10203: 洛谷P2939 - [USACO09FEB] Revamping Trails G
[Creator : ]
Description
约翰一共有 $N$ 个牧场.由 $M\ (1 \leq M \leq 50,000)$ 条布满尘埃的小径连接。小径可以双向通行。每天早上约翰从牧场 $1$ 出发到牧场 $N\ (1 \leq N \leq 10,000)$ 去给奶牛检查身体。
通过每条小径都需要消耗一定的时间。约翰打算升级其中 $K$ 条小径，使之成为高速公路。在高速公路上的通行几乎是瞬间完成的，所以高速公路的通行时间为 $0$。
请帮助约翰决定对哪些小径进行升级，使他每天从 $1$ 号牧场到第 $N$ 号牧场所花的时间最短。输出这一最短时间即可。
Farmer John dutifully checks on the cows every day. He traverses some of the $M\ (1 \leq M \leq 50,000)$ trails conveniently numbered 1..M from pasture 1 all the way out to pasture N (a journey which is always possible for trail maps given in the test data). The $N\ (1 \leq N \leq 10,000)$ pastures conveniently numbered 1..N on Farmer John's farm are currently connected by bidirectional dirt trails. Each trail i connects pastures $P1_i,P2_i\ (1 \leq P1_i \leq N;\ 1 \leq P2_i \leq N)$ and requires $T_i\ (1 \leq T_i \leq 1,000,000)$ units of time to traverse.
He wants to revamp some of the trails on his farm to save time on his long journey. Specifically, he will choose $K\ (1 \leq K \leq 20)$ trails to turn into highways, which will effectively reduce the trail's traversal time to 0. Help FJ decide which trails to revamp to minimize the resulting time of getting from pasture 1 to N.
通过每条小径都需要消耗一定的时间。约翰打算升级其中 $K$ 条小径，使之成为高速公路。在高速公路上的通行几乎是瞬间完成的，所以高速公路的通行时间为 $0$。
请帮助约翰决定对哪些小径进行升级，使他每天从 $1$ 号牧场到第 $N$ 号牧场所花的时间最短。输出这一最短时间即可。
Farmer John dutifully checks on the cows every day. He traverses some of the $M\ (1 \leq M \leq 50,000)$ trails conveniently numbered 1..M from pasture 1 all the way out to pasture N (a journey which is always possible for trail maps given in the test data). The $N\ (1 \leq N \leq 10,000)$ pastures conveniently numbered 1..N on Farmer John's farm are currently connected by bidirectional dirt trails. Each trail i connects pastures $P1_i,P2_i\ (1 \leq P1_i \leq N;\ 1 \leq P2_i \leq N)$ and requires $T_i\ (1 \leq T_i \leq 1,000,000)$ units of time to traverse.
He wants to revamp some of the trails on his farm to save time on his long journey. Specifically, he will choose $K\ (1 \leq K \leq 20)$ trails to turn into highways, which will effectively reduce the trail's traversal time to 0. Help FJ decide which trails to revamp to minimize the resulting time of getting from pasture 1 to N.
Input
Line 1: Three space-separated integers: $N, M, K$
Lines 2..M+1: Line i+1 describes trail i with three space-separated integers: $P1_i, P2_i, T_i$
Lines 2..M+1: Line i+1 describes trail i with three space-separated integers: $P1_i, P2_i, T_i$
Output
Line 1: The length of the shortest path after revamping no more than $K$ edges
Sample 1 Input
4 4 1
1 2 10
2 4 10
1 3 1
3 4 100
Sample 1 Output
1
K is 1; revamp trail 3->4 to take time 0 instead of 100. The new shortest path is 1->3->4, total traversal time now 1.