2060: [USACO 2025 January Contest, Gold] Problem 1. Median Heap

Farmer John has a binary tree with N nodes where the nodes are numbered from 1 to N (1 ≤ N < 2⋅10⁵ and N is odd). For i > 1, the parent of node i is ⌊i / 2⌋. Each node has an initial integer value a_i, and a cost c_i to change the initial value to any other integer value ( 0 ≤ a_i , c_i ≤ 10⁹ ).

He has been tasked by the Federal Bovine Intermediary (FBI) with finding an approximate median value within this tree, and has devised a clever algorithm to do so.

He starts at the last node N and works his way backward. At every step of the algorithm, if a node would not be the median of it and its two children, he swaps the values of the current node and the child value that would be the median. At the end of this algorithm, the value at node 1 (the root) is the median approximation.

The FBI has also given Farmer John a list of Q (1 ≤ Q ≤ 2⋅10⁵) independent queries each specified by a target value m (0 ≤ m ≤ 10⁹). For each query, FJ will first change some of the node's initial values, and then execute the median approximation algorithm. For each query, determine the minimum possible total cost for FJ to make the output of the algorithm equal to m.

The next N lines each contain two integers a_i and c_i.

The next line contains Q.

The next Q lines each contain a target value m.

5
10 10000
30 1000
20 100
50 10
40 1
11
55
50
45
40
35
30
25
20
15
10
5

111
101
101
100
100
100
100
0
11
11
111