| Summary: | 碩士 === 國立高雄大學 === 應用數學系碩博士班 === 107 === The model of classical group testing is: given n items, at most d of them are defective, we want to find out all of the defectives using the minimum number of tests that are applied on a subset of these items. In a nonadaptive algorithm, also called a pooling design, one must decide all tests before any testing occurs. Pooling designs play an important role in the field of group testing, there are many applications such as blood testing, quality controlling and DNA screening.
Error-correcting pooling design uses (d; z)-disjunct matrices. A binary matrix M is called (d; z)-disjunct if given any d + 1 columns of M with one designated, there are z rows intersecting the designated column and none of the other d columns. A (d; z)-disjunct matrix is called completely (d; z)-disjunct if it is not (d; z1)-disjunct whenever z1 > z. Let M(m,m,d) be the 01-matrix whose rows are indexed by all d-matchings on K2m and whose columns are indexed by all perfect matchings on K2m. M(m,m,d) has a 1 in row i and column j if and only if the i-th d-matching is contained in the j-th m-matching.
In this thesis, we study a pooling design M(m,m,d) and find its corresponding error-tolerance capabilities. Our results are divided into three parts: when m-d = 1 (resp. m-d = 2), M(m,m,d) is completely (d; z)-disjunct with z = m (resp. z =("m" ¦"2" )-d); when m≥d+3 and 1 ≤ d ≤ ⌊"m" /"2" ⌋, M(m,m,d) is completely (d; "2" ^"d" )-disjunct; when m≥d+3 and ⌊"m" /"2" ⌋< d < m, M(m,m,d) is completely (d; z)-disjunct where z = O("4" ^"(2d−m)/3" ∙"3" ^"m−d" ).
|
|---|
