HBC24609区间价值[USACO 2011 Jan G]Bottleneck题解

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Farmer John is gathering the cows. His farm contains a network of N (1

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 P i , and currently contains C_i cows (1 <= C_i C i <= 1,000,000,000). In each time unit, no more than M_i M i (0 <= M_i M i <= 1,000,000,000) cows can travel from field i to field P_i P i (1 <= P_i P i <= N) (i.e., only M_i 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 T i (1 <= T_i T i <= 1,000,000,000), and he wants to know, for each T_i T i in the list, the maximum number of cows that can arrive at field 1 by T_i T i if scheduled to optimize this quantity. Consider an example where the tree is a straight line, and the T_i T i list contains only T_1 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.