(Analysis by Dhruv Rohatgi )

This problem can be solved with flood fill. We loop over all ice cream cells, and depth-first search or breadth-first search to label the entire blob containing the cell. Ordinarily this would take $O(N^4)$ time, since each search could take $O(N^2)$ time and we perform $O(N^2)$ searches. However, if a cell has already been visited by a previous search (i.e. it is in the same component as a previously seen cell) then we can skip it, since the cells in its component have already been labelled. Thus, every cell is only visited once, so the complexity of all the searches together is only $O(N^2)$, which runs in time.

Now each ice cream cell is labelled with a "blob ID". Now we need to find the area and perimeter of each blob. Area is simply the number of cells labelled with this ID, and the perimeter is the sum over all cells in the region of the number of "border edges" of that cell: that is, the number of empty-space cells adjacent to that ice cream cell.

Hence, we can initialize the area and perimeter of each region to zero, and compute them incrementally by making a final pass over the grid.

Once all areas and perimeters are computed, it is a simple matter to find the largest area, and among those blobs with largest area, find the smallest perimeter.

Brian Dean's code is below.

#include <iostream>
#include <fstream>
#include <vector>
#include <stack>
using namespace std;

int N, R;
char grid[1002][1002]; // pad with .'s
int region[1002][1002], area[1000000], perimeter[1000000];

typedef pair<int,int> pii;
void visit(int i, int j, int r)
{
stack<pii> to_visit;
to_visit.push(make_pair(i,j));
while (!to_visit.empty()) {
pii current = to_visit.top();
to_visit.pop();
i = current.first; j = current.second;
if (region[i][j] != 0 || grid[i][j]=='.') continue;
region[i][j] = R;
area[R]++;
to_visit.push(make_pair(i-1,j));
to_visit.push(make_pair(i+1,j));
to_visit.push(make_pair(i,j-1));
to_visit.push(make_pair(i,j+1));
}
}

void find_perimeters(void)
{
for (int i=1; i<=N; i++)
for (int j=1; j<=N; j++) {
int r = region[i][j];
if (r == 0) continue;
if (region[i-1][j]==0) perimeter[r]++;
if (region[i+1][j]==0) perimeter[r]++;
if (region[i][j-1]==0) perimeter[r]++;
if (region[i][j+1]==0) perimeter[r]++;
}
}

int main(void)
{
ifstream fin ("perimeter.in");
fin >> N;
string s;
for (int i=0; i<N+2; i++) grid[0][i] = grid[N+1][i] = '.';
for (int i=1; i<=N; i++) {
grid[i][0] = grid[i][N+1] = '.';
fin >> s;
for (int j=1; j<=N; j++) grid[i][j] = s[j-1];
}

for (int i=1; i<=N; i++)
for (int j=1; j<=N; j++)
if (grid[i][j] == '#' && region[i][j] == 0) visit(i,j,++R);

find_perimeters();

int best_a=0, best_p=0;
for (int i=1; i<=R; i++)
if (area[i] > best_a || (area[i] == best_a && perimeter[i] < best_p)) {
best_a = area[i];
best_p = perimeter[i];
}

ofstream fout ("perimeter.out");
fout << best_a << " " << best_p << "\n";
return 0;
}