Best way to efficiently find high density regions

Question

Over the course of my coding, I have come across a problem as follows: Find the a region of fixed size in a 2D space that has the highest density of particles. The particles can

Answer 1

I don't know what brute force method you use, but the most brute force way would be O(n^2 d^2), by iterating over every region in O(n^2) time, then count the number of particles in that region in O(d^2) time where d is the size of your region.

This problem is exactly the same as this problem: Rat Attack, since the region area is fixed, and so the density is the same as the count, for which the solution is O(n^2 + k*d^2), where

1. n is the size of the whole area (length of the side)
2. k is the number of particles
3. d is the size of each region (length of the side)

by this algorithm:

1. For each particle, update the count of the O(d^2) regions affected by this particle
2. Iterate over all O(n^2) possible regions, find the maximum

as shown in this code, I copy the relevant part here for your reference:

using namespace std;

int mat [1024 + 3] [1024 + 3]; // Here n is assumed to be 1024

int main ()
{
    int testCases; scanf ("%d", &testCases);

    while ( testCases-- ) {

        Set(mat, 0);

        int d; scanf ("%d", &d); // d is the size of the region
        int k; scanf ("%d", &k); // k is the number of particles

        int x, y, cost;

        for ( int i = 0; i < k; i++ ) {
            scanf ("%d %d %d", &x, &y, &cost); // Read each particle position

            // Update the count of the d^2 region affected by this particle
            for ( int j = max (0, x - d); j <= min (x + d, 1024); j++ ) {
                for ( int k = max (0, y - d); k <= min (y + d, 1024); k++ ) mat [j] [k] += cost;
            }
        }

        int resX, resY, maxi = -1;

        // Find the maximum count over all regions
        for ( int i = 0; i < 1025; i++ ) {
            for ( int j = 0; j < 1025; j++ ) {
                if ( maxi < mat [i] [j] ) {
                    maxi = mat [i] [j];
                    resX = i;
                    resY = j;
                }
            }
        }

        printf ("%d %d %d\n", resX, resY, maxi);

    }
    return 0;
}

I've put my comments in the code to explain it to you.