Puzzles

题面翻译

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.

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:

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 $10^{-6}$ .

样例 #1

样例输入 #1

7
1 2 1 1 4 4

样例输出 #1

1.0 4.0 5.0 3.5 4.5 5.0 5.0

样例 #2

样例输入 #2

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

样例输出 #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