APIO '09 P1 - Digging for Oil

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Points: 20 (partial)
Time limit: 0.75s
Memory limit: 128M

Problem type

The Government of Siruseri has decided to auction off land in its oil-rich Navalur province to private contractors to set up oil wells. The entire area that is being auctioned off has been divided up into an M \times N rectangular grid of smaller plots.

The Geological Survey of Siruseri has data on the estimated oil reserves in Navalur. This information is published as an M \times N grid of non-negative integers, giving the estimated reserves in each of the plots. In order to prevent a monopoly, the government has ruled that any contractor may bid for only one K \times K square block of contiguous plots. The AoE oil cartel consists of a group of 3 colluding contractors who would like to choose 3 disjoint blocks so as to maximize their total yield. Suppose that the estimated oil reserves are as described below:

1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1
1 8 8 8 8 8 1 1 1
1 8 8 8 8 8 1 1 1
1 8 8 8 8 8 1 1 1
1 1 1 1 8 8 8 1 1
1 1 1 1 1 1 8 8 8
1 1 1 1 1 1 9 9 9
1 1 1 1 1 1 9 9 9

If K = 2, the AoE cartel can take over plots with a combined estimated reserve of 100 units, whereas if K = 3 they can take over plots with a combined estimated reserve of 208 units. AoE has hired you to write a program to help them identify the maximum estimated oil reserves that they can take over.

Input Specification

The first line of the input contains three integers M, N and K, where M and N are the number of rows and columns in the rectangular grid of plots and K is the size of the square block for which bids can be made. The next M lines contain N non-negative integers — each line describes the estimated oil reserves for one row of plots.

Output Specification

A single line with a single integer indicating the maximum estimated oil reserves that can be taken over by the AoE cartel.

Test Data

You may assume that K \le M and K \le N and that at least three disjoint K \times K blocks are available. In 30% of the inputs, M, N \le 12. In all inputs, M, N \le 1500. The estimated oil reserve for each plot is always non-negative and never exceeds 500.

Sample Input

9 9 3
1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1
1 8 8 8 8 8 1 1 1
1 8 8 8 8 8 1 1 1
1 8 8 8 8 8 1 1 1
1 1 1 1 8 8 8 1 1
1 1 1 1 1 1 8 8 8
1 1 1 1 1 1 9 9 9
1 1 1 1 1 1 9 9 9

Sample Output

208

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