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Moreover, f ( p ) = 1 {\displaystyle f({\textbf {p}})=1} if and only if there is some index, n {\displaystyle n} , where a n = p n = 1 {\displaystyle {\textbf {a}}_{n}={\textbf {p}}_{n}=1} . Thus this problem is analogous to a group-testing problem with d {\displaystyle d} defectives and n {\displaystyle n} total items. The entries of a {\displaystyle {\textbf {a}}} are the items, which are defective if they are 1 {\displaystyle 1} , p {\displaystyle {\textbf {p}}} specifies a test, and a test is positive if and only if f ( p ) = 1 {\displaystyle f({\textbf {p}})=1} .
In reality, one will often be interested in functions that are more complicated, such as f : C N → C {\displaystyle f:\mathbb {C} ^{N}\to \mathbb {C} } , again where f ( x ) = a ⋅ x {\displaystyle f({\textbf {x}})={\textbf {a}}\cdot {\textbf {x}}} .
This section formally defines the notions and terms relating to group testing.
The input vector, x = ( x 1 , x 2 , ... , x n ) {\displaystyle \mathbf {x} =(x_{1},x_{2},\dots ,x_{n})} , is defined to be a binary vector of length n {\displaystyle n} (that is, x ∈ { 0 , 1 } n {\displaystyle \mathbf {x} \in \{0,1\}^{n}} ), with the j-th item being called defective if and only if x j = 1 {\displaystyle x_{j}=1} . Further, any non-defective item is referred called a 'good' item.
x {\displaystyle \mathbf {x} } is intended to describe the (unknown) set of defective items. The key property of x {\displaystyle \mathbf {x} } is that it is an implicit input.
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1 2 3 4 5 Goodrich, Michael T.; Atallah, Mikhail J.; Tamassia, Roberto (7 June 2005). Indexing information for data forensics. Lecture Notes in Computer Science. 3531. 206–221. doi:10.1007/11496137_15. ISBN 978-3-540-26223-7
↑ Chlebus, B. S. (2001). "Randomized communication in radio networks". In: Pardalos, P. M.; Rajasekaran, S.; Reif, J.; Rolim, J. D. P. (Eds.), Handbook of Randomized Computing, Vol. I, p.401–456. Kluwer Academic Publishers, Dordrecht.
↑ Takhar, D.; Laska, J. N.; Wakin, M. B.; Duarte, M. F.; Baron, D.; Sarvotham, S.; Kelly, K. F.; Baraniuk, R. G.
For n / d {\displaystyle n/d} large, it can be shown that T → d log 2  ( n / d ) {\displaystyle T\rightarrow d\log _{2}(n/d)} , which compares favorably to the t = e log 2  e d log 2  ( n d ) {\displaystyle t={\frac {e}{\log _{2}e}}d\log _{2}\left({\frac {n}{d}}\right)} tests required for Li's s {\displaystyle s} -stage algorithm. In fact, the generalised binary-splitting algorithm is close to optimal in the following sense. When d ≥ 2 {\displaystyle d\geq 2} it can be shown that T − B I ( d , n ) ≤ ( d − 1 ) {\displaystyle T-B_{I}(d,n)\leq (d-1)} , where B I ( d , n ) = ⌈ log 2  ∑ i = 0 d ( n i ) ⌉ {\displaystyle B_{I}(d,n)=\left\lceil \log _{2}\sum _{i=0}^{d}{n \choose i}\right\rceil } is the information lower bound.