Monge array

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 $\scriptstyle i,\, j,\, k,\, \ell$ such that

$1\le i < k\le m\text{ and }1\le j < \ell\le n$

one obtains^[1]

$A[i,j] + A[k,\ell] \le A[i,\ell] + A[k,j].\,$

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:

$\begin{bmatrix} 10 & 17 & 13 & 28 & 23 \\ 17 & 22 & 16 & 29 & 23 \\ 24 & 28 & 22 & 34 & 24 \\ 11 & 13 & 6 & 17 & 7 \\ 45 & 44 & 32 & 37 & 23 \\ 36 & 33 & 19 & 21 & 6 \\ 75 & 66 & 51 & 53 & 34 \end{bmatrix}$

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

$\begin{bmatrix} 17 & 23\\ 11 & 7 \end{bmatrix}$

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 $A[i,j] + A[i+1,j+1]\le A[i,j+1] + A[i+1,j]$ for all $1\le i < m$ and $1\le j < n$.

• 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 $f(x) = \arg\min_{i\in \{1,\ldots,m\}} A[x,i]$, then $f(j)\le f(j+1)$ for all $1\le j < n$. 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 $A[i,i] + A[r,s]\le A[i,s] + A[r,i]$ only for all $1\le i < r,s\le n$.
• 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

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.

• Deineko, Vladimir G.; Woeginger, Gerhard J. (October 2006). "Some problems around travelling salesmen, dart boards, and euro-coins" (PDF). Bulletin of the European Association for Theoretical Computer Science. EATCS. 90: 43–52. ISSN 0252-9742.