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 super-imposed codes or d-cover-free families.

According to Chen and Hwang (2006),

• 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 $\bar{d}$-separable (that's d with an overline) 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.

The following relationships are "well-known":

• Every $\overline{d+1}$-separable matrix is also $d$-disjunct.
• Every $d$-disjunct matrix is also $\bar{d}$-separable.
• Every $\bar{d}$-separable matrix is also $d$-separable (by definition).

Concrete examples

The following $6\times 8$ 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 $110000\vee 001100=111100$; 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 $111111$) equals the sum of columns 1, 4, and 5.

This matrix is also not $\bar{2}$-separable, because the sum of columns 1 and 8 (namely $110000$) equals the sum of column 1 alone. In fact, no matrix with an all-zero column can possibly be $\bar{d}$-separable for any $d\ge 1$.

$\left[\begin{array}{cccccccc}1& 0& 0& 1& 1& 0& 0& 0\\ 1& 0& 0& 0& 0& 1& 1& 0\\ 0& 1& 0& 1& 0& 1& 0& 0\\ 0& 1& 0& 0& 1& 0& 1& 0\\ 0& 0& 1& 0& 1& 1& 0& 0\\ 0& 0& 1& 1& 0& 0& 1& 0\end{array}\right]$

The following $6\times 4$ matrix is $\bar{3}$-separable (and thus 2-disjunct) but not 3-disjunct.

$\left[\begin{array}{cccc}1& 0& 0& 1\\ 1& 0& 1& 0\\ 0& 1& 1& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$

There are 15 possible ways to choose 3-or-fewer columns from this matrix, and each choice leads to a different boolean sum:

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 (namely $111111$) is a superset of column 1 (namely $110000$), which means that this matrix is not 3-disjunct.

Application of d-separability to group testing

Template:Main 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 $d$-separable matrix with $t$ rows and $n$ 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 $d$-disjunct matrix (or, more generally, any $\bar{d}$-separable matrix) with $t$ rows and $n$ 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 $t\times n$ matrix may (according to current knowledge) be smaller than the number of rows t in the smallest d-disjunct $t\times n$ 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 (that is, a boolean sum such as $111100$) 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 $O(nt)$. For arbitrary d-separable but non-d-disjunct matrices, the best known decoding algorithms are exponential-time.

Porat and Rothschild (2008) present a deterministic $O(nt)$-time algorithm for constructing a d-disjoint matrix with n columns and $\mathrm{\Theta }(d^2\ln n)$ rows.

See also

• Group testing
• Concatenated code
• Compressed sensing

Further reading

• Atri Rudra's book on Error Correcting Codes: Combinatorics, Algorithms, and Applications (Spring 201), Chapter 17 Template:Webarchive
• Dýachkov, A. G., & Rykov, V. V. (1982). Bounds on the length of disjunctive codes. Problemy Peredachi Informatsii [Problems of Information Transmission], 18(3), 7–13.
• Dýachkov, A. G., Rashad, A. M., & Rykov, V. V. (1989). Superimposed distance codes. Problemy Upravlenija i Teorii Informacii [Problems of Control and Information Theory], 18(4), 237–250.
• Template:Cite journal