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In mathematics applied to
computer science Computer science is the study of computation, information, and automation. Computer science spans Theoretical computer science, theoretical disciplines (such as algorithms, theory of computation, and information theory) to Applied science, ...
, Monge arrays, or Monge matrices, are mathematical objects named for their discoverer, the French mathematician
Gaspard Monge Gaspard Monge, Comte de Péluse (; 9 May 1746 – 28 July 1818) was a French mathematician, commonly presented as the inventor of descriptive geometry, (the mathematical basis of) technical drawing, and the father of differential geometry. Dur ...
. An ''m''-by-''n''
matrix Matrix (: matrices or matrixes) or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the m ...
is said to be a ''Monge array'' if, for all \scriptstyle i,\, j,\, k,\, \ell such that :1\le i < k\le m\text1\le j < \ell\le n one obtains :A ,j+ A ,\ell\le A ,\ell+ A ,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 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 For example, take the intersection of rows 2 and 4 with columns 1 and 5. The four elements are: : \begin 17 & 23\\ 11 & 7 \end : 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 In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
A ,j+ A +1,j+1le A ,j+1+ A +1,j/math> 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 In mathematics, a linear combination or superposition is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' a ...
with non-negative coefficients of Monge arrays is itself a Monge array. *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. 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_ A ,i/math>, 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+ A ,sle A ,s+ A ,i/math> only for all 1\le i < r,s\le n. *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

Monge matrices has applications in
combinatorial optimization Combinatorial optimization is a subfield of mathematical optimization that consists of finding an optimal object from a finite set of objects, where the set of feasible solutions is discrete or can be reduced to a discrete set. Typical combina ...
problems: *When the
traveling salesman problem In the theory of computational complexity, the travelling salesman problem (TSP) asks the following question: "Given a list of cities and the distances between each pair of cities, what is the shortest possible route that visits each city exac ...
has a cost matrix which is a Monge matrix it can be solved in quadratic time. *A square Monge matrix which is also symmetric about its main diagonal is called a '' Supnick matrix'' (after Fred Supnick). Any linear combination of Supnick matrices is itself a Supnick matrix, and when the cost matrix in a traveling salesman problem is Supnick, the optimal solution is a predetermined route, unaffected by the specific values within the 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 , author2-link = Gerhard J. Woeginger , journal = Bulletin of the European Association for Theoretical Computer Science , publisher = EATCS , volume = 90 , date=October 2006 , issn = 0252-9742 , pages = 43–52 , url = http://alexandria.tue.nl/openaccess/Metis211810.pdf Theoretical computer science