Contest Results

(Analysis by Andi Qu, Benjamin Qi, Danny Mittal)

For any subrectangle we can construct a graph where there exists an edge separating two cells if they are of different colors. For example, the graph for

AAB
BBA
BBB

would be

._._._.
|   | |
._._._.
|   | |
. . ._.
|     |
._._._.

The main idea is to use Euler's formula for planar graphs. This equation states that $F = E - V + C + 1$ , where

$F\implies$ # of faces
$E\implies$ # of edges
$V\implies$ # of vertices
$C\implies$ # of connected components

So in the above example,

$F=4+1=5,$ where $1$ comes from the face consisting of everything outside of the subrectangle.
$E=18,$ the total number of cell borders (horizontal and vertical unit segments).
$V=4\cdot 4=16,$ one for each cell corner (dot).
$C=2,$ one for the vertex that has no edges coming out of it and one for the remaining vertices.

So the answer for each candidate is just $E - V + C.$ $V$ depends only on the dimensions of the subrectangle, and $E$ can be found in $O(1)$ time using prefix sums. The main challenge is finding $C$ , the number of connected components completely inside the subrectangle (plus one for the component corresponding to the border).

We can use the same concept as Romanian IOI Selection 2017 "Rooms", which was the inspiration for this problem. For each component, call its topmost vertex "special". Then the number of special vertices that lie strictly within the border of the subrectangle is a good approximation for the number of components completely inside the subrectangle, and can also be computed via prefix sums.

However, it is possible that a component does not lie completely within the subrectangle but its special vertex does, leading to an overcount. Notice that such components must have at least one vertex on the border of the subrectangle, so simply traverse the border of the subrectangle and delete all such components from the count. This solution takes $\mathcal{O}(MN + Q(M + N))$ time in total.

/*
Paint by Letters
Andi Qu
- F = E - V + C + 1
- Every corner of a cell is a node
- Every side of a cell is an edge if it is between 2 different colours
- DFS to find connected components
- To check if a component is completely inside a query rectangle, traverse
  the perimeter of the rectangle
- Complexity: O(MN + Q(M + N))
*/
 
#include <bits/stdc++.h>
#define FOR(i, x, y) for (int i = x; i < y; i++)
using namespace std;
 
int n, m, q;
 
char grid[2001][2001];
pair<int, int> special[2002][2002], d[4]{{1, 0}, {-1, 0}, {0, 1}, {0, -1}};
int s_pref[2002][2002], v_pref[2002][2002], h_pref[2002][2002];
bool deleted[2002][2002];
 
bool inside(int x, int y, int nx, int ny) {
    if (!(~nx && ~ny && nx <= n + 1 && ny <= m + 1)) return false;
    if (nx == x + 1) return grid[x][y] != grid[x][y - 1];
    if (nx == x - 1) return grid[nx][y] != grid[nx][y - 1];
    if (ny == y + 1) return grid[x][y] != grid[x - 1][y];
    return grid[x][ny] != grid[x - 1][ny];
}
 
bool outside(int x, int y, int x1, int y1, int x2, int y2) {
    int a = special[x][y].first, b = special[x][y].second;
    if (a > x1 && a <= x2 && b > y1 && b <= y2 && !deleted[a][b]) {
        deleted[a][b] = true;
        return true;
    }
    return false;
}
 
void deactivate(int x, int y) {
    int a = special[x][y].first, b = special[x][y].second;
    deleted[a][b] = false;
}
 
void dfs(int x, int y) {
    FOR(i, 0, 4) {
        int nx = x + d[i].first, ny = y + d[i].second;
        if (inside(x, y, nx, ny)) {
            if (nx == x + 1) v_pref[x][y] = 1;
            else if (nx == x - 1) v_pref[nx][y] = 1;
            else if (ny == y + 1) h_pref[x][y] = 1;
            else h_pref[x][ny] = 1;
 
            if (!special[nx][ny].first) {
                special[nx][ny] = special[x][y];
                dfs(nx, ny);
            }
        }
    }
}
 
int main() {
    ios_base::sync_with_stdio(0);
    cin.tie(0);
    // freopen("paint.in", "r", stdin);
    // freopen("paint.out", "w", stdout);
 
    cin >> n >> m >> q;
    FOR(i, 1, n + 1) FOR(j, 1, m + 1) cin >> grid[i][j];
    FOR(i, 1, n + 2) FOR(j, 1, m + 2) if (!special[i][j].first) {
        s_pref[i][j] = 1;
        special[i][j] = {i, j};
        dfs(i, j);
    }
    FOR(i, 1, n + 2) FOR(j, 1, m + 2) {
        s_pref[i][j] += s_pref[i - 1][j] + s_pref[i][j - 1] - s_pref[i - 1][j - 1];
        v_pref[i][j] += v_pref[i - 1][j] + v_pref[i][j - 1] - v_pref[i - 1][j - 1];
        h_pref[i][j] += h_pref[i - 1][j] + h_pref[i][j - 1] - h_pref[i - 1][j - 1];
    }
 
    while (q--) {
        int x1, x2, y1, y2;
        cin >> x1 >> y1 >> x2 >> y2;
        if (x1 > x2) swap(x1, x2);
        if (y1 > y2) swap(y1, y2);
 
        int E = v_pref[x2][y2] - v_pref[x2][y1] - v_pref[x1 - 1][y2] + v_pref[x1 - 1][y1] +
                h_pref[x2][y2] - h_pref[x2][y1 - 1] - h_pref[x1][y2] + h_pref[x1][y1 - 1];
        int V = (x2 - x1) * (y2 - y1);
        int C = s_pref[x2][y2] - s_pref[x2][y1] - s_pref[x1][y2] + s_pref[x1][y1];
 
        // Delete connected components not entirely inside the query rectangle
        FOR(i, x1, x2 + 2) {
            if (outside(i, y2 + 1, x1, y1, x2, y2)) C--;
            if (outside(i, y1, x1, y1, x2, y2)) C--;
        }
        FOR(i, y1, y2 + 2) {
            if (outside(x2 + 1, i, x1, y1, x2, y2)) C--;
            if (outside(x1, i, x1, y1, x2, y2)) C--;
        }
        FOR(i, x1, x2 + 2) {
            deactivate(i, y2 + 1);
            deactivate(i, y1);
        }
        FOR(i, y1, y2 + 2) {
            deactivate(x2 + 1, i);
            deactivate(x1, i);
        }
 
        cout << E - V + C + 1 << '\n';
    }
    return 0;
}

A similar approach involves treating cells as vertices, two adjacent cells of the same color as an edge, and a cycle of a single color (as described in the problem statement) which contains no smaller cycles as a face. Then we can use Euler's formula to count the number of connected components ( $V-E+F$ ). Partial credit corresponded to no faces, faces of a single type (a $2\times 2$ square of a single color), and faces of two types (in addition to the one previously mentioned, a $3\times 3$ square with the border of a single color and the center square of a different color).

Some additional problems for which you can use Euler's formula can be found here.

A completely different approach is to simply optimize the naive brute-force-every-query approach by using a two dimensional segment tree. To build the segment tree, in each two dimensional "segment", we use either DFS or union find to determine its connected components, and then store the number of components along with the connected component that each cell on the segment's boundary belongs to. This takes $O(MN(\log M + \log N))$ if implemented properly.

To answer queries, we find the corresponding segments, summing the amount of components in each one, and then merge them along their boundaries (again using either DFS or union find), subtracting 1 from our result each time two components become connected. This takes $O(M\log N + N\log M)$ per query, yielding an overall runtime of $O(MN(\log M + \log N) + Q(M\log N + N\log M))$ , or simply $O(N^2\log N + QN\log N)$ when $M = N$ .

Note that the above complexities should be multiplied by $\alpha(N)$ when using union find, though in practice union find is faster than DFS for the given constraints.

The time and memory limits were raised to allow such solutions to pass (though depending on your implementation, you might need to spend some time optimizing the constant factor).

Danny Mittal's code:

import java.io.BufferedReader;
import java.io.IOException;
import java.io.InputStreamReader;
import java.util.*;
 
public class PaintByLettersPlatinum2 {
 
    public static void main(String[] args) throws IOException {
        BufferedReader in = new BufferedReader(new InputStreamReader(System.in));
        StringTokenizer tokenizer = new StringTokenizer(in.readLine());
        int n = Integer.parseInt(tokenizer.nextToken());
        int m = Integer.parseInt(tokenizer.nextToken());
        int q = Integer.parseInt(tokenizer.nextToken());
        char[][] canvas = new char[n + 1][];
        for (int y = 1; y <= n; y++) {
            canvas[y] = (" " + in.readLine()).toCharArray();
        }
        SegmentTree segTree = new SegmentTree(n, m, canvas);
        StringBuilder out = new StringBuilder();
        for (int t = 1; t <= q; t++) {
            tokenizer = new StringTokenizer(in.readLine());
            int y1 = Integer.parseInt(tokenizer.nextToken());
            int x1 = Integer.parseInt(tokenizer.nextToken());
            int y2 = Integer.parseInt(tokenizer.nextToken());
            int x2 = Integer.parseInt(tokenizer.nextToken());
            out.append(segTree.query(y1, y2, x1, x2)).append('\n');
        }
        System.out.print(out);
    }
 
    static class SegmentTree {
        static final int LIMIT = 512;
        
        final int n;
        final int m;
        final char[][] canvas;
        final int[][][] top = new int[LIMIT][LIMIT][];
        final int[][][] bottom = new int[LIMIT][LIMIT][];
        final int[][][] left = new int[LIMIT][LIMIT][];
        final int[][][] right = new int[LIMIT][LIMIT][];
        final int[][] amt = new int[LIMIT][LIMIT];
        final int[][] component;
        final int[] seen;
        final int[] union;
        int t = 0;
        int res = 0;
 
        int find(int u) {
            if (seen[u] < t) {
                seen[u] = t;
                union[u] = u;
            } else if (union[union[u]] != union[u]) {
                union[u] = find(union[u]);
            }
            return union[u];
        }
 
        void merge(int u, int v) {
            u = find(u);
            v = find(v);
            if (u != v) {
                union[u] = v;
                res--;
            }
        }
 
        SegmentTree(int n, int m, char[][] canvas) {
            this.n = n;
            this.m = m;
            this.canvas = canvas;
            this.component = new int[n + 1][m + 1];
            this.seen = new int[(n * m) + m + 1];
            this.union = new int[(n * m) + m + 1];
            construct(1, 1, n);
        }
 
        void construct(int nodeY, int segFromY, int segToY) {
            if (segFromY != segToY) {
                int mid = (segFromY + segToY) / 2;
                construct(2 * nodeY, segFromY, mid);
                construct((2 * nodeY) + 1, mid + 1, segToY);
            }
            construct(nodeY, segFromY, segToY, 1, 1, m);
        }
 
        void construct(int nodeY, int segFromY, int segToY, int nodeX, int segFromX, int segToX) {
            if (segFromX != segToX) {
                int mid = (segFromX + segToX) / 2;
                construct(nodeY, segFromY, segToY, 2 * nodeX, segFromX, mid);
                construct(nodeY, segFromY, segToY, (2 * nodeX) + 1, mid + 1, segToX);
            }
            if (nodeY < LIMIT && nodeX < LIMIT) {
                t++;
                if ((segFromX == segToX || 2 * nodeX >= LIMIT) && (segFromY == segToY || 2 * nodeY >= LIMIT)) {
                    res = (segToY - segFromY + 1) * (segToX - segFromX + 1);
                    for (int y = segFromY; y <= segToY; y++) {
                        for (int x = segFromX; x <= segToX; x++) {
                            int c = (y * m) + x;
                            if (y > segFromY && canvas[y - 1][x] == canvas[y][x]) {
                                merge(c, c - m);
                            }
                            if (x > segFromX && canvas[y][x - 1] == canvas[y][x]) {
                                merge(c, c - 1);
                            }
                        }
                    }
                    amt[nodeY][nodeX] = res;
                    top[nodeY][nodeX] = new int[segToX - segFromX + 1];
                    bottom[nodeY][nodeX] = new int[segToX - segFromX + 1];
                    for (int x = segFromX; x <= segToX; x++) {
                        top[nodeY][nodeX][x - segFromX] = find((segFromY * m) + x);
                        bottom[nodeY][nodeX][x - segFromX] = find((segToY * m) + x);
                    }
                    left[nodeY][nodeX] = new int[segToY - segFromY + 1];
                    right[nodeY][nodeX] = new int[segToY - segFromY + 1];
                    for (int y = segFromY; y <= segToY; y++) {
                        left[nodeY][nodeX][y - segFromY] = find((y * m) + segFromX);
                        right[nodeY][nodeX][y - segFromY] = find((y * m) + segToX);
                    }
                } else if (segFromY == segToY || 2 * nodeY >= LIMIT) {
                    res = amt[nodeY][2 * nodeX] + amt[nodeY][(2 * nodeX) + 1];
                    int mid = (segFromX + segToX) / 2;
                    for (int y = segFromY; y <= segToY; y++) {
                        if (canvas[y][mid] == canvas[y][mid + 1]) {
                            merge(right[nodeY][2 * nodeX][y - segFromY], left[nodeY][(2 * nodeX) + 1][y - segFromY]);
                        }
                    }
                    amt[nodeY][nodeX] = res;
                    left[nodeY][nodeX] = new int[segToY - segFromY + 1];
                    right[nodeY][nodeX] = new int[segToY - segFromY + 1];
                    for (int y = segFromY; y <= segToY; y++) {
                        left[nodeY][nodeX][y - segFromY] = find(left[nodeY][2 * nodeX][y - segFromY]);
                        right[nodeY][nodeX][y - segFromY] = find(right[nodeY][(2 * nodeX) + 1][y - segFromY]);
                    }
                    top[nodeY][nodeX] = new int[segToX - segFromX + 1];
                    bottom[nodeY][nodeX] = new int[segToX - segFromX + 1];
                    for (int x = segFromX; x <= segToX; x++) {
                        if (x <= mid) {
                            top[nodeY][nodeX][x - segFromX] = find(top[nodeY][2 * nodeX][x - segFromX]);
                            bottom[nodeY][nodeX][x - segFromX] = find(bottom[nodeY][2 * nodeX][x - segFromX]);
                        } else {
                            top[nodeY][nodeX][x - segFromX] = find(top[nodeY][(2 * nodeX) + 1][x - mid - 1]);
                            bottom[nodeY][nodeX][x - segFromX] = find(bottom[nodeY][(2 * nodeX) + 1][x - mid - 1]);
                        }
                    }
                } else {
                    res = amt[2 * nodeY][nodeX] + amt[(2 * nodeY) + 1][nodeX];
                    int mid = (segFromY + segToY) / 2;
                    for (int x = segFromX; x <= segToX; x++) {
                        if (canvas[mid][x] == canvas[mid + 1][x]) {
                            merge(bottom[2 * nodeY][nodeX][x - segFromX], top[(2 * nodeY) + 1][nodeX][x - segFromX]);
                        }
                    }
                    amt[nodeY][nodeX] = res;
                    top[nodeY][nodeX] = new int[segToX - segFromX + 1];
                    bottom[nodeY][nodeX] = new int[segToX - segFromX + 1];
                    for (int x = segFromX; x <= segToX; x++) {
                        top[nodeY][nodeX][x - segFromX] = find(top[2 * nodeY][nodeX][x - segFromX]);
                        bottom[nodeY][nodeX][x - segFromX] = find(bottom[(2 * nodeY) + 1][nodeX][x - segFromX]);
                    }
                    left[nodeY][nodeX] = new int[segToY - segFromY + 1];
                    right[nodeY][nodeX] = new int[segToY - segFromY + 1];
                    for (int y = segFromY; y <= segToY; y++) {
                        if (y <= mid) {
                            left[nodeY][nodeX][y - segFromY] = find(left[2 * nodeY][nodeX][y - segFromY]);
                            right[nodeY][nodeX][y - segFromY] = find(right[2 * nodeY][nodeX][y - segFromY]);
                        } else {
                            left[nodeY][nodeX][y - segFromY] = find(left[(2 * nodeY) + 1][nodeX][y - mid - 1]);
                            right[nodeY][nodeX][y - segFromY] = find(right[(2 * nodeY) + 1][nodeX][y - mid - 1]);
                        }
                    }
                }
            }
        }
 
        int query(int fromY, int toY, int fromX, int toX) {
            t++;
            res = 0;
            query(fromY, toY, fromX, toX, 1, 1, n);
            return res;
        }
 
        void query(int fromY, int toY, int fromX, int toX, int nodeY, int segFromY, int segToY) {
            if (segFromY > toY || segToY < fromY) {
 
            } else if (fromY <= segFromY && segToY <= toY) {
                query(fromY, toY, fromX, toX, nodeY, segFromY, segToY, 1, 1, m);
            } else {
                int mid = (segFromY + segToY) / 2;
                query(fromY, toY, fromX, toX, 2 * nodeY, segFromY, mid);
                query(fromY, toY, fromX, toX, (2 * nodeY) + 1, mid + 1, segToY);
            }
        }
 
        void query(int fromY, int toY, int fromX, int toX, int nodeY, int segFromY, int segToY, int nodeX, int segFromX, int segToX) {
            if (segFromX > toX || segToX < fromX) {
 
            } else if (fromX <= segFromX && segToX <= toX) {
                if (nodeY < LIMIT && nodeX < LIMIT) {
                    for (int x = segFromX; x <= segToX; x++) {
                        int c = top[nodeY][nodeX][x - segFromX];
                        if (segFromY > fromY && canvas[segFromY - 1][x] == canvas[segFromY][x]) {
                            merge(c, component[segFromY - 1][x]);
                        }
                    }
                    for (int x = segFromX; x <= segToX; x++) {
                        int c = bottom[nodeY][nodeX][x - segFromX];
                        component[segToY][x] = c;
                    }
                    for (int y = segFromY; y <= segToY; y++) {
                        int c = left[nodeY][nodeX][y - segFromY];
                        if (segFromX > fromX && canvas[y][segFromX - 1] == canvas[y][segFromX]) {
                            merge(c, component[y][segFromX - 1]);
                        }
                    }
                    for (int y = segFromY; y <= segToY; y++) {
                        int c = right[nodeY][nodeX][y - segFromY];
                        component[y][segToX] = c;
                    }
                    res += amt[nodeY][nodeX];
                } else {
                    for (int y = segFromY; y <= segToY; y++) {
                        for (int x = segFromX; x <= segToX; x++) {
                            int c = (y * m) + x;
                            component[y][x] = c;
                            if (y > fromY && canvas[y - 1][x] == canvas[y][x]) {
                                merge(c, component[y - 1][x]);
                            }
                            if (x > fromX && canvas[y][x - 1] == canvas[y][x]) {
                                merge(c, component[y][x - 1]);
                            }
                        }
                    }
                    res += (segToY - segFromY + 1) * (segToX - segFromX + 1);
                }
            } else {
                int mid = (segFromX + segToX) / 2;
                query(fromY, toY, fromX, toX, nodeY, segFromY, segToY, 2 * nodeX, segFromX, mid);
                query(fromY, toY, fromX, toX, nodeY, segFromY, segToY, (2 * nodeX) + 1, mid + 1, segToX);
            }
        }
    }
}

In C++:

#include <bits/stdc++.h>
 
using namespace std;
 
struct SegmentTree {
    static const int LIMIT = 512;
    
    int n;
    int m;
    vector<string> canvas;
    vector<int> top[LIMIT][LIMIT];
    vector<int> bottom[LIMIT][LIMIT];
    vector<int> left[LIMIT][LIMIT];
    vector<int> right[LIMIT][LIMIT];
    int amt[LIMIT][LIMIT];
    int component[1001][1001];
    vector<int> seen, uni;
    int t = 0;
    int res = 0;
 
    int find(int u) {
        if (seen[u] < t) {
            seen[u] = t;
            uni[u] = u;
        } else if (uni[uni[u]] != uni[u]) {
            uni[u] = find(uni[u]);
        }
        return uni[u];
    }
 
    void merge(int u, int v) {
        u = find(u);
        v = find(v);
        if (u != v) {
            uni[u] = v;
            res--;
        }
    }
 
    SegmentTree(int n, int m, vector<string> canvas) {
        this->n = n;
        this->m = m;
        this->canvas = canvas;
        seen = uni = vector<int>((n * m) + m + 1);
        construct(1, 1, n);
    }
 
    void construct(int nodeY, int segFromY, int segToY) {
        if (segFromY != segToY) {
            int mid = (segFromY + segToY) / 2;
            construct(2 * nodeY, segFromY, mid);
            construct((2 * nodeY) + 1, mid + 1, segToY);
        }
        construct(nodeY, segFromY, segToY, 1, 1, m);
    }
 
    void construct(int nodeY, int segFromY, int segToY, int nodeX, int segFromX, int segToX) {
        if (segFromX != segToX) {
            int mid = (segFromX + segToX) / 2;
            construct(nodeY, segFromY, segToY, 2 * nodeX, segFromX, mid);
            construct(nodeY, segFromY, segToY, (2 * nodeX) + 1, mid + 1, segToX);
        }
        if (nodeY < LIMIT && nodeX < LIMIT) {
            t++;
            if ((segFromX == segToX || 2 * nodeX >= LIMIT) && (segFromY == segToY || 2 * nodeY >= LIMIT)) {
                res = (segToY - segFromY + 1) * (segToX - segFromX + 1);
                for (int y = segFromY; y <= segToY; y++) {
                    for (int x = segFromX; x <= segToX; x++) {
                        int c = (y * m) + x;
                        if (y > segFromY && canvas[y - 1][x] == canvas[y][x]) {
                            merge(c, c - m);
                        }
                        if (x > segFromX && canvas[y][x - 1] == canvas[y][x]) {
                            merge(c, c - 1);
                        }
                    }
                }
                amt[nodeY][nodeX] = res;
                top[nodeY][nodeX] = vector<int>(segToX - segFromX + 1);
                bottom[nodeY][nodeX] = vector<int>(segToX - segFromX + 1);
                for (int x = segFromX; x <= segToX; x++) {
                    top[nodeY][nodeX][x - segFromX] = find((segFromY * m) + x);
                    bottom[nodeY][nodeX][x - segFromX] = find((segToY * m) + x);
                }
                left[nodeY][nodeX] = vector<int>(segToY - segFromY + 1);
                right[nodeY][nodeX] = vector<int>(segToY - segFromY + 1);
                for (int y = segFromY; y <= segToY; y++) {
                    left[nodeY][nodeX][y - segFromY] = find((y * m) + segFromX);
                    right[nodeY][nodeX][y - segFromY] = find((y * m) + segToX);
                }
            } else if (segFromY == segToY || 2 * nodeY >= LIMIT) {
                res = amt[nodeY][2 * nodeX] + amt[nodeY][(2 * nodeX) + 1];
                int mid = (segFromX + segToX) / 2;
                for (int y = segFromY; y <= segToY; y++) {
                    if (canvas[y][mid] == canvas[y][mid + 1]) {
                        merge(right[nodeY][2 * nodeX][y - segFromY], left[nodeY][(2 * nodeX) + 1][y - segFromY]);
                    }
                }
                amt[nodeY][nodeX] = res;
                left[nodeY][nodeX] = vector<int>(segToY - segFromY + 1);
                right[nodeY][nodeX] = vector<int>(segToY - segFromY + 1);
                for (int y = segFromY; y <= segToY; y++) {
                    left[nodeY][nodeX][y - segFromY] = find(left[nodeY][2 * nodeX][y - segFromY]);
                    right[nodeY][nodeX][y - segFromY] = find(right[nodeY][(2 * nodeX) + 1][y - segFromY]);
                }
                top[nodeY][nodeX] = vector<int>(segToX - segFromX + 1);
                bottom[nodeY][nodeX] = vector<int>(segToX - segFromX + 1);
                for (int x = segFromX; x <= segToX; x++) {
                    if (x <= mid) {
                        top[nodeY][nodeX][x - segFromX] = find(top[nodeY][2 * nodeX][x - segFromX]);
                        bottom[nodeY][nodeX][x - segFromX] = find(bottom[nodeY][2 * nodeX][x - segFromX]);
                    } else {
                        top[nodeY][nodeX][x - segFromX] = find(top[nodeY][(2 * nodeX) + 1][x - mid - 1]);
                        bottom[nodeY][nodeX][x - segFromX] = find(bottom[nodeY][(2 * nodeX) + 1][x - mid - 1]);
                    }
                }
            } else {
                res = amt[2 * nodeY][nodeX] + amt[(2 * nodeY) + 1][nodeX];
                int mid = (segFromY + segToY) / 2;
                for (int x = segFromX; x <= segToX; x++) {
                    if (canvas[mid][x] == canvas[mid + 1][x]) {
                        merge(bottom[2 * nodeY][nodeX][x - segFromX], top[(2 * nodeY) + 1][nodeX][x - segFromX]);
                    }
                }
                amt[nodeY][nodeX] = res;
                top[nodeY][nodeX] = vector<int>(segToX - segFromX + 1);
                bottom[nodeY][nodeX] = vector<int>(segToX - segFromX + 1);
                for (int x = segFromX; x <= segToX; x++) {
                    top[nodeY][nodeX][x - segFromX] = find(top[2 * nodeY][nodeX][x - segFromX]);
                    bottom[nodeY][nodeX][x - segFromX] = find(bottom[(2 * nodeY) + 1][nodeX][x - segFromX]);
                }
                left[nodeY][nodeX] = vector<int>(segToY - segFromY + 1);
                right[nodeY][nodeX] = vector<int>(segToY - segFromY + 1);
                for (int y = segFromY; y <= segToY; y++) {
                    if (y <= mid) {
                        left[nodeY][nodeX][y - segFromY] = find(left[2 * nodeY][nodeX][y - segFromY]);
                        right[nodeY][nodeX][y - segFromY] = find(right[2 * nodeY][nodeX][y - segFromY]);
                    } else {
                        left[nodeY][nodeX][y - segFromY] = find(left[(2 * nodeY) + 1][nodeX][y - mid - 1]);
                        right[nodeY][nodeX][y - segFromY] = find(right[(2 * nodeY) + 1][nodeX][y - mid - 1]);
                    }
                }
            }
        }
    }
 
    int query(int fromY, int toY, int fromX, int toX) {
        t++;
        res = 0;
        query(fromY, toY, fromX, toX, 1, 1, n);
        return res;
    }
 
    void query(int fromY, int toY, int fromX, int toX, int nodeY, int segFromY, int segToY) {
        if (segFromY > toY || segToY < fromY) {
 
        } else if (fromY <= segFromY && segToY <= toY) {
            query(fromY, toY, fromX, toX, nodeY, segFromY, segToY, 1, 1, m);
        } else {
            int mid = (segFromY + segToY) / 2;
            query(fromY, toY, fromX, toX, 2 * nodeY, segFromY, mid);
            query(fromY, toY, fromX, toX, (2 * nodeY) + 1, mid + 1, segToY);
        }
    }
 
    void query(int fromY, int toY, int fromX, int toX, int nodeY, int segFromY, int segToY, int nodeX, int segFromX, int segToX) {
        if (segFromX > toX || segToX < fromX) {
 
        } else if (fromX <= segFromX && segToX <= toX) {
            if (nodeY < LIMIT && nodeX < LIMIT) {
                for (int x = segFromX; x <= segToX; x++) {
                    int c = top[nodeY][nodeX][x - segFromX];
                    if (segFromY > fromY && canvas[segFromY - 1][x] == canvas[segFromY][x]) {
                        merge(c, component[segFromY - 1][x]);
                    }
                }
                for (int x = segFromX; x <= segToX; x++) {
                    int c = bottom[nodeY][nodeX][x - segFromX];
                    component[segToY][x] = c;
                }
                for (int y = segFromY; y <= segToY; y++) {
                    int c = left[nodeY][nodeX][y - segFromY];
                    if (segFromX > fromX && canvas[y][segFromX - 1] == canvas[y][segFromX]) {
                        merge(c, component[y][segFromX - 1]);
                    }
                }
                for (int y = segFromY; y <= segToY; y++) {
                    int c = right[nodeY][nodeX][y - segFromY];
                    component[y][segToX] = c;
                }
                res += amt[nodeY][nodeX];
            } else {
                for (int y = segFromY; y <= segToY; y++) {
                    for (int x = segFromX; x <= segToX; x++) {
                        int c = (y * m) + x;
                        component[y][x] = c;
                        if (y > fromY && canvas[y - 1][x] == canvas[y][x]) {
                            merge(c, component[y - 1][x]);
                        }
                        if (x > fromX && canvas[y][x - 1] == canvas[y][x]) {
                            merge(c, component[y][x - 1]);
                        }
                    }
                }
                res += (segToY - segFromY + 1) * (segToX - segFromX + 1);
            }
        } else {
            int mid = (segFromX + segToX) / 2;
            query(fromY, toY, fromX, toX, nodeY, segFromY, segToY, 2 * nodeX, segFromX, mid);
            query(fromY, toY, fromX, toX, nodeY, segFromY, segToY, (2 * nodeX) + 1, mid + 1, segToX);
        }
    }
};
 
 
int main() {
    int n,m,q; cin >> n >> m >> q;
    vector<string> canvas(n+1);
    for (int y = 1; y <= n; y++) {
        string s; cin >> s;
        canvas[y] = " " + s;
    }
    SegmentTree segTree(n, m, canvas);
    for (int t = 1; t <= q; t++) {
        int y1,x1,y2,x2; cin >> y1 >> x1 >> y2 >> x2;
        cout << segTree.query(y1, y2, x1, x2) << "\n";
    }
}