In computer science

Computer science

Computer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...

, Monge arrays, or Monge matrices, are mathematical objects named for their discoverer, the French mathematician Gaspard Monge

Gaspard Monge

Gaspard Monge, Comte de Péluse was a French mathematician, revolutionary, and was inventor of descriptive geometry. During the French Revolution, he was involved in the complete reorganization of the educational system, founding the École Polytechnique...

.

An m-by-n matrix

Matrix (mathematics)

In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...

 is said to be a Monge array if, for all such that


one obtains


So whenever we pick two rows and two columns of a Monge array (a 2 × 2 sub-matrix) and consider the four elements at the intersection points, the sum of the upper-left and lower right elements (across the main diagonal) is less than or equal to the sum of the lower-left and upper-right elements (across the antidiagonal).

This matrix is a Monge array:

For example, take the intersection of rows 2 and 4 with columns 1 and 5.
The four elements are:

17 + 7 = 24

23 + 11 = 34

The sum of the upper-left and lower right elements is less than or equal to the sum of the lower-left and upper-right elements.

Properties

• The above definition is equivalent to the statement

A matrix is a Monge array if and only if

If and only if

In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....

  for all and .

• Any subarray produced by selecting certain rows and columns from an original Monge array will itself be a Monge array.

• Any linear combination
  Linear combination
  In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results...
  
   with non-negative coefficients of Monge arrays is itself a Monge array.

• One interesting property of Monge arrays is that if you mark with a circle the leftmost minimum of each row, you will discover that your circles march downward to the right; that is to say, if , then for all . Symmetrically, if you mark the uppermost minimum of each column, your circles will march rightwards and downwards. The row and column maxima march in the opposite direction: upwards to the right and downwards to the left.

• The notion of weak Monge arrays has been proposed; a weak Monge array is a square n-by-n matrix which satisfies the Monge property only for all .

Applications

• A square Monge matrix which is also symmetric about its main diagonal is called a Supnick matrix
  Supnick matrix
  A Supnick matrix or Supnick array – named after Fred Supnick of the City College of New York, who introduced the notion in 1957 – is a Monge array which is also a symmetric matrix.- Mathematical definition :...
  
  (after Fred Supnick); this kind of matrix has applications to the traveling salesman problem (namely, that the problem admits of easy solutions when the distance matrix
  Distance matrix
  In mathematics, computer science and graph theory, a distance matrix is a matrix containing the distances, taken pairwise, of a set of points...
  
  can be written as a Supnick matrix). Note that any linear combination of Supnick matrices is itself a Supnick matrix.