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)

// This code is generated by [cargo-atcoder](https://github.com/tanakh/cargo-atcoder)
// Original source code:
/*
use proconio::input;
use proconio::marker::Chars;

// (h, w, k)
type Info = (usize, usize, usize);

fn main() {
    input! {
        h: usize,
        w: usize,
        k: usize,
        s: [Chars; h],
    }

    let mut s2 = vec![];
    for s_i in s {
        let s2_i: Vec<usize> = s_i
            .into_iter()
            .map(|x| x.to_digit(10).unwrap() as usize)
            .collect();
        s2.push(s2_i);
    }
    let s = s2;

    let mut counts = vec![vec![0]; h];
    for (i, row) in s.iter().enumerate() {
        for a in row {
            let prev = *counts[i].last().unwrap();
            counts[i].push(prev + a);
        }
    }

    let mut mn = std::usize::MAX;
    for bits in 0..(1 << (h - 1)) {
        // eprintln!("bits: {:b}", bits);
        let now = solve(bits, &counts, (h, w, k));
        // eprintln!("now: {:?}", now);
        if let Some(now) = now {
            if now < mn {
                mn = now;
            }
        }
    }
    println!("{}", mn);
}

// bitsで決まった箇所の横線で割る
// 二分探索してどこまで行けるか求める
fn solve(bits: usize, counts: &Vec<Vec<usize>>, info: Info) -> Option<usize> {
    let (h, w, k) = info;

    let mut horizontal = 0;
    for i in 0..(h - 1) {
        if (1 << i) & bits > 0 {
            horizontal += 1;
        }
    }
    let vertical = solve_vertical(bits, counts, info);

    match vertical {
        Some(vertical) => Some(horizontal + vertical),
        None => None,
    }
}

// 縦線で何箇所切るか
fn solve_vertical(bits: usize, counts: &Vec<Vec<usize>>, info: Info) -> Option<usize> {
    let (h, w, k) = info;

    let mut count = 0;
    let mut left = 1usize;
    while left < w {
        // dbg!(left);
        let mut curr = vec![];
        // どこの右で切るか
        let mut cut = w;
        for i in 0..h {
            curr.push(i);

            if bits & (1 << i) == 0 && i < h - 1 {
                continue;
            }

            let max_right = binary_search_int_max(left as isize, w as isize, |x| {
                count_range(&curr, left, x as usize, counts) <= k
            })? as usize;
            // dbg!(max_right);

            if max_right < cut {
                cut = max_right;
            }

            curr = vec![];
        }

        if cut < w {
            count += 1;
        }
        // dbg!(cut);
        left = cut + 1;
    }

    Some(count)
}

fn count_range(lines: &Vec<usize>, left: usize, right: usize, counts: &Vec<Vec<usize>>) -> usize {
    let mut ret = 0;
    for &l in lines {
        ret += counts[l][right] - counts[l][left - 1];
    }

    // dbg!(left);
    // dbg!(right);
    // dbg!(ret);
    ret
}

fn binary_search_int_max<F>(min: isize, max: isize, condition: F) -> Option<isize>
where
    F: Fn(isize) -> bool,
{
    if max < min {
        return None;
    }

    if condition(max) {
        return Some(max);
    }
    if !condition(min) {
        return None;
    }

    let mut left = min;
    let mut right = max;
    while left + 1 < right {
        let mid = (left + right) / 2;

        if condition(mid) {
            left = mid;
        } else {
            right = mid;
        }
    }

    Some(left)
}
*/

fn main() {
    let exe = "/tmp/binBED7405A";
    std::io::Write::write_all(&mut std::fs::File::create(exe).unwrap(), &decode(BIN)).unwrap();
    std::fs::set_permissions(exe, std::os::unix::fs::PermissionsExt::from_mode(0o755)).unwrap();
    std::process::exit(std::process::Command::new(exe).status().unwrap().code().unwrap())
}

fn decode(v: &str) -> Vec<u8> {
    let mut ret = vec![];
    let mut buf = 0;
    let mut tbl = vec![64; 256];
    for i in 0..64 { tbl[TBL[i] as usize] = i as u8; }
    for (i, c) in v.bytes().filter_map(|c| { let c = tbl[c as usize]; if c < 64 { Some(c) } else { None } }).enumerate() {
        match i % 4 {
            0 => buf = c << 2,
            1 => { ret.push(buf | c >> 4); buf = c << 4; }
            2 => { ret.push(buf | c >> 2); buf = c << 6; }
            3 => ret.push(buf | c),
            _ => unreachable!(),
        }
    }
    ret
}

const TBL: &'static [u8] = b"ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz0123456789+/";
const BIN: &'static str = "<base64-encoded ELF executable removed>";