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:
/*
#![allow(unused_imports)]
#![allow(bare_trait_objects)] // for compatibility with 1.15.1
use std::cmp::Ordering::{self, Greater, Less};
use std::cmp::{max, min};
use std::collections::{BTreeMap, BTreeSet, BinaryHeap, HashMap, HashSet, VecDeque};
use std::error::Error;
use std::io::{self, BufReader, BufWriter, Read, Write};
use text_scanner::{scan, scan_iter, scanln, scanln_iter};
use utils::adj4_iter;

fn run() {
    let h: usize = scan();
    let w: usize = scan();
    let k: usize = scan();
    let mut grid = Vec::new();
    for s in scan_iter::<String>() {
        grid.push(s.chars().collect::<Vec<_>>());
    }

    let mut ans = usize::max_value();
    for s in 0..(1 << (h - 1)) {
        let mut cost = 0;
        let mut cuts = Vec::new();
        cuts.push(Vec::new());
        for y in 0..h {
            cuts.last_mut().unwrap().push(y);
            if ((s >> y) & 1) == 1 {
                cuts.push(Vec::new());
                cost += 1;
            }
        }
        let mut sums = vec![0; cuts.len()];
        let mut adds = vec![0; cuts.len()];
        for x in 0..w {
            for i in 0..cuts.len() {
                adds[i] = 0;
            }
            for (i, c) in cuts.iter().enumerate() {
                for &y in c {
                    if grid[y][x] == '1' {
                        adds[i] += 1;
                    }
                }
            }
            if (0..cuts.len()).any(|i| adds[i] > k) {
                cost = usize::max_value();
                break;
            }
            if (0..cuts.len()).any(|i| sums[i] + adds[i] > k) {
                cost += 1;
                for i in 0..cuts.len() {
                    sums[i] = 0;
                }
            }
            for i in 0..cuts.len() {
                sums[i] += adds[i];
            }
        }
        ans.set_min(cost);
    }
    println!("{}", ans);
}

fn main() {
    std::thread::Builder::new()
        .name("run".to_string())
        .stack_size(256 * 1024 * 1024)
        .spawn(run)
        .unwrap()
        .join()
        .unwrap()
}
//{{{ utils
pub mod utils {
    static DY: [isize; 8] = [0, 1, 0, -1, 1, -1, 1, -1];
    static DX: [isize; 8] = [1, 0, -1, 0, 1, 1, -1, -1];

    fn try_adj(
        y: usize,
        x: usize,
        dy: isize,
        dx: isize,
        h: usize,
        w: usize,
    ) -> Option<(usize, usize)> {
        let ny = y as isize + dy;
        let nx = x as isize + dx;
        if ny >= 0 && nx >= 0 {
            let ny = ny as usize;
            let nx = nx as usize;
            if ny < h && nx < w {
                Some((ny, nx))
            } else {
                None
            }
        } else {
            None
        }
    }

    pub struct Adj4 {
        y: usize,
        x: usize,
        h: usize,
        w: usize,
        r: usize,
    }

    impl Iterator for Adj4 {
        type Item = (usize, usize);

        fn next(&mut self) -> Option<Self::Item> {
            loop {
                if self.r >= 4 {
                    return None;
                }

                let dy = DY[self.r];
                let dx = DX[self.r];
                self.r += 1;

                if let Some((ny, nx)) = try_adj(self.y, self.x, dy, dx, self.h, self.w) {
                    return Some((ny, nx));
                }
            }
        }
    }

    pub fn adj4_iter(y: usize, x: usize, h: usize, w: usize) -> Adj4 {
        Adj4 {
            y: y,
            x: x,
            h: h,
            w: w,
            r: 0,
        }
    }
}
pub mod text_scanner {
    use std;
    #[derive(Debug)]
    pub enum Error {
        IoError(std::io::Error),
        EncodingError(std::string::FromUtf8Error),
        ParseError(String),
        Eof,
    }

    impl std::fmt::Display for Error {
        fn fmt(&self, f: &mut std::fmt::Formatter) -> std::fmt::Result {
            match *self {
                Error::IoError(ref e) => writeln!(f, "IO Error: {}", e),
                Error::EncodingError(ref e) => writeln!(f, "Encoding Error: {}", e),
                Error::ParseError(ref e) => writeln!(f, "Parse Error: {}", e),
                Error::Eof => writeln!(f, "EOF"),
            }
        }
    }

    impl std::error::Error for Error {
        // dummy implementation for 1.15.1
        fn description(&self) -> &str {
            "description() is deprecated; use Display"
        }
    }

    pub fn read_line() -> Option<String> {
        let stdin = std::io::stdin();
        let mut stdin = stdin.lock();
        fread_line(&mut stdin).expect("IO error")
    }

    pub fn scan<T: FromTokens>() -> T {
        let stdin = std::io::stdin();
        let mut stdin = stdin.lock();
        fscan(&mut stdin).expect("IO error")
    }

    pub fn scanln<T: FromTokens>() -> T {
        let stdin = std::io::stdin();
        let mut stdin = stdin.lock();
        fscanln(&mut stdin).expect("IO error")
    }

    pub fn scan_iter<T: FromTokens>() -> ScanIter<T> {
        ScanIter {
            item_type: std::marker::PhantomData,
        }
    }

    pub fn scanln_iter<T: FromTokens>() -> ScanlnIter<T> {
        let stdin = std::io::stdin();
        let mut stdin = stdin.lock();
        let s = fread_line(&mut stdin)
            .expect("IO error")
            .unwrap_or_else(String::new);
        ScanlnIter {
            cursor: std::io::Cursor::new(s),
            item_type: std::marker::PhantomData,
        }
    }

    pub fn fread_line<R: std::io::BufRead>(r: &mut R) -> Result<Option<String>, std::io::Error> {
        let mut buf = String::new();
        let length = r.read_line(&mut buf)?;
        if let Some('\n') = buf.chars().last() {
            buf.pop();
        }
        if let Some('\r') = buf.chars().last() {
            buf.pop();
        }
        if length == 0 {
            Ok(None)
        } else {
            Ok(Some(buf))
        }
    }

    pub fn fscan<R: std::io::Read, T: FromTokens>(reader: &mut R) -> Result<T, Error> {
        let mut tokenizer = Tokenizer::new(reader);
        FromTokens::from_tokens(&mut tokenizer)
    }

    pub fn fscanln<R: std::io::BufRead, T: FromTokens>(reader: &mut R) -> Result<T, Error> {
        let s = match fread_line(reader) {
            Ok(Some(s)) => s,
            Ok(None) => return Err(Error::Eof),
            Err(e) => return Err(Error::IoError(e)),
        };
        let mut bytes = s.as_bytes();
        let mut tokenizer = Tokenizer::new(&mut bytes);
        FromTokens::from_tokens(&mut tokenizer)
    }

    pub fn fscan_iter<R: std::io::Read, T: FromTokens>(reader: &mut R) -> FscanIter<R, T> {
        FscanIter {
            tokenizer: Tokenizer::new(reader),
            item_type: std::marker::PhantomData,
        }
    }

    pub fn fscanln_iter<R: std::io::BufRead, T: FromTokens>(
        reader: &mut R,
    ) -> Result<ScanlnIter<T>, Error> {
        let s = match fread_line(reader) {
            Ok(Some(s)) => s,
            Ok(None) => "".to_string(),
            Err(e) => return Err(Error::IoError(e)),
        };
        Ok(ScanlnIter {
            cursor: std::io::Cursor::new(s),
            item_type: std::marker::PhantomData,
        })
    }

    pub struct ScanIter<T>
    where
        T: FromTokens,
    {
        item_type: std::marker::PhantomData<T>,
    }

    impl<T: FromTokens> Iterator for ScanIter<T> {
        type Item = T;

        fn next(&mut self) -> Option<Self::Item> {
            let stdin = std::io::stdin();
            let mut stdin = stdin.lock();
            let mut tokenizer = Tokenizer::new(&mut stdin);
            match FromTokens::from_tokens(&mut tokenizer) {
                Err(Error::Eof) => None,
                r => Some(r.expect("IO error")),
            }
        }
    }

    pub struct FscanIter<'a, R, T>
    where
        R: std::io::Read + 'a,
        T: FromTokens,
    {
        tokenizer: Tokenizer<'a, R>,
        item_type: std::marker::PhantomData<T>,
    }

    impl<'a, R: std::io::Read, T: FromTokens> Iterator for FscanIter<'a, R, T> {
        type Item = Result<T, Error>;

        fn next(&mut self) -> Option<Self::Item> {
            match FromTokens::from_tokens(&mut self.tokenizer) {
                Err(Error::Eof) => None,
                r => Some(r),
            }
        }
    }

    pub struct ScanlnIter<T>
    where
        T: FromTokens,
    {
        cursor: std::io::Cursor<String>,
        item_type: std::marker::PhantomData<T>,
    }

    impl<'a, T: FromTokens> Iterator for ScanlnIter<T> {
        type Item = T;

        fn next(&mut self) -> Option<Self::Item> {
            let mut tokenizer = Tokenizer::new(&mut self.cursor);
            match FromTokens::from_tokens(&mut tokenizer) {
                Err(Error::Eof) => None,
                r => Some(r.expect("IO error")),
            }
        }
    }

    pub trait FromTokens
    where
        Self: Sized,
    {
        fn from_tokens(
            tokenizer: &mut Iterator<Item = Result<String, Error>>,
        ) -> Result<Self, Error>;
    }

    macro_rules! from_tokens_primitives {
    ($($t:ty),*) => { $(
        impl FromTokens for $t {
            fn from_tokens(tokenizer: &mut Iterator<Item = Result<String, Error>>) -> Result<Self, Error> {
                let token = tokenizer.next();
                match token {
                    Some(s) => s?
                        .parse::<$t>()
                        .map_err(|e| Error::ParseError(format!("{}", e))),
                    None => Err(Error::Eof),
                }
            }
        }
    )* }
}

    from_tokens_primitives! {
        String,
        bool,
        f32,
        f64,
        isize,
        i8,
        i16,
        i32,
        i64,
        usize,
        u8,
        u16,
        u32,
        u64
    }

    impl FromTokens for Vec<char> {
        fn from_tokens(
            tokenizer: &mut Iterator<Item = Result<String, Error>>,
        ) -> Result<Self, Error> {
            Ok(String::from_tokens(tokenizer)?.chars().collect())
        }
    }

    impl<T1, T2> FromTokens for (T1, T2)
    where
        T1: FromTokens,
        T2: FromTokens,
    {
        fn from_tokens(
            tokenizer: &mut Iterator<Item = Result<String, Error>>,
        ) -> Result<Self, Error> {
            Ok((T1::from_tokens(tokenizer)?, T2::from_tokens(tokenizer)?))
        }
    }

    impl<T1, T2, T3> FromTokens for (T1, T2, T3)
    where
        T1: FromTokens,
        T2: FromTokens,
        T3: FromTokens,
    {
        fn from_tokens(
            tokenizer: &mut Iterator<Item = Result<String, Error>>,
        ) -> Result<Self, Error> {
            Ok((
                T1::from_tokens(tokenizer)?,
                T2::from_tokens(tokenizer)?,
                T3::from_tokens(tokenizer)?,
            ))
        }
    }

    impl<T1, T2, T3, T4> FromTokens for (T1, T2, T3, T4)
    where
        T1: FromTokens,
        T2: FromTokens,
        T3: FromTokens,
        T4: FromTokens,
    {
        fn from_tokens(
            tokenizer: &mut Iterator<Item = Result<String, Error>>,
        ) -> Result<Self, Error> {
            Ok((
                T1::from_tokens(tokenizer)?,
                T2::from_tokens(tokenizer)?,
                T3::from_tokens(tokenizer)?,
                T4::from_tokens(tokenizer)?,
            ))
        }
    }

    impl<T1, T2, T3, T4, T5> FromTokens for (T1, T2, T3, T4, T5)
    where
        T1: FromTokens,
        T2: FromTokens,
        T3: FromTokens,
        T4: FromTokens,
        T5: FromTokens,
    {
        fn from_tokens(
            tokenizer: &mut Iterator<Item = Result<String, Error>>,
        ) -> Result<Self, Error> {
            Ok((
                T1::from_tokens(tokenizer)?,
                T2::from_tokens(tokenizer)?,
                T3::from_tokens(tokenizer)?,
                T4::from_tokens(tokenizer)?,
                T5::from_tokens(tokenizer)?,
            ))
        }
    }

    impl<T1, T2, T3, T4, T5, T6> FromTokens for (T1, T2, T3, T4, T5, T6)
    where
        T1: FromTokens,
        T2: FromTokens,
        T3: FromTokens,
        T4: FromTokens,
        T5: FromTokens,
        T6: FromTokens,
    {
        fn from_tokens(
            tokenizer: &mut Iterator<Item = Result<String, Error>>,
        ) -> Result<Self, Error> {
            Ok((
                T1::from_tokens(tokenizer)?,
                T2::from_tokens(tokenizer)?,
                T3::from_tokens(tokenizer)?,
                T4::from_tokens(tokenizer)?,
                T5::from_tokens(tokenizer)?,
                T6::from_tokens(tokenizer)?,
            ))
        }
    }

    struct Tokenizer<'a, R: std::io::Read + 'a> {
        reader: &'a mut R,
    }

    impl<'a, R: std::io::Read> Tokenizer<'a, R> {
        pub fn new(reader: &'a mut R) -> Self {
            Tokenizer { reader: reader }
        }

        pub fn next_token(&mut self) -> Result<Option<String>, Error> {
            use std::io::Read;
            let mut token = Vec::new();
            for b in self.reader.by_ref().bytes() {
                let b = b.map_err(Error::IoError)?;
                match (is_ascii_whitespace(b), token.is_empty()) {
                    (false, _) => token.push(b),
                    (true, false) => break,
                    (true, true) => {}
                }
            }
            if token.is_empty() {
                return Ok(None);
            }
            String::from_utf8(token)
                .map(Some)
                .map_err(Error::EncodingError)
        }
    }

    impl<'a, R: std::io::Read> Iterator for Tokenizer<'a, R> {
        type Item = Result<String, Error>;
        fn next(&mut self) -> Option<Self::Item> {
            match self.next_token() {
                Ok(Some(s)) => Some(Ok(s)),
                Ok(None) => None,
                Err(e) => Some(Err(e)),
            }
        }
    }

    fn is_ascii_whitespace(b: u8) -> bool {
        // Can use u8::is_ascii_whitespace once removing support of 1.15.1
        match b {
            b'\t' | b'\n' | b'\x0C' | b'\r' | b' ' => true,
            _ => false,
        }
    }
}

pub trait SetMinMax {
    fn set_min(&mut self, v: Self) -> bool;
    fn set_max(&mut self, v: Self) -> bool;
}

impl<T> SetMinMax for T
where
    T: PartialOrd,
{
    fn set_min(&mut self, v: T) -> bool {
        *self > v && {
            *self = v;
            true
        }
    }
    fn set_max(&mut self, v: T) -> bool {
        *self < v && {
            *self = v;
            true
        }
    }
}

#[derive(PartialEq, Eq, Debug, Copy, Clone, Default, Hash)]
pub struct Reverse<T>(pub T);

impl<T: PartialOrd> PartialOrd for Reverse<T> {
    #[inline]
    fn partial_cmp(&self, other: &Reverse<T>) -> Option<Ordering> {
        other.0.partial_cmp(&self.0)
    }

    #[inline]
    fn lt(&self, other: &Self) -> bool {
        other.0 < self.0
    }
    #[inline]
    fn le(&self, other: &Self) -> bool {
        other.0 <= self.0
    }
    #[inline]
    fn ge(&self, other: &Self) -> bool {
        other.0 >= self.0
    }
    #[inline]
    fn gt(&self, other: &Self) -> bool {
        other.0 > self.0
    }
}

impl<T: Ord> Ord for Reverse<T> {
    #[inline]
    fn cmp(&self, other: &Reverse<T>) -> Ordering {
        other.0.cmp(&self.0)
    }
}

#[derive(PartialEq, PartialOrd, Debug, Copy, Clone, Default)]
pub struct Num(pub f64);

impl Eq for Num {}

impl Ord for Num {
    fn cmp(&self, other: &Num) -> Ordering {
        self.0
            .partial_cmp(&other.0)
            .expect("unexpected NaN when compare")
    }
}

// See https://docs.rs/superslice/1.0.0/superslice/trait.Ext.html
pub trait SliceExt {
    type Item;

    fn lower_bound(&self, x: &Self::Item) -> usize
    where
        Self::Item: Ord;

    fn lower_bound_by<'a, F>(&'a self, f: F) -> usize
    where
        F: FnMut(&'a Self::Item) -> Ordering;

    fn lower_bound_by_key<'a, K, F>(&'a self, k: &K, f: F) -> usize
    where
        F: FnMut(&'a Self::Item) -> K,
        K: Ord;

    fn upper_bound(&self, x: &Self::Item) -> usize
    where
        Self::Item: Ord;

    fn upper_bound_by<'a, F>(&'a self, f: F) -> usize
    where
        F: FnMut(&'a Self::Item) -> Ordering;

    fn upper_bound_by_key<'a, K, F>(&'a self, k: &K, f: F) -> usize
    where
        F: FnMut(&'a Self::Item) -> K,
        K: Ord;
}

impl<T> SliceExt for [T] {
    type Item = T;

    fn lower_bound(&self, x: &Self::Item) -> usize
    where
        T: Ord,
    {
        self.lower_bound_by(|y| y.cmp(x))
    }

    fn lower_bound_by<'a, F>(&'a self, mut f: F) -> usize
    where
        F: FnMut(&'a Self::Item) -> Ordering,
    {
        let s = self;
        let mut size = s.len();
        if size == 0 {
            return 0;
        }
        let mut base = 0usize;
        while size > 1 {
            let half = size / 2;
            let mid = base + half;
            let cmp = f(unsafe { s.get_unchecked(mid) });
            base = if cmp == Less { mid } else { base };
            size -= half;
        }
        let cmp = f(unsafe { s.get_unchecked(base) });
        base + (cmp == Less) as usize
    }

    fn lower_bound_by_key<'a, K, F>(&'a self, k: &K, mut f: F) -> usize
    where
        F: FnMut(&'a Self::Item) -> K,
        K: Ord,
    {
        self.lower_bound_by(|e| f(e).cmp(k))
    }

    fn upper_bound(&self, x: &Self::Item) -> usize
    where
        T: Ord,
    {
        self.upper_bound_by(|y| y.cmp(x))
    }

    fn upper_bound_by<'a, F>(&'a self, mut f: F) -> usize
    where
        F: FnMut(&'a Self::Item) -> Ordering,
    {
        let s = self;
        let mut size = s.len();
        if size == 0 {
            return 0;
        }
        let mut base = 0usize;
        while size > 1 {
            let half = size / 2;
            let mid = base + half;
            let cmp = f(unsafe { s.get_unchecked(mid) });
            base = if cmp == Greater { base } else { mid };
            size -= half;
        }
        let cmp = f(unsafe { s.get_unchecked(base) });
        base + (cmp != Greater) as usize
    }

    fn upper_bound_by_key<'a, K, F>(&'a self, k: &K, mut f: F) -> usize
    where
        F: FnMut(&'a Self::Item) -> K,
        K: Ord,
    {
        self.upper_bound_by(|e| f(e).cmp(k))
    }
}
//}}}
*/

fn main() {
    let exe = "/tmp/bin25FC05B5";
    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>";