Disjunct matrix
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 superimposed codes or d-cover-free families.
According to Chen and Hwang,
- A matrix is said to be d-separable if no two sets of d columns have the same boolean sum.
- A matrix is said to be -separable if no two sets of d-or-fewer columns have the same boolean sum.
- A matrix is said to be d-disjunct if no set of d columns has a boolean sum which is a superset of any other single column.
- Every -separable matrix is also -disjunct.
- Every -disjunct matrix is also -separable.
- Every -separable matrix is also -separable.
Concrete examples
The following matrix is 2-separable, because each pair of columns has a distinct sum. For example, the boolean sum 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 equals the sum of columns 1, 4, and 5.
This matrix is also not -separable, because the sum of columns 1 and 8 equals the sum of column 1 alone. In fact, no matrix with an all-zero column can possibly be -separable for any.
The following matrix is -separable but not 3-disjunct.
There are 15 possible ways to choose 3-or-fewer columns from this matrix, and each choice leads to a different boolean sum:
| columns | boolean sum | columns | boolean sum | |
| none | 000000 | 2,3 | 011110 | |
| 1 | 110000 | 2,4 | 101101 | |
| 2 | 001100 | 3,4 | 111011 | |
| 3 | 011010 | 1,2,3 | 111110 | |
| 4 | 100001 | 1,2,4 | 111101 | |
| 1,2 | 111100 | 1,3,4 | 111011 | |
| 1,3 | 111010 | 2,3,4 | 111111 | |
| 1,4 | 110001 |
However, the sum of columns 2, 3, and 4 is a superset of column 1, which means that this matrix is not 3-disjunct.
Application of ''d''-separability to group testing
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.A -separable matrix with rows and 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.
A -disjunct matrix with rows and 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.
Practical concerns and published results
For a given n and d, the number of rows t in the smallest d-separable matrix may 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. Additionally, if the matrix is to be used for practical testing, some algorithm is needed that can "decode" a test result into the indices of the defective items. For arbitrary d-disjunct matrices, polynomial-time decoding algorithms are known; the naïve algorithm is. For arbitrary d-separable but non-d-disjunct matrices, the best known decoding algorithms are exponential-time.Porat and Rothschild present a deterministic -time algorithm for constructing a d-disjoint matrix with n columns and rows.