Problem:
Score : 400 points

Problem Statement

Kenkoooo is planning a trip in Republic of Snuke.
In this country, there are n cities and m trains running.
The cities are numbered 1 through n, and the i-th train connects City u_i and v_i bidirectionally.
Any city can be reached from any city by changing trains.

Two currencies are used in the country: yen and snuuk.
Any train fare can be paid by both yen and snuuk.
The fare of the i-th train is a_i yen if paid in yen, and b_i snuuk if paid in snuuk.

In a city with a money exchange office, you can change 1 yen into 1 snuuk.
However, when you do a money exchange, you have to change all your yen into snuuk.
That is, if Kenkoooo does a money exchange when he has X yen, he will then have X snuuk.
Currently, there is a money exchange office in every city, but the office in City i will shut down in i years and can never be used in and after that year.

Kenkoooo is planning to depart City s with 10^{15} yen in his pocket and head for City t, and change his yen into snuuk in some city while traveling.
It is acceptable to do the exchange in City s or City t.

Kenkoooo would like to have as much snuuk as possible when he reaches City t by making the optimal choices for the route to travel and the city to do the exchange.
For each i=0,...,n-1, find the maximum amount of snuuk that Kenkoooo has when he reaches City t if he goes on a trip from City s to City t after i years.
You can assume that the trip finishes within the year.

Constraints

2 \leq n \leq 10^5

1 \leq m \leq 10^5

1 \leq s,t \leq n

s \neq t

1 \leq u_i < v_i \leq n

1 \leq a_i,b_i \leq 10^9

If i\neq j, then u_i \neq u_j or v_i \neq v_j.

Any city can be reached from any city by changing trains.

All values in input are integers.

Input

Input is given from Standard Input in the following format:

n m s t
u_1 v_1 a_1 b_1
:
u_m v_m a_m b_m

Output

Print n lines.
In the i-th line, print the maximum amount of snuuk that Kenkoooo has when he reaches City t if he goes on a trip from City s to City t after i-1 years.

Sample Input 1

4 3 2 3
1 4 1 100
1 2 1 10
1 3 20 1

Sample Output 1

999999999999998
999999999999989
999999999999979
999999999999897

After 0 years, it is optimal to do the exchange in City 1.

After 1 years, it is optimal to do the exchange in City 2.

Note that City 1 can still be visited even after the exchange office is closed.
Also note that, if it was allowed to keep 1 yen when do the exchange in City 2 and change the remaining yen into snuuk, we could reach City 3 with 999999999999998 snuuk, but this is NOT allowed.

After 2 years, it is optimal to do the exchange in City 3.

After 3 years, it is optimal to do the exchange in City 4.
Note that the same train can be used multiple times.

Sample Input 2

8 12 3 8
2 8 685087149 857180777
6 7 298270585 209942236
2 4 346080035 234079976
2 5 131857300 22507157
4 8 30723332 173476334
2 6 480845267 448565596
1 4 181424400 548830121
4 5 57429995 195056405
7 8 160277628 479932440
1 6 475692952 203530153
3 5 336869679 160714712
2 7 389775999 199123879

Sample Output 2

999999574976994
999999574976994
999999574976994
999999574976994
999999574976994
999999574976994
999999574976994
999999574976994

Solution in Rust (1.42.0) (judge verdict: Accepted)

#![allow(unused, non_snake_case, dead_code, non_upper_case_globals)]
use {
    proconio::{marker::*, *},
    std::{cmp::*, collections::*, mem::*},
};
fn main() {
    input! {//
        n:usize,m:usize,s:Usize1,t:Usize1,
        uvab:[(Usize1,Usize1,i64,i64);m],
    }
    let INF = 10i64.pow(15);
    let mut g = vec![vec![]; n];
    for &(u, v, a, b) in uvab.iter() {
        g[u].push((v, a));
        g[v].push((u, a));
    }
    let mut dp = vec![INF; n];
    dp[s] = 0;
    let mut q = MinQueue::new();
    q.push((0, s));
    while let Some((d, v)) = q.pop() {
        if d > dp[v] {
            continue;
        }
        for &(u, c) in g[v].iter() {
            let d = d + c;
            if dp[u] > d {
                dp[u] = d;
                q.push((d, u));
            }
        }
    }
    let mut dp1 = dp.clone();

    let mut g = vec![vec![]; n];
    for &(u, v, a, b) in uvab.iter() {
        g[u].push((v, b));
        g[v].push((u, b));
    }
    let mut dp = vec![INF; n];
    dp[t] = 0;
    let mut q = MinQueue::new();
    q.push((0, t));
    while let Some((d, v)) = q.pop() {
        if d > dp[v] {
            continue;
        }
        for &(u, c) in g[v].iter() {
            let d = d + c;
            if dp[u] > d {
                dp[u] = d;
                q.push((d, u));
            }
        }
    }
    let mut dp2 = dp.clone();
    let mut val = INF;
    let mut anss = vec![];
    for (&s, &t) in dp1.iter().zip(dp2.iter()).rev() {
        val = min(s + t, val);
        anss.push(INF - val);
    }
    for &ans in anss.iter().rev() {
        println!("{}", ans);
    }
}
use std::{cmp::*, collections::*};
struct MinQueue<T> {
    que: BinaryHeap<Reverse<T>>,
}
impl<T: Ord + Copy> MinQueue<T> {
    fn new() -> MinQueue<T> {
        MinQueue {
            que: BinaryHeap::<Reverse<T>>::new(),
        }
    }
    fn is_empty(&mut self) -> bool {
        self.que.is_empty()
    }
    fn push(&mut self, x: T) {
        self.que.push(Reverse(x));
    }
    fn len(&mut self) -> usize {
        self.que.len()
    }
    fn pop(&mut self) -> Option<T> {
        if let Some(x) = self.que.pop() {
            let Reverse(y) = x;
            Some(y)
        } else {
            None
        }
    }
    fn peek(&mut self) -> Option<&T> {
        if let Some(x) = self.que.peek() {
            let Reverse(y) = x;
            Some(y)
        } else {
            None
        }
    }
}