Legacy
Contest: Codeforces Round 406 (Div. 1)
Difficulty rating: 2300
Tags: data structures, graphs, shortest paths

Rick and his co-workers have made a new radioactive formula and a lot of bad guys are after them. So Rick wants to give his legacy to Morty before bad guys catch them.

There are n planets in their universe numbered from 1 to n. Rick is in planet number s (the earth) and he doesn't know where Morty is. As we all know, Rick owns a portal gun. With this gun he can open one-way portal from a planet he is in to any other planet (including that planet). But there are limits on this gun because he's still using its free trial.

By default he can not open any portal by this gun. There are q plans in the website that sells these guns. Every time you purchase a plan you can only use it once but you can purchase it again if you want to use it more.

Plans on the website have three types:

1. With a plan of this type you can open a portal from planet v to planet u.
2. With a plan of this type you can open a portal from planet v to any planet with index in range [l, r].
3. With a plan of this type you can open a portal from any planet with index in range [l, r] to planet v.

Rick doesn't known where Morty is, but Unity is going to inform him and he wants to be prepared for when he finds and start his journey immediately. So for each planet (including earth itself) he wants to know the minimum amount of money he needs to get from earth to that planet.

Input format:
The first line of input contains three integers n, q and s (1 ≤ n, q ≤ 105, 1 ≤ s ≤ n) — number of planets, number of plans and index of earth respectively.

The next q lines contain the plans. Each line starts with a number t, type of that plan (1 ≤ t ≤ 3). If t = 1 then it is followed by three integers v, u and w where w is the cost of that plan (1 ≤ v, u ≤ n, 1 ≤ w ≤ 109). Otherwise it is followed by four integers v, l, r and w where w is the cost of that plan (1 ≤ v ≤ n, 1 ≤ l ≤ r ≤ n, 1 ≤ w ≤ 109).

Output format:
In the first and only line of output print n integers separated by spaces. i-th of them should be minimum money to get from earth to i-th planet, or  - 1 if it's impossible to get to that planet.

Constraints:
Time limit: 2.0 s
Memory limit: 256.0 MB

Note:
In the first sample testcase, Rick can purchase 4th plan once and then 2nd plan in order to get to get to planet number 2.

Example input:
3 5 1
2 3 2 3 17
2 3 2 2 16
2 2 2 3 3
3 3 1 1 12
1 3 3 17

Example output:
0 28 12

Example input:
4 3 1
3 4 1 3 12
2 2 3 4 10
1 2 4 16

Example output:
0 -1 -1 12

Editorial:
Consider a weighted directed graph (initially it has n vertices and no edges). We will construct a segment tree to handle queries of second type (and one for the third type but with similar approach).
Build a segment tree on number 1, ..., n. For each node of segment tree consider a vertex in the graph. For each leaf in this tree (like one with interval [l, l + 1)), add an edge with weight equal to 0 from vertex corresponding to this node to vertex l in the original graph. And for each non-leaf node, add an edge with weight equal to 0 from vertex corresponding to this node to the vertex corresponding to node of each of its children.
So we're adding about 4n vertices and edges to the graph. For each query of second type, we will add an edge from v to each maximal node of segment tree that [l, r) contains (lg(n) nodes for each query) with weight equal to w.
And construct a segment tree in the same way for queries of third type. Finally run Dijkstra's algorithm on this graph.
Time complexity: $${\mathcal {O}}(n\lg ^{2}(n))$$