Question

1434. Number of Ways to Wear Different Hats to Each Other

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Description

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:

$$ f[i][j]=f[i-1][j]+ \sum_{k \in like[i]} f[i-1][j \oplus 2^k] $$

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.

class Solution:
  def numberWays(self, hats: List[List[int]]) -> int:
    g = defaultdict(list)
    for i, h in enumerate(hats):
      for v in h:
        g[v].append(i)
    mod = 10**9 + 7
    n = len(hats)
    m = max(max(h) for h in hats)
    f = [[0] * (1 << n) for _ in range(m + 1)]
    f[0][0] = 1
    for i in range(1, m + 1):
      for j in range(1 << n):
        f[i][j] = f[i - 1][j]
        for k in g[i]:
          if j >> k & 1:
            f[i][j] = (f[i][j] + f[i - 1][j ^ (1 << k)]) % mod
    return f[m][-1]

class Solution {
  public int numberWays(List<List<Integer>> hats) {
    int n = hats.size();
    int m = 0;
    for (var h : hats) {
      for (int v : h) {
        m = Math.max(m, v);
      }
    }
    List<Integer>[] g = new List[m + 1];
    Arrays.setAll(g, k -> new ArrayList<>());
    for (int i = 0; i < n; ++i) {
      for (int v : hats.get(i)) {
        g[v].add(i);
      }
    }
    final int mod = (int) 1e9 + 7;
    int[][] f = new int[m + 1][1 << n];
    f[0][0] = 1;
    for (int i = 1; i <= m; ++i) {
      for (int j = 0; j < 1 << n; ++j) {
        f[i][j] = f[i - 1][j];
        for (int k : g[i]) {
          if ((j >> k & 1) == 1) {
            f[i][j] = (f[i][j] + f[i - 1][j ^ (1 << k)]) % mod;
          }
        }
      }
    }
    return f[m][(1 << n) - 1];
  }
}

class Solution {
public:
  int numberWays(vector<vector<int>>& hats) {
    int n = hats.size();
    int m = 0;
    for (auto& h : hats) {
      m = max(m, *max_element(h.begin(), h.end()));
    }
    vector<vector<int>> g(m + 1);
    for (int i = 0; i < n; ++i) {
      for (int& v : hats[i]) {
        g[v].push_back(i);
      }
    }
    const int mod = 1e9 + 7;
    int f[m + 1][1 << n];
    memset(f, 0, sizeof(f));
    f[0][0] = 1;
    for (int i = 1; i <= m; ++i) {
      for (int j = 0; j < 1 << n; ++j) {
        f[i][j] = f[i - 1][j];
        for (int k : g[i]) {
          if (j >> k & 1) {
            f[i][j] = (f[i][j] + f[i - 1][j ^ (1 << k)]) % mod;
          }
        }
      }
    }
    return f[m][(1 << n) - 1];
  }
};

func numberWays(hats [][]int) int {
  n := len(hats)
  m := 0
  for _, h := range hats {
    m = max(m, slices.Max(h))
  }
  g := make([][]int, m+1)
  for i, h := range hats {
    for _, v := range h {
      g[v] = append(g[v], i)
    }
  }
  const mod = 1e9 + 7
  f := make([][]int, m+1)
  for i := range f {
    f[i] = make([]int, 1<<n)
  }
  f[0][0] = 1
  for i := 1; i <= m; i++ {
    for j := 0; j < 1<<n; j++ {
      f[i][j] = f[i-1][j]
      for _, k := range g[i] {
        if j>>k&1 == 1 {
          f[i][j] = (f[i][j] + f[i-1][j^(1<<k)]) % mod
        }
      }
    }
  }
  return f[m][(1<<n)-1]
}

function numberWays(hats: number[][]): number {
  const n = hats.length;
  const m = Math.max(...hats.flat());
  const g: number[][] = Array.from({ length: m + 1 }, () => []);
  for (let i = 0; i < n; ++i) {
    for (const v of hats[i]) {
      g[v].push(i);
    }
  }
  const f: number[][] = Array.from({ length: m + 1 }, () =>
    Array.from({ length: 1 << n }, () => 0),
  );
  f[0][0] = 1;
  const mod = 1e9 + 7;
  for (let i = 1; i <= m; ++i) {
    for (let j = 0; j < 1 << n; ++j) {
      f[i][j] = f[i - 1][j];
      for (const k of g[i]) {
        if (((j >> k) & 1) === 1) {
          f[i][j] = (f[i][j] + f[i - 1][j ^ (1 << k)]) % mod;
        }
      }
    }
  }
  return f[m][(1 << n) - 1];
}