[USACO11JAN] Bottleneck G

题面翻译

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

题目描述

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:

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

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.

## 输入格式

\* 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

## 输出格式

\* 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.

## 样例 #1

### 样例输入 #1

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

### 样例输出 #1

25