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In [[computer science]], '''Monge arrays''', or '''Monge matrices''', are mathematical objects named for their discoverer, the French mathematician [[Gaspard Monge]].
 
An ''m''-by-''n'' [[matrix (mathematics)|matrix]] is said to be a ''Monge array'' if, for all <math>\scriptstyle i,\, j,\, k,\, \ell</math> such that
 
:<math>1\le i < k\le m\text{ and }1\le j < \ell\le n</math>
 
one obtains
 
:<math>A[i,j] + A[k,\ell] \le A[i,\ell] + A[k,j].\,</math>
 
So whenever we pick two rows and two columns of a Monge array (a 2&nbsp;&times;&nbsp;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:
:<math>
\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}</math>
 
For example, take the intersection of rows 2 and 4 with columns 1 and 5.
The four elements are:
:<math>
\begin{bmatrix}
17 & 23\\
11 & 7 \end{bmatrix}</math>
 
: 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]] <math>A[i,j] + A[i+1,j+1]\le A[i,j+1] + A[i+1,j]</math> for all <math>1\le i < m</math> and <math>1\le j < n</math>.
 
*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 <math>f(x) = \arg\min_{i\in 1\ldots m} A[x,i]</math>, then <math>f(j)\le f(j+1)</math> for all <math>1\le j < n</math>. 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 <math>A[i,i] + A[r,s]\le A[i,s] + A[r,i]</math> only for all <math>1\le i < r,s\le n</math>.
*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]].
 
==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 ==
* {{cite journal | title = Some problems around travelling salesmen, dart boards, and euro-coins | first1 = Vladimir G. | last1 = Deineko | first2 =  Gerhard J. | last2 = Woeginger | journal = Bulletin of the European Association for Theoretical Computer Science | publisher = [[European Association for Theoretical Computer Science|EATCS]] | volume = 90 |date=October 2006 | issn = 0252-9742 | pages = 43–52 | url = http://alexandria.tue.nl/openaccess/Metis211810.pdf | format = PDF }}
 
[[Category:Operations research]]
[[Category:Theoretical computer science]]

Latest revision as of 20:27, 12 January 2015

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