Problem:
Score : 500 points

Problem Statement

We have a chocolate bar partitioned into H horizontal rows and W vertical columns of squares.

The square (i, j) at the i-th row from the top and the j-th column from the left is dark if S_{i,j} is 0, and white if S_{i,j} is 1.

We will cut the bar some number of times to divide it into some number of blocks. In each cut, we cut the whole bar by a line running along some boundaries of squares from end to end of the bar.

How many times do we need to cut the bar so that every block after the cuts has K or less white squares?

Constraints

1 \leq H \leq 10

1 \leq W \leq 1000

1 \leq K \leq H \times W

S_{i,j} is 0 or 1.

Input

Input is given from Standard Input in the following format:

H W K
S_{1,1}S_{1,2}...S_{1,W}
:
S_{H,1}S_{H,2}...S_{H,W}

Output

Print the number of minimum times the bar needs to be cut so that every block after the cuts has K or less white squares.

Sample Input 1

3 5 4
11100
10001
00111

Sample Output 1

2

For example, cutting between the 1-st and 2-nd rows and between the 3-rd and 4-th columns - as shown in the figure to the left - works.

Note that we cannot cut the bar in the ways shown in the two figures to the right.

Sample Input 2

3 5 8
11100
10001
00111

Sample Output 2

0

No cut is needed.

Sample Input 3

4 10 4
1110010010
1000101110
0011101001
1101000111

Sample Output 3

3

Solution in C (GCC 5.4.1) (judge verdict: Accepted)

#include <stdio.h>
#include <string.h>
#include <stdlib.h>
#include <stdint.h> // uint64_t

#define HEIGHT_MAX 10
#define WIDTH_MAX 1000
#define BUF_SIZE (WIDTH_MAX+50)

// size: specify sizeof(str)
int get_str(char *str, int size) {
    if(!fgets(str, size, stdin)) return -1;
    return 0;
}

int get_int3(int *a1, int *a2, int *a3) {
#ifdef BUF_SIZE
  char line[BUF_SIZE];
  if(!fgets(line, BUF_SIZE, stdin)) return -1;
  sscanf(line, "%d %d %d", a1, a2, a3);
#else
#error
#endif
  return 0;
}

#define min(a,b) ((a) > (b) ? (b) : (a))

int main(void) {
    int height, width, ulimit;
    int i, j;
    static char map[HEIGHT_MAX][WIDTH_MAX];

    get_int3(&height, &width, &ulimit);
    
    for(i = 0; i < height; i++) {
        char buf[BUF_SIZE];
        get_str(buf, BUF_SIZE);
        for(j = 0; j < width; j++) {
            map[i][j] = (buf[j] - '0');
        }
    }
    int pat, bit;
    int ans = (WIDTH_MAX+HEIGHT_MAX);
    for(pat = 0; pat <(1<<height); pat+=2) {
        int res = __builtin_popcount(pat);
        if(res >= ans) continue;
        int sum[11] = {0};
        int group[11] = {0};
        int idx = 1;
        int wid;
        int moved = 0;
        // create groups
        for(bit = 0; bit <= height; bit++) {
            if(pat & (1<<bit)) group[idx++] = bit;
        }
        group[idx++] = height;
#ifdef DEBUG
        for(i = 0; i < idx; i++) printf("%d ", group[i]);
#endif

        // pat = 0: [0, 5]
        for(wid = 0; wid < width;) {
            int flag = 1;
            int new[11] = {0};
            for(i = 0; i < idx-1; i++) {
                for(bit = group[i]; bit < group[i+1]; bit++) {
                    new[i] += map[bit][wid];
                }
#ifdef DEBUG
                for(j = 0; j < idx-1; j++) { printf("(%d+%d) vs %d\n", new[i], sum[i], ulimit); }
#endif
                // must split
                if(sum[i] + new[i] > ulimit) { flag = 0; break; }
            }
#ifdef DEBUG
            printf("flag: %d ", flag);
#endif
            if(!flag) {
                if(!moved) { break; }
                res++;
                moved = 0;
                for(i = 0; i < idx-1; i++) sum[i] = 0;
            } else {
                moved = 1;
                wid++;
                for(i = 0; i < idx-1; i++) sum[i] += new[i];
            }
        }
#ifdef DEBUG
        printf("[pat: %d] possible?: %d ans: %d\n", pat, moved, res);
#endif
        if(!moved) continue;
        ans = min(ans, res);
    }
    printf("%d\n", ans);
    return 0;
}