Roads and Planes (roadplane2.txt)

[[ This is last week's "roadplane problem" -- but now, with
   LARGE sets of test data!  Use Dijkstra's algorithm, expanding
   nodes ordered first by the "road-component," then by distance. ]]

Farmer John is conducting research for a new milk contract in a new
territory. He intends to distribute milk to T (1 <= T <= 25,000)
towns conveniently numbered 1..T which are connected by up to R (1
<= R <= 50,000) roads conveniently numbered 1..R and/or P (1 <= P
<= 50,000) airplane flights conveniently numbered 1..P.

Either road i or plane i connects town A_i (1 <= A_i <= T) to town
B_i (1 <= B_i <= T) with traversal cost C_i. For roads, 0 <= C_i
<= 10,000; however, due to the strange finances of the airlines,
the cost for planes can be quite negative (-10,000 <= C_i <= 10,000).

Roads are bidirectional and can be traversed from A_i to B_i or B_i
to A_i for the same cost; the cost of a road must be non-negative.

Planes, however, can only be used in the direction from A_i to B_i
specified in the input. In fact, if there is a plane from A_i to
B_i it is guaranteed that there is no way to return from B_i to A_i
with any sequence of roads and planes due to recent antiterror
regulation.

Farmer John is known around the world as the source of the world's
finest dairy cows. He has in fact received orders for his cows from
every single town. He therefore wants to find the cheapest price
for delivery to each town from his distribution center in town S
(1 <= S <= T) or to know that it is not possible if this is the
case.

PROBLEM NAME: roadplane2

INPUT FORMAT:

* Line 1: Four space separated integers: T, R, P, and S

* Lines 2..R+1: Three space separated integers describing a road: A_i,
        B_i and C_i

* Lines R+2..R+P+1: Three space separated integers describing a plane:
        A_i, B_i and C_i

SAMPLE INPUT:

6 3 3 4
1 2 5
3 4 5
5 6 10
3 5 -100
4 6 -100
1 3 -10

INPUT DETAILS:

6 towns.  There are roads between town 1 and town 2, town 3 and town 4, and 
town 5 and town 6 with costs 5, 5 and 10; there are planes from town 3 to 
town 5, from town 4 to town 6, and from town 1 to town 3 with costs -100, -
100 and -10.  FJ is based in town 4.

OUTPUT FORMAT:

* Lines 1..T: The minimum cost to get from town S to town i, or 'NO
        PATH' if this is not possible

SAMPLE OUTPUT:

NO PATH
NO PATH
5
0
-95
-100

OUTPUT DETAILS:

FJ's cows begin at town 4, and can get to town 3 on the road.  They
can get to towns 5 and 6 using planes from towns 3 and 4.  However,
there is no way to get to towns 1 and 2, since they cannot go
backwards on the plane from 1 to 3.