Problem:
Score : 600 points

Problem Statement

We will create an artwork by painting black some squares in a white square grid with 10^9 rows and N columns.

The current plan is as follows: for the i-th column from the left, we will paint the H_i bottommost squares and will not paint the other squares in that column.

Before starting to work, you can choose at most K columns (possibly zero) and change the values of H_i for these columns to any integers of your choice between 0 and 10^9 (inclusive).

Different values can be chosen for different columns.

Then, you will create the modified artwork by repeating the following operation:

Choose one or more consecutive squares in one row and paint them black. (Squares already painted black can be painted again, but squares not to be painted according to the modified plan should not be painted.)

Find the minimum number of times you need to perform this operation.

Constraints

1 \leq N \leq 300

0 \leq K \leq N

0 \leq H_i \leq 10^9

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N K
H_1 H_2 ... H_N

Output

Print the minimum number of operations required.

Sample Input 1

4 1
2 3 4 1

Sample Output 1

3

For example, by changing the value of H_3 to 2, you can create the modified artwork by the following three operations:

Paint black the 1-st through 4-th squares from the left in the 1-st row from the bottom.

Paint black the 1-st through 3-rd squares from the left in the 2-nd row from the bottom.

Paint black the 2-nd square from the left in the 3-rd row from the bottom.

Sample Input 2

6 2
8 6 9 1 2 1

Sample Output 2

7

Sample Input 3

10 0
1 1000000000 1 1000000000 1 1000000000 1 1000000000 1 1000000000

Sample Output 3

4999999996

Solution in Rust (1.15.1) (judge verdict: Accepted)

use self::mod_int::ModInt;
use std::cmp;
use std::collections::{BTreeMap, BTreeSet, VecDeque};

const MOD: usize = 1e9 as usize + 7;
const INF: i64 = 1 << 60;

fn main() {
    let (r, w) = (std::io::stdin(), std::io::stdout());
    let mut sc = IO::new(r.lock(), w.lock());

    let n: usize = sc.read();
    let k: usize = sc.read();
    let mut dp = BTreeMap::new();
    dp.insert((0, 0), 0);
    for _ in 0..n {
        let next_height: i64 = sc.read();
        let mut next = BTreeMap::new();
        for ((cur_height, used_k), min_cost) in dp.into_iter() {
            if used_k < k {
                let cur = next.entry((cur_height, used_k + 1)).or_insert(min_cost);
                update_min(cur, min_cost);
            }
            let cost = cmp::max(0, next_height - cur_height);
            let cur = next.entry((next_height, used_k)).or_insert(min_cost + cost);
            update_min(cur, min_cost + cost);
        }

        dp = next;
    }
    let ans = dp.into_iter().map(|(_, v)| v).min().unwrap();
    println!("{}", ans);
}

fn update_min<T: PartialOrd>(a: &mut T, b: T) {
    if *a > b {
        *a = b;
    }
}

pub mod mod_int {
    use super::MOD;
    use std::ops::{Add, AddAssign, Div, DivAssign, Mul, MulAssign, Sub, SubAssign};

    type Num = usize;

    #[derive(Clone, Copy)]
    pub struct ModInt<T: Copy + Clone>(pub T);

    impl Add<ModInt<Num>> for ModInt<Num> {
        type Output = ModInt<Num>;
        fn add(self, rhs: ModInt<Num>) -> ModInt<Num> {
            self + rhs.0
        }
    }

    impl Add<Num> for ModInt<Num> {
        type Output = ModInt<Num>;
        fn add(self, mut rhs: Num) -> ModInt<Num> {
            if rhs >= MOD {
                rhs %= MOD;
            }
            let mut t = rhs + self.0;
            if t >= MOD {
                t = t - MOD;
            }
            ModInt(t)
        }
    }

    impl Sub<Num> for ModInt<Num> {
        type Output = ModInt<Num>;
        fn sub(self, rhs: Num) -> ModInt<Num> {
            let rhs = if rhs >= MOD { rhs % MOD } else { rhs };
            let value = if self.0 < rhs { self.0 + MOD } else { self.0 };
            ModInt(value - rhs)
        }
    }

    impl Sub<ModInt<Num>> for ModInt<Num> {
        type Output = ModInt<Num>;
        fn sub(self, rhs: ModInt<Num>) -> ModInt<Num> {
            self - rhs.0
        }
    }

    impl AddAssign<Num> for ModInt<Num> {
        fn add_assign(&mut self, other: Num) {
            *self = *self + other;
        }
    }
    impl AddAssign<ModInt<Num>> for ModInt<Num> {
        fn add_assign(&mut self, other: ModInt<Num>) {
            *self = *self + other;
        }
    }

    impl SubAssign<Num> for ModInt<Num> {
        fn sub_assign(&mut self, other: Num) {
            *self = *self - other;
        }
    }

    impl SubAssign<ModInt<Num>> for ModInt<Num> {
        fn sub_assign(&mut self, other: ModInt<Num>) {
            *self = *self - other;
        }
    }

    impl Div<Num> for ModInt<Num> {
        type Output = ModInt<Num>;
        fn div(self, mut rhs: Num) -> ModInt<Num> {
            if rhs >= MOD {
                rhs %= MOD;
            }
            self * ModInt(rhs).pow(MOD - 2)
        }
    }

    impl Div<ModInt<Num>> for ModInt<Num> {
        type Output = ModInt<Num>;
        fn div(self, rhs: ModInt<Num>) -> ModInt<Num> {
            self / rhs.0
        }
    }

    impl DivAssign<Num> for ModInt<Num> {
        fn div_assign(&mut self, rhs: Num) {
            *self = *self / rhs
        }
    }
    impl DivAssign<ModInt<Num>> for ModInt<Num> {
        fn div_assign(&mut self, rhs: ModInt<Num>) {
            *self = *self / rhs
        }
    }

    impl Mul<ModInt<Num>> for ModInt<Num> {
        type Output = ModInt<Num>;

        fn mul(self, rhs: ModInt<Num>) -> ModInt<Num> {
            self * rhs.0
        }
    }
    impl Mul<Num> for ModInt<Num> {
        type Output = ModInt<Num>;

        fn mul(self, mut rhs: Num) -> ModInt<Num> {
            if rhs >= MOD {
                rhs %= MOD;
            }
            let t = (self.0 * rhs) % MOD;
            ModInt(t)
        }
    }

    impl MulAssign<Num> for ModInt<Num> {
        fn mul_assign(&mut self, rhs: Num) {
            *self = *self * rhs;
        }
    }

    impl MulAssign<ModInt<Num>> for ModInt<Num> {
        fn mul_assign(&mut self, rhs: ModInt<Num>) {
            *self = *self * rhs;
        }
    }

    impl ModInt<Num> {
        pub fn pow(self, e: Num) -> ModInt<Num> {
            let mut result = ModInt(1);
            let mut cur = self;
            let mut e = e;
            while e > 0 {
                if e & 1 == 1 {
                    result *= cur;
                }
                e >>= 1;
                cur *= cur;
            }
            result
        }
    }
}

pub struct IO<R, W: std::io::Write>(R, std::io::BufWriter<W>);

impl<R: std::io::Read, W: std::io::Write> IO<R, W> {
    pub fn new(r: R, w: W) -> Self {
        IO(r, std::io::BufWriter::new(w))
    }
    pub fn write<S: ToString>(&mut self, s: S) {
        use std::io::Write;
        self.1.write(s.to_string().as_bytes()).unwrap();
    }
    pub fn read<T: std::str::FromStr>(&mut self) -> T {
        use std::io::Read;
        let buf = self
            .0
            .by_ref()
            .bytes()
            .map(|b| b.unwrap())
            .skip_while(|&b| b == b' ' || b == b'\n' || b == b'\r' || b == b'\t')
            .take_while(|&b| b != b' ' && b != b'\n' && b != b'\r' && b != b'\t')
            .collect::<Vec<_>>();
        unsafe { std::str::from_utf8_unchecked(&buf) }
            .parse()
            .ok()
            .expect("Parse error.")
    }
    pub fn vec<T: std::str::FromStr>(&mut self, n: usize) -> Vec<T> {
        (0..n).map(|_| self.read()).collect()
    }
    pub fn chars(&mut self) -> Vec<char> {
        self.read::<String>().chars().collect()
    }
}