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In computer science, an m-by-n array of real numbers is a Monge array if for all i, j, k, l such that:
- <math> 1 \le i < k \le m <math> and <math> 1 \le j < l \le n <math>
one obtains:
- <math>A[i,j] + A[k,l] \le A[i,l] + A[k,j] <math>
So whenever we pick two rows and two columns of a Monge array and consider the four elements at the intersection points, 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.
This array 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
It holds that 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.
Monge arrays are useful for keeping growth of functions in O(nlog n) time or less.
See also:
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