Problem Description

In this problem, we are working on a project management scenario where we have a list of skills required for a project (req_skills) and a group of people where each person has a set of skills. Our goal is to form the smallest possible team such that each required skill for the project is covered by at least one team member's skill set. The people are represented by indices, and we need to find a subset of these indices such that the corresponding team members collectively possess all the required skills. This set should be as small as possible and the function should return any optimal solution.

Intuition

The intuition behind solving this problem lies in understanding it as a bit manipulation and dynamic programming problem. Each person's skill set can be represented as a bitmask, where each bit corresponds to a particular skill. For example, if we have 5 skills, a person with skills 1 and 3 would be represented as 10100 in binary (or 20 in decimal).

• Bitmask Representation: This conversion to bitmasks allows us to use bitwise operations to efficiently combine skill sets and check if all required skills are met.

• Dynamic Programming Approach: We use dynamic programming to track the optimal solution for each possible combination of skills. We initialize an array f where f[i] represents the smallest size of a team that can cover the skill set represented by the bitmask i. Initially, only the empty skill set has a team size of 0.

• Iterative Solution Building: Then we iterate over all possible combinations of skills. For each combination, we consider adding each person to the team one by one and see if adding them would provide a better solution (i.e., a smaller team size) for the new combination of skills that would result from adding this person's skills to the current combination.

• Backtracking to Find Team Members: After we populate the dynamic programming table with the minimum team sizes for all combinations, we backtrack from the solution that covers all required skills to find which people make up this optimal team. The arrays g and h help in this backtracking step; g stores the last person added to achieve this combination of skills, and h stores the previous combination of skills before the last person was added.

By using bit manipulation and dynamic programming, we can ensure that we cover all possible teams and arrive at a sufficient team with the smallest possible size.

Learn more about Dynamic Programming and Bitmask patterns.

Solution Approach

The problem is tackled with a combination of bit manipulation and dynamic programming, and here's the detailed walk-through:

1. Mapping Skills to Bits: First, a dictionary d is created that maps each skill to a unique index. This helps in converting the list of skills for each person into a bitmask where the bit at the index corresponding to a skill is set to 1 if the person has that skill. Initializing people’s skills involves iterating through each person and setting the relevant bits to 1 in their skill representation p[].

2. Dynamic Programming Initialization: A list f is initialized with inf (infinity) values, where each index of f corresponds to a bitmask of skills. The value at f[i] will store the minimum team size needed to cover the skill set represented by the bitmask i. The entry f[0], which corresponds to a team with no skills, is initially set to 0, as no team members are required to cover no skills.

3. Building the DP Table: The main loop iterates over all combinations of skills i and for each, checks if it's already been reached (i.e., f[i] is not inf). Then for each person j, it calculates the combination of skills that would result if this person is added to the team (i | p[j]). If this new combination results in a smaller team size than previously recorded, the algorithm updates f[i | p[j]] to this smaller size and records in g the index of the person added, and in h the previous skill combination.

4. Backtracking to Build the Smallest Team: The solution set bitmask (1 << m) - 1 represents all skills being covered. Starting from this final state, the algorithm uses the arrays g and h to backtrack and construct the smallest sufficient team. The array g contains the indices of the people added, and h tells us the previous state before the person in g was added. By backtracking from the state representing all skills covered, the algorithm adds the indices of the necessary team members into the ans array.

5. Return the Result: Finally, the array ans containing the indices of the smallest necessary team is returned.

To summarize, we use:

• Bit Manipulation: To efficiently combine skill sets and evaluate team compositions.
• Dynamic Programming: To build the minimum team sizes for all possible skill combinations iteratively.
• Backtracking: To construct the actual team by retracing which people were added at each step.

The algorithm ensures by its design that it will return one of the possible smallest sufficient teams.

Example Walkthrough

Let's consider a simplified example to illustrate the solution approach:

• The project requires 3 skills, so we need a team to cover these skills: req_skills = ["coding", "math", "data"].
• There are 3 people available, each with their own set of skills:
  • Person 0 has "coding".
  • Person 1 has "math".
  • Person 2 has "coding" and "data".

Using the solution approach described previously:

1. Mapping Skills to Bits: We map each skill to a bit position:

   • "coding" -> bit 0
   • "math" -> bit 1
   • "data" -> bit 2

   Now, we represent each person's skills as a bitmask:

   • Person 0: "coding" -> 001 binary -> 1 decimal
   • Person 1: "math" -> 010 binary -> 2 decimal
   • Person 2: "coding", "data" -> 101 binary -> 5 decimal

2. Dynamic Programming Initialization: We initialize an array f with four elements (since 2^3 skills = 8 combinations, from 000 to 111 in binary):

   • f = [inf, inf, inf, inf, inf, inf, inf, inf]
   • We set f[0] to 0 because no team is needed for no skills: f = [0, inf, inf, inf, inf, inf, inf, inf]

3. Building the DP Table: We iterate over combinations of skills and update the f table:

   • For bitmask 1 (Person 0): f[1] = 1 because Person 0 can cover the "coding" skill.
   • For bitmask 2 (Person 1): f[2] = 1 because Person 1 can cover the "math" skill.
   • For bitmask 5 (Person 2): f[5] = 1 because Person 2 can cover "coding" and "data".

4. Backtracking to Build the Smallest Team: We need to cover all skills (111 binary -> 7 decimal). The minimum value of f[7] would be found by considering the combination that includes Person 2 (5) and Person 1 (2):

   • We check f[7] = min(f[7], f[5] + 1) and f[7] = min(f[7], f[2] + 1) because adding either Person 2 or Person 1 to an already covered skill set can potentially meet all requirements.
   • In this case, we see that f[7] is not inf, and the value should be 2. This signifies we need at least 2 team members to cover all the skills.

5. Return the Result: The smallest team that can cover all skills includes Person 1 and Person 2. They collectively cover all three required skills.

From this example, we start with individual skills, build up our dynamic programming table by iteratively considering each person, and use backtracking to find the actual people that make up the smallest team that covers all necessary skills. The result is a team of two people with indices [1, 2], which successfully covers the required skills for the project.

Solution Implementation

Time and Space Complexity

Time Complexity

The given code implements a dynamic programming solution to find the smallest sufficient team for the required skills. Here's a step-by-step analysis of the time complexity:

• The dictionary creation d = {s: i for i, s in enumerate(req_skills)} takes O(m) time where m is the number of required skills.
• Preparing the array p requires iterating over each person and each skill of that person. If we denote the average number of skills per person as s_avg, this step takes O(n * s_avg) where n is the number of people.
• Initializing the arrays f, g, and h each takes O(1 << m) time.
• The nested loops iterate over each combination of skills (i) and each person (j). For each person, we might update the state, so the complexity of these nested loops is O(n * 2^m).
• Finally, there's a while loop to reconstruct the solution. In the worst case, it could take O(n) time since we might select each person once.

Given the above, the overall time complexity of the solution is dominated by O(n * 2^m).

Space Complexity

Now let's analyze the space complexity:

• The dictionary d has as many entries as there are unique skills, resulting in O(m) space.
• The array p has a size of n, giving O(n).
• The arrays f, g, and h each consume space proportional to 2^m, therefore each taking O(2^m).

Summing up the space used by these elements, the overall space complexity is O(2^m + m + n).

In conclusion, the final computational complexity for the code is:

• Time Complexity: O(n * 2^m)
• Space Complexity: O(2^m + m + n)

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