In mathematics, a logical matrix may be described as d-disjunct and/or d-separable.
These concepts play a pivotal role in the mathematical area of non-adaptive group testing.
In the mathematical literature, d-disjunct matrices may also be called super-imposed codes[1] or d-cover-free families.
[2] According to Chen and Hwang (2006),[3] The following relationships are "well-known":[3] The following
matrix is 2-separable, because each pair of columns has a distinct sum.
For example, the boolean sum (that is, the bitwise OR) of the first two columns is
; that sum is not attainable as the sum of any other pair of columns in the matrix.
However, this matrix is not 3-separable, because the sum of columns 1, 2, and 3 (namely
) equals the sum of columns 1, 4, and 5.
-separable, because the sum of columns 1 and 8 (namely
) equals the sum of column 1 alone.
In fact, no matrix with an all-zero column can possibly be
{\displaystyle \quad \left[{\begin{array}{cccccccc}1&0&0&1&1&0&0&0\\1&0&0&0&0&1&1&0\\0&1&0&1&0&1&0&0\\0&1&0&0&1&0&1&0\\0&0&1&0&1&1&0&0\\0&0&1&1&0&0&1&0\\\end{array}}\right]}
{\displaystyle \quad \left[{\begin{array}{cccc}1&0&0&1\\1&0&1&0\\0&1&1&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\\\end{array}}\right]}
There are 15 possible ways to choose 3-or-fewer columns from this matrix, and each choice leads to a different boolean sum: However, the sum of columns 2, 3, and 4 (namely
) is a superset of column 1 (namely
), which means that this matrix is not 3-disjunct.
The non-adaptive group testing problem postulates that we have a test which can tell us, for any set of items, whether that set contains a defective item.
We are asked to come up with a series of groupings that can exactly identify all the defective items in a batch of n total items, some d of which are defective.
columns concisely describes how to use t tests to find the defective items in a batch of n, where the number of defective items is known to be exactly d.
-disjunct matrix (or, more generally, any
columns concisely describes how to use t tests to find the defective items in a batch of n, where the number of defective items is known to be no more than d. For a given n and d, the number of rows t in the smallest d-separable
matrix may (according to current knowledge) be smaller than the number of rows t in the smallest d-disjunct
matrix, but in asymptotically they are within a constant factor of each other.
[3] Additionally, if the matrix is to be used for practical testing, some algorithm is needed that can "decode" a test result (that is, a boolean sum such as
) into the indices of the defective items (that is, the unique set of columns that produce that boolean sum).
For arbitrary d-disjunct matrices, polynomial-time decoding algorithms are known; the naïve algorithm is
[4] For arbitrary d-separable but non-d-disjunct matrices, the best known decoding algorithms are exponential-time.
[3] Porat and Rothschild (2008) present a deterministic
-time algorithm for constructing a d-disjoint matrix with n columns and