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 PyPy3 (2.4.0) (judge verdict: Accepted)

import sys
 
readline = sys.stdin.readline
MOD = 10 ** 9 + 7
INF = float('INF')
sys.setrecursionlimit(10 ** 5)
 
 
def main():
    def divide_choco(bit):
        divrow_num = bin(bit).count("1") + 1  # 割った回数+1
        choco_current = [[0] * w for _ in range(divrow_num)]
 
        row_divided_cur = 0
 
        for col, piece in enumerate(choco[0]):
            choco_current[0][col] = piece
 
        for row in range(h - 1):
            if bit >> row & 1:
                row_divided_cur += 1
            for col, piece in enumerate(choco[row + 1]):
                choco_current[row_divided_cur][col] += piece
 
        return choco_current, divrow_num
 
    h, w, k = list(map(int, readline().split()))
    choco = []
 
    for i in range(h):
        choco_in = input()
        choco_add = [int(c) for c in choco_in]
        choco.append(choco_add)
 
    ans = INF
 
    for bit in range(1 << (h - 1)):
        choco_current, divrow_num = divide_choco(bit)
        choco_cursum = [0] * divrow_num
        is_exceed = False  # 超えてる場合True
 
        ans_temp = bin(bit).count("1")
        if divrow_num > 1:
            for col in range(w):
                choco_currow = [choco_current[dr][col] for dr in range(divrow_num)]
                for drow, pieces in enumerate(choco_currow):
                    if pieces > k:
                        ans_temp += INF
                    elif choco_cursum[drow] + pieces > k:
                        is_exceed = True
 
                if is_exceed:
                    choco_cursum = [cc for cc in choco_currow]
                    ans_temp += 1
                    is_exceed = False
                else:
                    choco_cursum = [cc + cs for cc, cs in zip(choco_currow, choco_cursum)]
        else:
            choco_cursum = 0
            for pieces in choco_current[0]:
                if pieces > k:
                    ans_temp += INF
                elif choco_cursum + pieces > k:
                    choco_cursum = pieces
                    ans_temp += 1
                else:
                    choco_cursum = choco_cursum + pieces
 
        ans = min(ans, ans_temp)
 
    print(ans)
 
 
if __name__ == '__main__':
    main()