Bomb dropping algorithm

Question

I have an n x m matrix consisting of non-negative integers. For example:

2 3 4 7 1
1 5 2 6 2
4 3 4 2 1
2 1 2 4 1
3 1 3 4 1
2 1 4 3 2
6 9 1 6 4
         


        

Answer 1

There is no need to transform the problem to linear sub-problems.

Instead use a simple greedy heuristic, which is to bomb the corners, starting with the largest one.

In the given example there are four corners, { 2, 1, 6, 4 }. For each corner there is no better move than to bomb the cell diagonal to the corner, so we know for a fact our first 2+1+6+4 = 13 bombings must be in these diagonal cells. After doing the bombing we are left with a new matrix:

2 3 4 7 1      0 1 1 6 0      0 1 1 6 0     1 1 6 0     0 0 5     0 0 0 
1 5 2 6 2      0 3 0 5 1      0 3 0 5 1  => 1 0 4 0  => 0 0 3  => 0 0 0  
4 3 4 2 1      2 1 1 1 0      2 1 1 1 0     0 0 0 0     0 0 0     0 0 3  
2 1 2 4 1  =>  2 1 2 4 1  =>  2 1 2 4 1     0 0 3 0     0 0 3      
3 1 3 4 1      0 0 0 0 0      0 0 0 0 0 
2 1 4 3 2      0 0 0 0 0      0 0 0 0 0 
6 9 1 6 4      0 3 0 2 0      0 0 0 0 0 

After the first 13 bombings we use the heuristic to eliminate 3 0 2 via three bombings. Now, we have 2 new corners, { 2, 1 } in the 4th row. We bomb those, another 3 bombings. We have reduced the matrix to 4 x 4 now. There is one corner, the upper left. We bomb that. Now we have 2 corners left, { 5, 3 }. Since 5 is the largest corner we bomb that first, 5 bombings, then finally bomb the 3 in the other corner. The total is 13+3+3+1+5+3 = 28.