is unknown. Bshouty and Haddad-Zaknoon proved an upper bound of $O(\ell\log^2 n)$ tests and a lower bound of $$Ω\!\left(\frac{\ell\log^2 n}{\log \ell+\log\log n}\right).$$ We prove the matching lower bound. More generally, we show that every randomized non-adaptive algorithm that succeeds with constant probability for every defective set must use $$Ω\!\left(\ell\log^2(n/\ell)\right)$$ tests. The proof is as follows. At a fixed value of $d$, finding $\ell$ defectives requires about $\ell\log(n/d)$ bits of information. On the other hand, one fixed group test is informative only when its size is tuned to the scale of $d$; across all logarithmic scales of $d$, a single test contributes only $O(1)$ bits. Summing over all scales gives the lower bound. We also record the matching upper bound $$O\!\left(\ell\log^2(n/\ell)\right),$$ obtained by running the known-$d$ algorithm in parallel over dyadic guesses for $d$. Thus the randomized non-adaptive complexity of unknown-$d$ partial detection is $Θ\!\left(\ell\log^2(n/\ell)\right)$ for constant success probability.
