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 Rust (1.15.1) (judge verdict: Accepted)

//! ----------------------------------------------
//! Framework <https://github.com/vain0x/procon>
//!
//! See the bottom of file for solution.
//! ----------------------------------------------

#![allow(unused_imports)]
#![allow(non_snake_case)]

use std::cell::RefCell;
use std::cmp::{max, min, Ordering};
use std::collections::*;
use std::fmt::{Debug, Display, Formatter, Write as FmtWrite};
use std::io::{stderr, stdin, BufRead, Write};
use std::mem::{replace, swap};
use std::ops::*;
use std::rc::Rc;

/// Print values to standard error if debug mode.
#[allow(unused_macros)]
macro_rules! debug {
    ($($e:expr),*) => {
        #[cfg(debug_assertions)]
        $({
            let (e, mut err) = (stringify!($e), stderr());
            writeln!(err, "\x1B[33m{}\x1B[0m = {:?}", e, $e).unwrap()
        })*
    };
}

/// Read from standard input and parse each word.
/// - `read!(T, U, ..)` parses a line as a tuple of words.
/// - `read![[T]]` parses a line as an array of words.
/// - `read![..; N]` parses `N` lines, using `read!(..)` repeatedly.
#[allow(unused_macros)]
macro_rules! read {
    ([$t:ty] ; $n:expr) =>
        ((0..$n).map(|_| read!([$t])).collect::<Vec<_>>());
    ($($t:ty),+ ; $n:expr) =>
        ((0..$n).map(|_| read!($($t),+)).collect::<Vec<_>>());
    ([$t:ty]) =>
        (rl().split_whitespace().map(|w| w.parse().unwrap()).collect::<Vec<$t>>());
    ($($t:ty),*) => {{
        let buf = rl();
        let mut w = buf.split_whitespace();
        ($(w.next().unwrap().parse::<$t>().unwrap()),*)
    }};
}

/// Read a line from standard input.
#[allow(dead_code)]
fn rl() -> String {
    let mut buf = String::new();
    stdin().read_line(&mut buf).unwrap();

    #[allow(deprecated)]
    buf.trim_right().to_owned()
}

// -----------------------------------------------
// Solution
// -----------------------------------------------

fn main() {
    let (H, W, K) = read!(usize, usize, usize);
    let T = read![String; H];

    let board = T
        .into_iter()
        .map(|row| row.chars().collect::<Vec<_>>())
        .collect::<Vec<_>>();

    let mut rev = vec![0; H];
    let mut whites = vec![0; H];
    let mut inc = vec![0; H];
    let mut min_k = std::usize::MAX;
    let mut k;
    let mut impossible;

    for s in 0..1 << (H - 1) {
        impossible = false;
        k = 0;

        let bn = {
            let mut bi = 0;
            for y in 0..H {
                rev[y] = bi;

                if (s & (1 << y)) != 0 {
                    bi += 1;
                    k += 1;
                }
            }
            bi + 1
        };

        debug!((s, &rev, bn));

        if k >= min_k {
            continue;
        }

        for bi in 0..bn {
            whites[bi] = 0;
            inc[bi] = 0;
        }

        for x in 0..W {
            debug_assert!((0..bn).all(|i| inc[i] == 0));

            for y in 0..H {
                if board[y][x] == '1' {
                    inc[rev[y]] += 1_usize;
                }
            }

            debug!((x, &whites, &inc));

            if (0..bn).any(|i| inc[i] > K) {
                impossible = true;
                break;
            }

            if (0..bn).any(|i| whites[i] + inc[i] > K) {
                k += 1;

                for bi in 0..bn {
                    whites[bi] = inc[bi];
                    inc[bi] = 0;
                }
                continue;
            }

            for bi in 0..bn {
                whites[bi] += inc[bi];
                inc[bi] = 0;
            }
        }

        if impossible {
            debug!("impossible");
            continue;
        }

        min_k = min(min_k, k);
    }

    println!("{}", min_k)
}