There are n people and 40 types of hats labeled from 1 to 40.
Given a 2D integer array hats, where hats[i] is a list of all hats preferred by the ith person.
Return the number of ways that the n people wear different hats to each other.
Since the answer may be too large, return it modulo 109 + 7.
Example 1:
Input: hats = [[3,4],[4,5],[5]]
Output: 1
Explanation: There is only one way to choose hats given the conditions.
First person choose hat 3, Second person choose hat 4 and last one hat 5.
Example 2:
Input: hats = [[3,5,1],[3,5]]
Output: 4
Explanation: There are 4 ways to choose hats:
(3,5), (5,3), (1,3) and (1,5)
Example 3:
Input: hats = [[1,2,3,4],[1,2,3,4],[1,2,3,4],[1,2,3,4]]
Output: 24
Explanation: Each person can choose hats labeled from 1 to 4.
Number of Permutations of (1,2,3,4) = 24.
Constraints:
n == hats.length
1 <= n <= 10
1 <= hats[i].length <= 40
1 <= hats[i][j] <= 40
hats[i] contains a list of unique integers.
Solutions
Solution 1: Dynamic Programming
We notice that $n$ is not greater than $10$, so we consider using DP with state compression to solve this problem.
We define $f[i][j]$ as the number of ways to assign the first $i$ hats to the people whose state is $j$. Here $j$ is a binary number, which represents a set of people. We have $f[0][0]=1$ at the beginning, and the answer is $f[m][2^n - 1]$, where $m$ is the maximum number of hats and $n$ is the number of people.
Consider $f[i][j]$. If we don't assign the $i$-th hat to anyone, then $f[i][j]=f[i-1][j]$; if we assign the $i$-th hat to the person $k$ who likes it, then $f[i][j]=f[i-1][j \oplus 2^k]$. Here $\oplus$ denotes the XOR operation. Therefore, we can get the state transition equation:
where $like[i]$ denotes the set of people who like the $i$-th hat.
The final answer is $f[m][2^n - 1]$, and the answer may be very large, so we need to take it modulo $10^9 + 7$.
Time complexity $O(m \times 2^n \times n)$, space complexity $O(m \times 2^n)$. Here $m$ is the maximum number of hats, which is no more than $40$ in this problem; and $n$ is the number of people, which is no more than $10$ in this problem.