Problem Description

In this problem, you are given n people and a fixed collection of 40 different types of hats, each type has its unique label ranging from 1 to 40. A 2D integer array hats represents the hat preferences for each person, such that hats[i] contains a list of all the hats preferred by the i-th person. Your task is to determine the total number of unique ways that these n people can wear these hats so that no two people are wearing the same type of hat.

This is a classic combinatorial problem with a restriction that ensures the uniqueness of the hat type for each individual. The problem asks you to return this total number of unique distributions modulo 10^9 + 7, which is a common technique to manage large output values in computational problems, ensuring the result stays within the limits of standard integer data types.

Intuition

At its core, the solution to this problem is rooted in combinatorics and dynamic programming (DP). The intuition behind using dynamic programming comes from two key observations:

1. The number of people n is small enough (maximum 10) to consider the problem using state space representation where each state represents a set of people who have already been assigned a hat. This allows us to compress the state into a binary number (for instance, a binary representation where each bit represents whether a person has been assigned a hat or not).

2. The solution builds upon subproblems where each subproblem considers one less hat. This helps in constructing the final solution iteratively as DP excels at optimizing problems where the solution can be built from smaller subproblems.

Given the small number of people, we can use bit masks to represent who has a hat already. The dynamic programming array f[i][j] signifies the number of ways to assign hats up to the i-th hat, considering the people configuration j. The bit representation of j has '1' in the positions of people who have already been assigned hats and '0' otherwise.

The DP approach then incrementally builds up the solution by considering cases where either the i-th hat is not used or it's assigned to one of the people who like it following the rules.

Each state transition thus involves either:

• Keeping the configuration as is (not assigning the new hat), which doesn't change the state, or
• Including a new assignment (giving the i-th hat to person k), which changes the state by updating the bit mask to reflect that person k now has a hat.

This way, the final answer is built up by considering all the possibilities, taking into account which hats can be worn by whom, until all hats up to the maximum value in the given lists are processed.

Learn more about Dynamic Programming and Bitmask patterns.

Solution Approach

The implementation of the solution for this problem utilizes dynamic programming (DP) with bit masking and involves a few steps:

1. Initialize the DP Table: A 2-dimensional DP table f is created with dimensions (m + 1) x (1 << n), where m is the maximum hat number preferred by any person and n is the number of people. Each entry f[i][j] will store the count of ways to distribute the first i types of hats across the people represented by state j.

2. Define the Base Case: We start with the base case f[0][0] = 1, which means there is one way to assign zero hats to zero people.

3. Map Preferences: We create a mapping g from each hat to a list of people who like that hat. The mapping is needed to quickly find which people can be considered when trying to distribute a particular hat.

4. Iterate Over Hats: For each hat type from 1 to m, we iterate to assign this hat to different people. For each state j that represents which people already have hats, we can have two possibilities:

   • Hat not assigned: If the current hat is not assigned to anyone, then the number of ways to distribute hats remains the same as the previous hat, so f[i][j] = f[i - 1][j].

   • Hat assigned: If the hat is assigned to one of the potential people who like it, each assignment will result in a new state from j to j ⊕ (1 << k) where ⊕ denotes the XOR operation and (1 << k) denotes a bit mask with only the k-th bit set (meaning assigning the i-th hat to the k-th person). This operation toggles the k-th bit of j, so we accumulate these new ways into f[i][j] i.e., f[i][j] = (f[i][j] + f[i - 1][j ⊕ (1 << k)]) % mod.

5. Modulo Operation: Since the number of ways can be large, we take every sum modulo 10^9 + 7 (stored in the variable mod) to keep the values within the limits of a 32-bit signed integer.

6. Return the Result: After processing all hats, the result will be in f[m][2^n - 1], which represents all n people having different hats.

This DP approach exploits the concept of state transitions while considering all permutations of hat assignments, respecting the individual preferences and ensuring unique ownership of hat types across all the people. The use of bit masking is a clever trick to represent a set of states succinctly when the universe is small, which is typically harder to do when dealing with large sets or where relationships between elements are complex.

The algorithm complexity primarily depends on the number of hats (which is at most 40) and the number of people (the size of the bitmask, which is 2^n states). Although it looks like a large number, the small limitation of n makes this algorithm feasible.

Example Walkthrough

Let's illustrate the solution approach with a small example.

Suppose we have n = 2 people and m = 3 different types of hats. The hats preferred by the people are represented by the 2D array hats such that hats[0] = [1, 2] and hats[1] = [2, 3]. Our goal is to find the number of unique ways to distribute hats so that both people wear different hats from their preference list.

Step by Step Implementation:

1. Initialize the DP Table: We create a 2-dimensional DP table f with dimensions (3 + 1) x (1 << 2) (since there are at most 3 types of hats and 2 people). The table is initialized with zeros, except for the base case f[0][0] = 1.

2. Define the Base Case: As defined, f[0][0] = 1 means one way to assign zero hats to zero people.

3. Map Preferences: We create a mapping g from each hat to the list of people who like that hat. From the preference list, we get g[1] = [0], g[2] = [0, 1], g[3] = [1].

4. Iterate Over Hats:

   • For hat 1: Only person 0 likes this hat. We update f[1][1] to 1, which indicates that there's one way to assign hat 1 to person 0.

   • For hat 2: Both person 0 and person 1 like this hat. We can assign this hat to person 0 if person 0 hasn't been given a hat already, which means updating f[2][1] to f[1][0]. Similarly, we can assign hat 2 to person 1 if person 1 hasn't been given a hat already, updating f[2][2].

   • For hat 3: Only person 1 likes this hat. We update f[3][2] to f[2][0] because we can give hat 3 to person 1 if they have no hat yet.

5. Modulo Operation: After each assignment, we perform f[i][j] = (f[i][j] + f[i - 1][j ⊕ (1 << k)]) % mod to keep numbers within the 32-bit signed integer limit.

6. Return the Result: The result is f[3][3], which represents both people having different hats. In this example, f[3][3] should give us 2, indicating there are two ways to give hats to the people according to their preferences: Person 0 can wear hat 1 and Person 1 can wear hat 3 or Person 0 can wear hat 2 and Person 1 can wear hat 3.

Thus, the example helps us see how the dynamic programming table builds up solutions with different combinations using bit masks to represent states. The table f is updated by considering each hat and each combination of which people already have hats (states). This results in a final count of the number of unique ways to distribute hats so that no two people are wearing the same type of hat.

Solution Implementation

Time and Space Complexity

The provided code defines a function numberWays which calculates the number of ways people can wear hats given certain constraints. The analysis of its time and space complexity is as follows:

Time Complexity

The time complexity of the function is O(m * 2^n * n). Here's a breakdown of why that's the case:

• m represents the maximum number of different hats, which can go up to 40 as per the problem constraints.
• n is the number of people, limited to 10 in this scenario.
• 2^n signifies the number of different states or combinations for the assignment of hats to the n people. Since each person can either wear a hat or not, there are 2^n possible states.

For each of the m hats, the algorithm iterates over all 2^n combinations, and for each combination, it can potentially iterate over all n people to update the state (f[i][j]). As a result, the time complexity amounts to the multiplicative product of these terms.

Space Complexity

The space complexity of the function is O(m * 2^n). This is due to the following reasons:

• An m + 1 by 2^n sized 2D list f is created to store the states of hat assignments, where each sub-list f[i] has a length of 2^n to represent all combinations of n people, and there are m + 1 such sub-lists (ranging from 0 to m).
• The additional data structures use negligible space compared to the size of f, so their contribution to space complexity is not considered dominant.

In summary, the algorithm requires a significant amount of space proportional to the number of hat combinations multiplied by the number of different hats.

Learn more about how to find time and space complexity quickly using problem constraints.