Battalion Strength

题意翻译

定义一个 $n$ 个数的序列 $p_1,p_2,\cdots,p_n$，若 $n \leq 1$，则该序列权值为 $0$，否则定义该序列的权值是将原序列排序后相邻两项积的和。即当 $a$ 是 $1 \sim n$ 的排列且满足 $p_{a_1}\leq p_{a_2}\leq \cdots \leq p_{a_n}$ 时，该序列的权值为 $\sum_{i=1}^{n-1} p_{a_i} \cdot p_{a_{i+1}}$。 现在给你一个 $n$ 个数的序列，从中随机选取一个子序列（即不要求连续），假设所有的子序列被选到的概率相等，问取出的子序列的权值的期望时多少，动态带单点值修改，询问原序列和每次操作后的子序列权值的期望 $\mod 10^9+7$ 的结果。

题目描述

There are $ n $ officers in the Army of Byteland. Each officer has some power associated with him. The power of the $ i $ -th officer is denoted by $ p_{i} $ . As the war is fast approaching, the General would like to know the strength of the army. The strength of an army is calculated in a strange way in Byteland. The General selects a random subset of officers from these $ n $ officers and calls this subset a battalion.(All $ 2^n $ subsets of the $ n $ officers can be chosen equally likely, including empty subset and the subset of all officers). The strength of a battalion is calculated in the following way: Let the powers of the chosen officers be $ a_{1},a_{2},\ldots,a_{k} $ , where $ a_1 \le a_2 \le \dots \le a_k $ . The strength of this battalion is equal to $ a_1a_2 + a_2a_3 + \dots + a_{k-1}a_k $ . (If the size of Battalion is $ \leq 1 $ , then the strength of this battalion is $ 0 $ ). The strength of the army is equal to the expected value of the strength of the battalion. As the war is really long, the powers of officers may change. Precisely, there will be $ q $ changes. Each one of the form $ i $ $ x $ indicating that $ p_{i} $ is changed to $ x $ . You need to find the strength of the army initially and after each of these $ q $ updates. Note that the changes are permanent. The strength should be found by modulo $ 10^{9}+7 $ . Formally, let $ M=10^{9}+7 $ . It can be shown that the answer can be expressed as an irreducible fraction $ p/q $ , where $ p $ and $ q $ are integers and $ q\not\equiv 0 \bmod M $ ). Output the integer equal to $ p\cdot q^{-1} \bmod M $ . In other words, output such an integer $ x $ that $ 0 \leq x < M $ and $ x ⋅ q \equiv p \bmod M $ ).

输入输出格式

输入格式

The first line of the input contains a single integer $ n $ ( $ 1 \leq n \leq 3⋅10^{5} $ ) — the number of officers in Byteland's Army. The second line contains $ n $ integers $ p_{1},p_{2},\ldots,p_{n} $ ( $ 1 \leq p_{i} \leq 10^{9} $ ). The third line contains a single integer $ q $ ( $ 1 \leq q \leq 3⋅10^{5} $ ) — the number of updates. Each of the next $ q $ lines contains two integers $ i $ and $ x $ ( $ 1 \leq i \leq n $ , $ 1 \leq x \leq 10^{9} $ ), indicating that $ p_{i} $ is updated to $ x $ .

输出格式

In the first line output the initial strength of the army. In $ i $ -th of the next $ q $ lines, output the strength of the army after $ i $ -th update.

输入输出样例

输入样例 #1

2
1 2
2
1 2
2 1

输出样例 #1

500000004
1
500000004

输入样例 #2

4
1 2 3 4
4
1 5
2 5
3 5
4 5

输出样例 #2

625000011
13
62500020
375000027
62500027

说明

In first testcase, initially, there are four possible battalions - {} Strength = $ 0 $ - { $ 1 $ } Strength = $ 0 $ - { $ 2 $ } Strength = $ 0 $ - { $ 1,2 $ } Strength = $ 2 $ So strength of army is $ \frac{0+0+0+2}{4} $ = $ \frac{1}{2} $ After changing $ p_{1} $ to $ 2 $ , strength of battallion { $ 1,2 $ } changes to $ 4 $ , so strength of army becomes $ 1 $ . After changing $ p_{2} $ to $ 1 $ , strength of battalion { $ 1,2 $ } again becomes $ 2 $ , so strength of army becomes $ \frac{1}{2} $ .