Problem:
Score : 600 points

Problem Statement

Snuke is standing on a two-dimensional plane. In one operation, he can move by 1 in the positive x-direction, or move by 1 in the positive y-direction.

Let us define a function f(r, c) as follows:

f(r,c) := (The number of paths from the point (0, 0) to the point (r, c) that Snuke can trace by repeating the operation above)

Given are integers r_1, r_2, c_1, and c_2.
Find the sum of f(i, j) over all pair of integers (i, j) such that r_1 ≤ i ≤ r_2 and c_1 ≤ j ≤ c_2, and compute this value modulo (10^9+7).

Constraints

1 ≤ r_1 ≤ r_2 ≤ 10^6

1 ≤ c_1 ≤ c_2 ≤ 10^6

All values in input are integers.

Input

Input is given from Standard Input in the following format:

r_1 c_1 r_2 c_2

Output

Print the sum of f(i, j) modulo (10^9+7).

Sample Input 1

1 1 2 2

Sample Output 1

14

For example, there are two paths from the point (0, 0) to the point (1, 1): (0,0) → (0,1) → (1,1) and (0,0) → (1,0) → (1,1), so f(1,1)=2.

Similarly, f(1,2)=3, f(2,1)=3, and f(2,2)=6. Thus, the sum is 14.

Sample Input 2

314 159 2653 589

Sample Output 2

602215194

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

#[allow(unused_imports)]
use std::cmp::{Ordering};

#[allow(unused_imports)]
use std::collections::{BTreeMap, BTreeSet, BinaryHeap, HashMap, HashSet, VecDeque};

#[allow(unused_imports)]
use std::io::{stdin, stdout, BufWriter, Write};

#[allow(unused_imports)]
use std::iter::FromIterator;

#[allow(unused_macros)]
macro_rules! min {
    ($x: expr) => ($x);
    ($x: expr, $($z: expr),+) => {{
        std::cmp::min($x, min!($($z),*))
    }}
}

#[allow(unused_macros)]
macro_rules! max {
    ($x: expr) => ($x);
    ($x: expr, $($z: expr),+) => {{
        std::cmp::max($x, max!($($z),*))
    }}
}

#[allow(unused_macros)]
macro_rules! get { 
    ([$t: ty]) => { 
        { 
            let mut line = String::new(); 
            stdin().read_line(&mut line).unwrap(); 
            line.split_whitespace().map(|t|t.parse::<$t>().unwrap()).collect::<Vec<_>>()
        }
    };
    ([$t: ty];$n: expr) => {
        (0..$n).map(|_|get!([$t])).collect::<Vec<_>>()
    };
    ($t: ty) => {
        {
            let mut line = String::new();
            stdin().read_line(&mut line).unwrap();
            line.trim().parse::<$t>().unwrap()
        }
    };
    ($($t: ty),*) => {
        { 
            let mut line = String::new();
            stdin().read_line(&mut line).unwrap();
            let mut iter = line.split_whitespace();
            ($(iter.next().unwrap().parse::<$t>().unwrap(),)*)
        }
    };
    ($t: ty; $n: expr) => {
        (0..$n).map(|_|get!($t)).collect::<Vec<_>>()
    };
    ($($t: ty),*; $n: expr) => {
        (0..$n).map(|_|get!($($t),*)).collect::<Vec<_>>()
    };
}

#[allow(unused_macros)]
macro_rules! debug {
    ($($a:expr),*) => {
        #[cfg(debug_assertions)]
        writeln!(&mut std::io::stderr(), concat!("[DEBUG] ", $(stringify!($a), "={:?} "),*), $($a),*).unwrap();
    }
}

const BIG_STACK_SIZE: bool = true;

#[allow(dead_code)]
fn main() {
    use std::thread;
    if BIG_STACK_SIZE {
        thread::Builder::new()
            .stack_size(32 * 1024 * 1024)
            .name("solve".into())
            .spawn(solve)
            .unwrap()
            .join()
            .unwrap();
    } else {
        solve();
    }
}

struct Comb {
    div: usize,
    g1: Vec<usize>,
    g2: Vec<usize>,
}
impl Comb {
#[allow(dead_code)]
    fn new(nmax: usize) -> Self {
        let mut g1 = vec![1; nmax + 1];
        let mut g2 = vec![1; nmax + 1];
        let mut inv = vec![0; nmax + 1];
        inv[1] = 1;
        let div = 1000000007;
        for i in 2..(nmax + 1) {
            g1[i] = g1[i - 1] * i % div;
            inv[i] = div - (inv[div % i] * (div / i)) % div;
            g2[i] = (g2[i - 1] * inv[i]) % div;
        }
        Comb {
            div: div,
            g1: g1,
            g2: g2,
        }
    }
#[allow(dead_code)]
    fn compute(&self, n: usize, k: usize) -> usize {
        (((self.g1[n] * self.g2[k]) % self.div) * self.g2[n - k]) % self.div
    }
#[allow(dead_code)]
    fn compute_dupc(&self, n: usize, k: usize) -> usize {
        self.compute(n + k - 1, n - 1)
    }
#[allow(dead_code)]
    fn compuute_perm(&self, n: usize, k: usize) -> usize {
        (self.g1[n] * self.g2[n - k]) % self.div
    }
}

use self::modint::ModInt;
mod modint {
    use std;
#[derive(Copy, Clone, PartialEq, Eq, PartialOrd, Ord)]
    pub struct ModInt(usize);
    impl ModInt {
        pub fn new(value: usize) -> Self {
            ModInt(value)
        }
        fn modulo() -> usize {
            1_000_000_007
        }
    }
    impl std::ops::Add for ModInt {
        type Output = Self;
        fn add(self, other: Self) -> Self {
            let mut value = self.0 + other.0;
            if value > Self::modulo() {
                value -= Self::modulo();
            }
            ModInt(value)
        }
    }
    impl std::ops::Add<usize> for ModInt {
        type Output = Self;
        fn add(self, other: usize) -> Self {
            let mut value = self.0 + other;
            if value > Self::modulo() {
                value -= Self::modulo();
            }
            ModInt(value)
        }
    }
    impl std::ops::Sub for ModInt {
        type Output = Self;
        fn sub(self, other: Self) -> Self {
            let mut value = self.0;
            if value < other.0 {
                value += Self::modulo();
            }
            value -= other.0;
            ModInt(value)
        }
    }
    impl std::ops::Sub<usize> for ModInt {
        type Output = Self;
        fn sub(self, other: usize) -> Self {
            let mut value = self.0;
            if value < other {
                value += Self::modulo();
            }
            value -= other;
            ModInt(value)
        }
    }
    impl std::ops::Mul for ModInt {
        type Output = Self;
        fn mul(self, other: Self) -> Self {
            let value = self.0 * other.0 % Self::modulo();
            ModInt(value)
        }
    }
    impl std::ops::Mul<usize> for ModInt {
        type Output = Self;
        fn mul(self, other: usize) -> Self {
            let value = self.0 * other % Self::modulo();
            ModInt(value)
        }
    }
    impl std::ops::Div for ModInt {
        type Output = Self;
        fn div(self, other: Self) -> Self {
            let mut exp = Self::modulo() - 2;
            let mut value = self.0;
            let mut ovalue = other.0;
            while exp > 0 {
                if exp % 2 == 1 {
                    value = value * ovalue % Self::modulo();
                }
                ovalue = ovalue * ovalue % Self::modulo();
                exp /= 2;
            }
            ModInt(value)
        }
    }
    impl std::ops::Div<usize> for ModInt {
        type Output = Self;
        fn div(self, other: usize) -> Self {
            let mut exp = Self::modulo() - 2;
            let mut value = self.0;
            let mut ovalue = other;
            while exp > 0 {
                if exp % 2 == 1 {
                    value = value * ovalue % Self::modulo();
                }
                ovalue = ovalue * ovalue % Self::modulo();
                exp /= 2;
            }
            ModInt(value)
        }
    }
    impl std::ops::AddAssign for ModInt {
        fn add_assign(&mut self, other: Self) {
            *self = *self + other;
        }
    }
    impl std::ops::AddAssign<usize> for ModInt {
        fn add_assign(&mut self, other: usize) {
            *self = *self + other;
        }
    }
    impl std::ops::SubAssign for ModInt {
        fn sub_assign(&mut self, other: Self) {
            *self = *self - other;
        }
    }
    impl std::ops::SubAssign<usize> for ModInt {
        fn sub_assign(&mut self, other: usize) {
            *self = *self - other;
        }
    }
    impl std::ops::MulAssign for ModInt {
        fn mul_assign(&mut self, other: Self) {
            *self = *self * other;
        }
    }
    impl std::ops::MulAssign<usize> for ModInt {
        fn mul_assign(&mut self, other: usize) {
            *self = *self * other;
        }
    }
    impl std::ops::DivAssign for ModInt {
        fn div_assign(&mut self, other: Self) {
            *self = *self / other;
        }
    }
    impl std::ops::DivAssign<usize> for ModInt {
        fn div_assign(&mut self, other: usize) {
            *self = *self / other;
        }
    }
    impl std::fmt::Debug for ModInt {
        fn fmt(&self, f: &mut std::fmt::Formatter) -> std::fmt::Result {
            write!(f, "{}", self.0)
        }
    }
    impl std::fmt::Display for ModInt {
        fn fmt(&self, f: &mut std::fmt::Formatter) -> std::fmt::Result {
            write!(f, "{}", self.0)
        }
    }
}

fn solve() {
    let (r1,c1,r2,c2) = get!(usize, usize, usize, usize);
    let comb = Comb::new(r2+c2+2);
    let fcr = |r: usize, c: usize| -> ModInt {
        (ModInt::new(r+2)*ModInt::new(comb.compute(r+c+2,c))-ModInt::new(c+1))/ModInt::new(c+1)
    };
    println!("{}", fcr(r2,c2)-fcr(r2,c1-1)-fcr(r1-1,c2)+fcr(r1-1,c1-1));
}