Puzzles

题意翻译

有一棵树,共有N($ 1<=n<=10^5$)个节点,他会使用下列DFS算法对该树进行遍历。 ``` starting_time是一个容量为n的数组 current_time = 0 dfs(v): current_time =current_time+1 starting_time[v] = current_time 将children[v]的顺序随机排列 (每个排列的概率相同) // children[v]v的直接儿子组成的数组 for u in children[v]: dfs(u) ``` 1是这棵树的根,Bob会从1出发,即运行dfs(1),现在他想知道每个点starting_time的期望值。 translator:kfhkx

题目描述

Barney lives in country USC (United States of Charzeh). USC has $ n $ cities numbered from $ 1 $ through $ n $ and $ n-1 $ roads between them. Cities and roads of USC form a rooted tree (Barney's not sure why it is rooted). Root of the tree is the city number $ 1 $ . Thus if one will start his journey from city $ 1 $ , he can visit any city he wants by following roads. ![](https://cdn.luogu.com.cn/upload/vjudge_pic/CF696B/ba3e1e6f719802c9fe4b318c8d8513024bc088eb.png)Some girl has stolen Barney's heart, and Barney wants to find her. He starts looking for in the root of the tree and (since he is Barney Stinson not a random guy), he uses a random DFS to search in the cities. A pseudo code of this algorithm is as follows: ```plain let starting_time be an array of length n current_time = 0 dfs(v): current_time = current_time + 1 starting_time[v] = current_time shuffle children[v] randomly (each permutation with equal possibility) // children[v] is vector of children cities of city v for u in children[v]: dfs(u) ``` As told before, Barney will start his journey in the root of the tree (equivalent to call dfs(1)). Now Barney needs to pack a backpack and so he wants to know more about his upcoming journey: for every city $ i $ , Barney wants to know the expected value of starting\_time\[i\]. He's a friend of Jon Snow and knows nothing, that's why he asked for your help.

输入输出格式

输入格式


The first line of input contains a single integer $ n $ ( $ 1<=n<=10^{5} $ ) — the number of cities in USC. The second line contains $ n-1 $ integers $ p_{2},p_{3},...,p_{n} $ ( $ 1<=p_{i}&lt;i $ ), where $ p_{i} $ is the number of the parent city of city number $ i $ in the tree, meaning there is a road between cities numbered $ p_{i} $ and $ i $ in USC.

输出格式


In the first and only line of output print $ n $ numbers, where $ i $ -th number is the expected value of starting\_time\[i\]. Your answer for each city will be considered correct if its absolute or relative error does not exceed $ 10^{-6} $ .

输入输出样例

输入样例 #1

7
1 2 1 1 4 4

输出样例 #1

1.0 4.0 5.0 3.5 4.5 5.0 5.0 

输入样例 #2

12
1 1 2 2 4 4 3 3 1 10 8

输出样例 #2

1.0 5.0 5.5 6.5 7.5 8.0 8.0 7.0 7.5 6.5 7.5 8.0