Monge array

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In mathematics applied to computer science, Monge arrays, or Monge matrices, are mathematical objects named for their discoverer, the French mathematician Gaspard Monge.

An m-by-n matrix is said to be a Monge array if, for all such that

one obtains[1]

So for any two rows and two columns of a Monge array (a 2 × 2 sub-matrix) the four elements at the intersection points have the property that 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[edit]

  • The above definition is equivalent to the statement
A matrix is a Monge array if and only if 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 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 .
  • Every Monge array is totally monotone, meaning that its row minima occur in a nondecreasing sequence of columns, and that the same property is true for every subarray. This property allows the row minima to be found quickly by using the SMAWK algorithm.
  • Monge matrix is just another name for submodular function of two discrete variables. Precisely, A is a Monge matrix if and only if A[i,j] is a submodular function of variables i,j.

Applications[edit]

References[edit]

  1. ^ Burkard, Rainer E.; Klinz, Bettina; Rudolf, Rüdiger (1996). "Perspectives of Monge properties in optimization". Discrete Applied Mathematics. ELSEVIER. 70 (2): 95–96. doi:10.1016/0166-218x(95)00103-x.