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| | Anybody who wrote the is called Eusebio. His friends say it's bad for him but precisely he loves doing is regarded as acting and he's has been doing it for some time now. Filing has been his profession as news got around. Massachusetts has always been his everyday life place and his family loves it. Go to his website locate out more: http://prometeu.net<br><br>My page: [http://prometeu.net Clash Of Clans Hack Tool] |
| In [[computer science]], '''Monge arrays''', or '''Monge matrices''', are mathematical objects named for their discoverer, the French mathematician [[Gaspard Monge]].
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| 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
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| :<math>1\le i < k\le m\text{ and }1\le j < \ell\le n</math>
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| one obtains
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| :<math>A[i,j] + A[k,\ell] \le A[i,\ell] + A[k,j].\,</math>
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| 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]]).
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| This matrix is a Monge array:
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| :<math>
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| \begin{bmatrix}
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| 10 & 17 & 13 & 28 & 23 \\
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| 17 & 22 & 16 & 29 & 23 \\
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| 24 & 28 & 22 & 34 & 24 \\
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| 11 & 13 & 6 & 17 & 7 \\
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| 45 & 44 & 32 & 37 & 23 \\
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| 36 & 33 & 19 & 21 & 6 \\
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| 75 & 66 & 51 & 53 & 34 \end{bmatrix}</math>
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| For example, take the intersection of rows 2 and 4 with columns 1 and 5.
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| The four elements are:
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| :<math>
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| \begin{bmatrix}
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| 17 & 23\\
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| 11 & 7 \end{bmatrix}</math>
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| : 17 + 7 = 24
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| : 23 + 11 = 34
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| 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.
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| ==Properties==
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| *The above definition is equivalent to the statement
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| :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>. | |
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| *Any subarray produced by selecting certain rows and columns from an original Monge array will itself be a Monge array.
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| *Any [[linear combination]] with non-negative coefficients of Monge arrays is itself a Monge array.
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| *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.
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| *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>.
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| *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]].
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| ==Applications==
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| *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.
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| == References ==
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| * {{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 }}
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| [[Category:Operations research]]
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| [[Category:Theoretical computer science]]
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