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)]
#![allow(non_snake_case)]
use std::cmp::*;
use std::collections::*;
use std::io::Write;

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

fn count(r: usize, c: usize, facts: &[Modulo], facts_inv: &[Modulo], summed: &[Modulo]) -> Modulo {
    let mut ret = summed[r + c - 1] - summed[r - 1] - summed[c - 1];
    for ri in 1..r {
        let coef = comb(ri + c - 1, ri, facts, facts_inv);
        ret -= coef * summed[r - ri - 1];
    }
    for ci in 1..c {
        let coef = comb(ci + r - 1, ci, facts, facts_inv);
        ret -= coef * summed[c - ci - 1];
    }
    ret
}

fn main() {
    let v = read_vec::<usize>();
    let (r1, c1, r2, c2) = (v[0] + 1, v[1] + 1, v[2] + 1, v[3] + 1);
    let mut facts = vec![Modulo(1); r2 + c2 + 1];
    for i in 2..r2 + c2 + 1 {
        facts[i] = facts[i - 1] * i as i64;
    }
    let mut facts_inv = vec![Modulo(1); r2 + c2 + 1];
    for i in 0..r2 + c2 + 1 {
        facts_inv[i] = facts[i].inv();
    }
    let mut coef = Modulo(1);
    let mut summed = vec![Modulo(0); r2 + c2];
    for i in 1..r2 + c2 {
        summed[i] = summed[i - 1] + coef;
        coef *= 2;
    }
    let ans = count(r2, c2, &facts, &facts_inv, &summed)
        - count(r2, c1 - 1, &facts, &facts_inv, &summed)
        - count(r1 - 1, c2, &facts, &facts_inv, &summed)
        + count(r1 - 1, c1 - 1, &facts, &facts_inv, &summed);
    println!("{}", ans);
}

fn read<T: std::str::FromStr>() -> T {
    let mut s = String::new();
    std::io::stdin().read_line(&mut s).ok();
    s.trim().parse().ok().unwrap()
}

fn read_vec<T: std::str::FromStr>() -> Vec<T> {
    read::<String>()
        .split_whitespace()
        .map(|e| e.parse().ok().unwrap())
        .collect()
}

#[derive(Clone, Copy, Debug, Default, Eq, Hash, Ord, PartialEq, PartialOrd)]
struct Modulo(i64);
static mut MODULUS: i64 = 1000_000_000 + 7;
impl Modulo {
    fn set_modulus(m: i64) {
        unsafe {
            MODULUS = m;
        }
    }
    fn get_modulus() -> i64 {
        unsafe { MODULUS }
    }
    fn new(x: i64) -> Modulo {
        let m = Modulo::get_modulus();
        if x < 0 {
            Modulo(x % m + m)
        } else if x < m {
            Modulo(x)
        } else {
            Modulo(x % m)
        }
    }
    fn pow(self, p: i64) -> Modulo {
        if p == 0 {
            Modulo(1)
        } else {
            let mut t = self.pow(p / 2);
            t *= t;
            if p & 1 == 1 {
                t *= self;
            }
            t
        }
    }
    fn inv(self) -> Modulo {
        self.pow(Modulo::get_modulus() - 2)
    }
}
impl std::fmt::Display for Modulo {
    fn fmt(&self, f: &mut std::fmt::Formatter) -> std::fmt::Result {
        self.0.fmt(f)
    }
}
impl std::ops::AddAssign for Modulo {
    fn add_assign(&mut self, other: Modulo) {
        let m = Modulo::get_modulus();
        self.0 += other.0;
        if self.0 >= m {
            self.0 -= m;
        }
    }
}
impl std::ops::MulAssign for Modulo {
    fn mul_assign(&mut self, other: Modulo) {
        let m = Modulo::get_modulus();
        self.0 *= other.0;
        self.0 %= m;
    }
}
impl std::ops::SubAssign for Modulo {
    fn sub_assign(&mut self, other: Modulo) {
        let m = Modulo::get_modulus();
        self.0 += m - other.0;
        if self.0 >= m {
            self.0 -= m;
        }
    }
}
macro_rules! impl_modulo_ops {
    ($imp:ident, $method:ident, $assign_imp:ident, $assign_method:ident) => {
        impl<'a> std::ops::$assign_imp<&'a Modulo> for Modulo {
            fn $assign_method(&mut self, other: &'a Modulo) {
                std::ops::$assign_imp::$assign_method(self, *other);
            }
        }
        impl std::ops::$imp for Modulo {
            type Output = Modulo;
            fn $method(self, other: Modulo) -> Modulo {
                let mut x = self;
                std::ops::$assign_imp::$assign_method(&mut x, other);
                x
            }
        }
        impl<'a> std::ops::$imp<Modulo> for &'a Modulo {
            type Output = Modulo;
            fn $method(self, other: Modulo) -> Modulo {
                std::ops::$imp::$method(*self, other)
            }
        }
        impl<'a> std::ops::$imp<&'a Modulo> for Modulo {
            type Output = Modulo;
            fn $method(self, other: &'a Modulo) -> Modulo {
                std::ops::$imp::$method(self, *other)
            }
        }
        impl<'a, 'b> std::ops::$imp<&'b Modulo> for &'a Modulo {
            type Output = Modulo;
            fn $method(self, other: &'b Modulo) -> Modulo {
                std::ops::$imp::$method(*self, *other)
            }
        }
        impl std::ops::$assign_imp<i64> for Modulo {
            fn $assign_method(&mut self, other: i64) {
                std::ops::$assign_imp::$assign_method(self, Modulo::new(other));
            }
        }
        impl<'a> std::ops::$assign_imp<&'a i64> for Modulo {
            fn $assign_method(&mut self, other: &'a i64) {
                std::ops::$assign_imp::$assign_method(self, *other);
            }
        }
        impl std::ops::$imp<i64> for Modulo {
            type Output = Modulo;
            fn $method(self, other: i64) -> Modulo {
                let mut x = self;
                std::ops::$assign_imp::$assign_method(&mut x, other);
                x
            }
        }
        impl<'a> std::ops::$imp<&'a i64> for Modulo {
            type Output = Modulo;
            fn $method(self, other: &'a i64) -> Modulo {
                std::ops::$imp::$method(self, *other)
            }
        }
        impl<'a> std::ops::$imp<i64> for &'a Modulo {
            type Output = Modulo;
            fn $method(self, other: i64) -> Modulo {
                std::ops::$imp::$method(*self, other)
            }
        }
        impl<'a, 'b> std::ops::$imp<&'b i64> for &'a Modulo {
            type Output = Modulo;
            fn $method(self, other: &'b i64) -> Modulo {
                std::ops::$imp::$method(*self, *other)
            }
        }
    };
}
impl_modulo_ops!(Add, add, AddAssign, add_assign);
impl_modulo_ops!(Mul, mul, MulAssign, mul_assign);
impl_modulo_ops!(Sub, sub, SubAssign, sub_assign);

fn comb(n: usize, k: usize, fact: &[Modulo], fact_inv: &[Modulo]) -> Modulo {
    assert!(n >= k);
    fact[n] * fact_inv[n - k] * fact_inv[k]
}