P2886 [USACO07NOV] Cow Relays G

For their physical fitness program, $N$ ($2 \le N \le 1,000,000$) cows have decided to run a relay race using the $T$ ($2 \le T \le 100$) cow trails throughout the pasture. Each trail connects two different intersections ($1 \le I_{1,i} \le 1,000$; $1 \le I_{2,i} \le 1,000$), each of which is the termination for at least two trails. The cows know the lengthi of each trail ($1 \le length_i  \le 1,000$), the two intersections the trail connects, and they know that no two intersections are directly connected by two different trails. The trails form a structure known mathematically as a graph. To run the relay, the $N$ cows position themselves at various intersections (some intersections might have more than one cow). They must position themselves properly so that they can hand off the baton cow-by-cow and end up at the proper finishing place. Write a program to help position the cows. Find the shortest path that connects the starting intersection ($S$) and the ending intersection ($E$) and traverses exactly $N$ cow trails.

\* Line 1: Four space-separated integers: $N$, $T$, $S$, and $E$ \* Lines 2..$T+1$: Line $i+1$ describes trail $i$ with three space-separated integers: lengthi , $I_{1,i}$ , and $I_{2,i}$

\* Line 1: A single integer that is the shortest distance from intersection $S$ to intersection $E$ that traverses exactly $N$ cow trails.