[USACO11JAN] Bottleneck G

WC正在召集奶牛,他的农场有一个包含 ***N*** 块农田的网络，编号为 **1 -- N** ，每个农场里有 $C_i$ 头牛。农场被 **N-1** 条 **单向** 边链接,（每个农场有通向$P_i$的路） 保证从任何点可以到达1号点。WC想让所有奶牛集中到1号农场。 **时间是离散的** 奶牛可以在1单位时间里走过任意多条道路，但是每条路有一个容纳上限 *$M_i$* 并且奶牛不会离开1号农场(农场没有容量上限) ### 每一个单位时间，奶牛可以选择如下几种行动 1. 留在当前的农场 2. 经过几条道路，向1号农场移动（需要满足$M_i$） WC想要知道有多少牛可以在某个特定的时刻到达1号农场， 他有一张列着 ***K*** 个时间（分别为$T_i$)的单子 ，他想知道在每个$T_i$, 采用最优策略在$T_i$结束最多能有多少牛到1号农场 ### 数据范围如下： $1 \le N \le 10^5$ $1 \le C_i \le 10^9$ $0 \le M_i \le 10^9$ $1 \le P_i \le N$ $1 \le K \le 10^4$ $1 \le T_i \le 10^9$ ## **输入输出格式** * 输入格式 *第一行：两个整数 N 和 K *第2—N行，第i行描述一块农场及它的路 $P_i \;C_i\;M_i$ *第N+1 - N+K行， 第N+i 一个整数 $T_i$ * 输出格式 *每行一个答案对应$T_i$ 感谢@ToBiChi 提供翻译

Farmer John is gathering the cows. His farm contains a network of N (1 <= N <= 100,000) fields conveniently numbered 1..N and connected by N-1 unidirectional paths that eventually lead to field 1. The fields and paths form a tree. Each field i > 1 has a single one-way, exiting path to field P\_i, and currently contains C\_i cows (1 <= C\_i <= 1,000,000,000). In each time unit, no more than M\_i (0 <= M\_i <= 1,000,000,000) cows can travel from field i to field P\_i (1 <= P\_i <= N) (i.e., only M\_i cows can traverse the path). Farmer John wants all the cows to congregate in field 1 (which has no limit on the number of cows it may have). Rules are as follows: \* Time is considered in discrete units. \* Any given cow might traverse multiple paths in the same time unit. However, no more than M\_i total cows can leave field i (i.e., traverse its exit path) in the same time unit. \* Cows never move \*away\* from field #1. In other words, every time step, each cow has the choice either to a) stay in its current field b) move through one or more fields toward field #1, as long as the bottleneck constraints for each path are not violated Farmer John wants to know how many cows can arrive in field 1 by certain times. In particular, he has a list of K (1 <= K <= 10,000) times T\_i (1 <= T\_i <= 1,000,000,000), and he wants to know, for each T\_i in the list, the maximum number of cows that can arrive at field 1 by T\_i if scheduled to optimize this quantity. Consider an example where the tree is a straight line, and the T\_i list contains only T\_1=5, and cows are distibuted as shown: ```cpp Locn: 1---2---3---4 <-- Pasture ID numbers C_i: 0 1 12 12 <-- Current number of cows M_i: 5 8 3 <-- Limits on path traversal; field 1 has no limit since it has no exit The solution is as follows; the goal is to move cows to field 1: ``` Tree: 1---2---3---4 ```cpp t=0 0 1 12 12 <-- Initial state t=1 5 4 7 9 <-- field 1 has cows from field 2 and 3 t=2 10 7 2 6 t=3 15 7 0 3 t=4 20 5 0 0 t=5 25 0 0 0 Thus, the answer is 25: all 25 cows can arrive at field 1 by time t=5.

\* Line 1: Two space-separated integers: N and K \* Lines 2..N: Line i (not i+1) describes field i with three space-separated integers: P\_i, C\_i, and M\_i \* Lines N+1..N+K: Line N+i contains a single integer: T\_i

\* Lines 1..K: Line i contains a single integer that is the maximum number of cows that can arrive at field 1 by time T\_i.

4 1 
1 1 5 
2 12 7 
3 12 3 
5 

25