Hiho #1075: 开锁魔法III

Problem Statement

描述

一日，崔克茜来到小马镇表演魔法。

其中有一个节目是开锁咒：舞台上有 n 个盒子，每个盒子中有一把钥匙，对于每个盒子而言有且仅有一把钥匙能打开它。初始时，崔克茜将会随机地选择 k 个盒子用魔法将它们打开。崔克茜想知道最后所有盒子都被打开的概率，你能帮助她回答这个问题吗？

输入

第一行一个整数 $T$ ( $T \leq 100$ )表示数据组数。对于每组数据，第一行有两个整数 $n$ 和 $k$ ( $1 \leq n \leq 300, 0 \leq k \leq n$ )。第二行有 $n$ 个整数 $a_i$ ，表示第 $i$ 个盒子中，装有可以打开第 $a_i$ 个盒子的钥匙。

输出

对于每组询问，输出一行表示对应的答案。要求相对误差不超过四位小数。

样例输入

4

5 1

2 5 4 3 1

5 2

2 5 4 3 1

5 3

2 5 4 3 1

5 4

2 5 4 3 1

样例输出

0.000000000

0.600000000

0.900000000

1.000000000

The problem is to compute the probability that use $k$ keys to open the $n$ boxes. In fact we only need to comupte the number of methods that successfully opening $n$ boxes by $k$ choices. Then dividing $C_n^k$ is the final result. So, let's focus on the more refined problem.

First, let's use some notations to express the problem.

Assume the key in box $i$ can open box $a[i]$ . Then, the boxes can be opend from box $1$ to $n$ is { $a[1], a[2], ..., a[n]$ }.

If we determine open box $i$ , then we'll use the key $a[i]$ to open box $a[i]$ which contains the $a[a[i]]$ box key.

So we can assume the keys in the $n$ boxes as a permutation of numbers { $1, 2, ..., n$ }. The math model here is just the permutation group in Abstract Algebra.

In order to open all $n$ boxes, we first need to check how many cycles in the permutation. Because the number of keys we need to open all boxes must be greater than or equal to the number of cycles in permutation.

So, if define the number of keys is $k$ , and the number of cycles in the $n$ -permutation is $m$ , the above states $k \geq m$ .

Now, we need to design an algorithm to solve the problem. The basical idea is Dynamic Programming (DP).

In general, the hard part of DP is to form a sub-problem. Here, based on the analysis of the permutation above, we'll set the sub-problem by cycles. Because there're $m$ cycles in the $n$ -permutation, we'll use $m$ steps to solve the problem.

Define: dp[i][j] = the number of methods that using $j$ keys to solve first $i$ cycles.

Thus the problem can be expressed as computing $dp[m][k]$ .

Next, let's construct the recursion. Assume we need to compute $dp[i][j]$ .

In order to solve first $i$ cycles, we can first solve $i-1$ cycles and then the $i^{th}$ cycle.

If we use $k_i$ keys to solve the $i^{th}$ cycle, we can use only $j-k_i$ keys to solve the first $i-1$ cycles.

$k_i$ can vary from $1$ to $j$ . (Because the initial status may not need key solving, thus $m$ can vary to $j$ .) And $k_i$ can't be greater than the size of $i^{th}$ cycle, denoted as $l_i$ . (Because every key is belong to one box, so the number of keys we choose can't be greater than the number of boxes in all.)

For every fixed $k_i$ , we just need to multiply the result of first $i-1$ cycles and the result of $i^{th}$ cycle, i.e. $dp[i-1][j-k_i] * C_{l_i}^{k_i}$
(Every $k_i$ keys can solve the $i^{th}$ cycle, so the result of solving $i^{th}$ cycle is $C_{l_i}^{k_i}$ .)

According to above statements, we can get the recursion equation. Here, we use array $comb[n][m]$ to denote the math combination $C_n^m$ .

$dp[i][j] = \sum_{m=1}^{j}(dp[i-1][j-m]*comb[cycle\_i\_length][m])$

So, the problem is done. But there're two additional problems we need to solve priori.

First, for efficiency, we can compute the combination numbers before we do the DP algorithm. The computing is also based on DP thinking:

Compute the combination number by DP, i.e. the simple math equation $C_i^j = C_{i-1}^j + C_{i-1}^{j-1}$ . Code is

for(int i = 0; i < 500; ++i)
    for(int j = 0; j <= i; ++j)
        comb[i][j] = 
        (0 == i || 0 == j) ? 1 : comb[i-1][j] + comb[i-1][j-1];

Second, we need to compute the $m$ sizes of cycles in the $n$ -permutation:

Get the cycle information in a $n$ -permutation, including number of cycles and the size of every cycle. Here we use array $perm$ to indicate the permutation of $n$ elements.

vector<int> cycles;    //store the cycle information
bool used[500];
memset(used, 0, sizeof used);
 
for (int i = 0; i < n; ++i){
    if(used[i] == true)
        continue;
 
    int num = 0;
    int idx = i;
    while(!used[idx])
    {
        ++num;
        used[idx] = true;
        idx = perm[idx];
    }
 
    cycles.push_back(num);          
}