Graph Game

题意翻译

题目描述 求基环树随机点分治总遍历次数期望 基环树随机点分治步骤： ①遍历当前分治区域所有点一次 ②在当前分治区域随机选择一个点$x$ ③将$x$删掉，产生的所有连通块递归处理。 输入数据 第一行一个正整数$n(1 \leq n \leq 3000)$表示树的大小，接下来$n$行每行两个非负整数$a_i , b_i(0 \leq a_i,b_i < n)$表示一条边 输出数据 一行一个浮点数表示期望，误差不超过$10^{-6}$。

题目描述

In computer science, there is a method called "Divide And Conquer By Node" to solve some hard problems about paths on a tree. Let's desribe how this method works by function: $ solve(t) $ ( $ t $ is a tree): 1. Chose a node $ x $ (it's common to chose weight-center) in tree $ t $ . Let's call this step "Line A". 2. Deal with all paths that pass $ x $ . 3. Then delete $ x $ from tree $ t $ . 4. After that $ t $ becomes some subtrees. 5. Apply $ solve $ on each subtree. This ends when $ t $ has only one node because after deleting it, there's nothing. Now, WJMZBMR has mistakenly believed that it's ok to chose any node in "Line A". So he'll chose a node at random. To make the situation worse, he thinks a "tree" should have the same number of edges and nodes! So this procedure becomes like that. Let's define the variable $ totalCost $ . Initially the value of $ totalCost $ equal to $ 0 $ . So, $ solve(t) $ (now $ t $ is a graph): 1. $ totalCost=totalCost+(size of t) $ . The operation "=" means assignment. $ (Size of t) $ means the number of nodes in $ t $ . 2. Choose a node $ x $ in graph $ t $ at random (uniformly among all nodes of $ t $ ). 3. Then delete $ x $ from graph $ t $ . 4. After that $ t $ becomes some connected components. 5. Apply $ solve $ on each component. He'll apply $ solve $ on a connected graph with $ n $ nodes and $ n $ edges. He thinks it will work quickly, but it's very slow. So he wants to know the expectation of $ totalCost $ of this procedure. Can you help him?

输入输出格式

输入格式

The first line contains an integer $ n $ ( $ 3<=n<=3000 $ ) — the number of nodes and edges in the graph. Each of the next $ n $ lines contains two space-separated integers $ a_{i},b_{i} $ $ (0<=a_{i},b_{i}<=n-1) $ indicating an edge between nodes $ a_{i} $ and $ b_{i} $ . Consider that the graph nodes are numbered from $ 0 $ to $ (n-1) $ . It's guaranteed that there are no self-loops, no multiple edges in that graph. It's guaranteed that the graph is connected.

输出格式

Print a single real number — the expectation of $ totalCost $ . Your answer 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

5
3 4
2 3
2 4
0 4
1 2

输出样例 #1

13.166666666666666

输入样例 #2

3
0 1
1 2
0 2

输出样例 #2

6.000000000000000

输入样例 #3

5
0 1
1 2
2 0
3 0
4 1

输出样例 #3

13.166666666666666

说明

Consider the second example. No matter what we choose first, the $ totalCost $ will always be $ 3+2+1=6 $ .

题意翻译

题目描述 求基环树随机点分治总遍历次数期望 基环树随机点分治步骤： ①遍历当前分治区域所有点一次 ②在当前分治区域随机选择一个点$x$ ③将$x$删掉，产生的所有连通块递归处理。 输入数据 第一行一个正整数$n(1 \leq n \leq 3000)$表示树的大小，接下来$n$行每行两个非负整数$a_i , b_i(0 \leq a_i,b_i < n)$表示一条边 输出数据 一行一个浮点数表示期望，误差不超过$10^{-6}$。