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 C++ (GCC 9.2.1) (judge verdict: Wrong Answer)

#include <bits/stdc++.h>
using namespace std;

const long long inf = 1000000037;
const long long yens = 1000000000000000;
const int maxn = 100005;

int n, m, s, t, u, v, w, a, b;
vector < pair < int, int > > ga[maxn];
vector < pair < int, int > > gb[maxn];
long long da[maxn], db[maxn], ans[maxn];
set < pair < long long, int > > dset;

signed main () {
	scanf("%d %d %d %d", &n, &m, &s, &t);
	for (int i = 0; i < m; ++i) {
		scanf("%d %d %d %d", &u, &v, &a, &b);
		ga[u].push_back(make_pair(v, a));
		ga[v].push_back(make_pair(u, a));
		gb[u].push_back(make_pair(v, b));
		gb[v].push_back(make_pair(u, b));
	}
	for (int i = 1; i <= n; ++i)
		da[i] = db[i] = inf;
	da[s] = 0LL;
	dset.insert(make_pair(da[s], s));
	while (!dset.empty()) {
		u = dset.begin() -> second;
		dset.erase(dset.begin());
		for (auto pr: ga[u]) {
			v = pr.first, w = pr.second;
			if (da[v] > da[u] + w) {
				dset.erase(make_pair(da[v], v));
				da[v] = da[u] + w;
				dset.insert(make_pair(da[v], v));
			}
		}
	}
	db[t] = 0LL;
	dset.insert(make_pair(db[t], t));
	while (!dset.empty()) {
		u = dset.begin() -> second;
		dset.erase(dset.begin());
		for (auto pr: gb[u]) {
			v = pr.first, w = pr.second;
			if (db[v] > db[u] + w) {
				dset.erase(make_pair(db[v], v));
				db[v] = db[u] + w;
				dset.insert(make_pair(db[v], v));
			}
		}
	}
	ans[n + 1] = inf;
	for (int i = n; i; --i) 
		ans[i] = min(ans[i + 1], da[i] + db[i]);
	for (int i = 1; i <= n; ++i)
		printf("%lld\n", yens - ans[i]);
}