Combinatorial Group Testing and Its Applications by Ding-Zhu Du, Frank Kwang Hwang

By Ding-Zhu Du, Frank Kwang Hwang

Staff checking out has been utilized in scientific, chemical and electric checking out, coding, drug screening, pollutants keep watch over, multiaccess channel administration, and extra lately in facts verification, clone library screening and AIDS checking out. The mathematical version could be both combinatorial or probabilistic. This paintings is a precis of all very important effects below the combinatorial version, and it demonstrates their functions in actual difficulties. another seek difficulties, together with the recognized counterfeit-coins challenge, also are studied intensive. This moment version is up-to-date and embraces the transforming into value of 2 issues: nonadaptive algorithms and blunder tolerance. new chapters, one on clone library screening and the opposite on mistakes tolerance, were further. additionally integrated is a brand new bankruptcy on counterfeit cash, and the chapters were reorganized into elements to supply focuses and views.

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Clearly, that resultant (d, n) algorithm is a nested algorithm. Therefore, it suffices to prove that every nested algorithm is convertible to a merging algorithm of R*-class. Let I = {Il, I2i • • • , In} be the set of items. Without loss of generality, assume that the first test is on the group G = {J , • • • , Ik} for some k < it If G is pure, relabel the unclassified items so that the current I has n - k items and repeat the procedure. If G is contaminated, then the next group to be tested must be a subset of G.

Iii) it is realizable. Proof. We show (i) (ii) = (iii) = (i). (i) (ii): Since S C So, clearly s fl so 2 S. So only s fl so g S needs to be shown. Let D E S fl So, then D E S and hence there is some D' E S such that D' C D CII S II. Suppose D ¢ S. Then D = Si , for some i . But D E S implies Si -a S and D' C D implies S -+ Si. 2. 3 Li's s-Stage Algorithm 23 (ii) (iii): If it is not realizable , then G, r contains a directed cycle . Since each Si is a single element , S; = Si and hence any directed cycle must contain S, say Sl _* S2 -+ ...

Proceeding like this , every group being tested consists of the first i items in the sequence of unclassified items for some i. Clearly each such test corresponds to a comparison of a1 vs. b;. The proof is complete. 5). 7 Number of Group Testing Algorithms One interesting problem is to count the number of group testing algorithms for S. This is the total number of binary trees with ISI leaves (labeled by members of S) which satisfy the group testing structure . While this problem remains open, Moon and Sobel [13] counted for a class of algorithms when the sample space S is the power set of n items.

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