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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.

Contents

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

1 ≤ i < k ≤ m  and  1 ≤ j < ℓ ≤ n

one obtains

A [ i , j ] + A [ k , ℓ ] ≤ A [ i , ℓ ] + A [ k , j ] .

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:

[ 10 17 13 28 23 17 22 16 29 23 24 28 22 34 24 11 13 6 17 7 45 44 32 37 23 36 33 19 21 6 75 66 51 53 34 ]

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

[ 17 23 11 7 ] 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 A [ i , j ] + A [ i + 1 , j + 1 ] ≤ A [ i , j + 1 ] + A [ i + 1 , j ] for all 1 ≤ i < m and 1 ≤ j < n .
  • 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 f ( x ) = arg ⁡ min i ∈ { 1 , … , m } A [ x , i ] , then f ( j ) ≤ f ( j + 1 ) for all 1 ≤ j < n . 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 A [ i , i ] + A [ r , s ] ≤ A [ i , s ] + A [ r , i ] only for all 1 ≤ i < r , s ≤ n .
  • 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

  • A square Monge matrix which is also symmetric about its main diagonal is called a Supnick matrix (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 can be written as a Supnick matrix). Note that any linear combination of Supnick matrices is itself a Supnick matrix.

    • References

    Monge array Wikipedia
    (Text) CC BY-SA