Minimum Number of Groups to Create a Valid Assignment

Minimum Number of Groups to Create a Valid Assignment

1. What is this problem about?

The Minimum Number of Groups to Create a Valid Assignment coding problem asks you to organize items into groups based on their frequency. You are given an array of integers (ids). You must group identical ids together, but the groups must satisfy a size constraint: every group must have a size either S or S-1 for some integer S. Your goal is to find the minimum number of groups required to satisfy this for any valid S.

2. Why is this asked in interviews?

Amazon and BNY Mellon use this question to test a candidate's ability to combine frequency maps with mathematical optimization. It requires a two-step approach: first, counting the occurrences, and second, iterating through possible group sizes to find the global minimum. It's a great test of efficiency and logical partitioning.

3. Algorithmic pattern used

This utilizes the Array, Hash Table, Greedy interview pattern.

1. Use a Hash Table to count the frequency of each id.
2. The possible group size S-1 can be at most the minimum frequency found.
3. Iterate S from min_frequency + 1 down to 1.
4. For each S, check if every frequency f can be partitioned into groups of size S and S-1. This is possible if f // (S-1) >= f % (S-1).
5. The first S that works for all frequencies will give you the minimum number of groups (because a larger S means fewer groups).

4. Example explanation

Frequencies: [10, 10, 11] Min frequency is 10. Try S-1 = 10 (so S = 11).

• Can 10 be split into 11s and 10s? Yes (one 10).
• Can 11 be split? Yes (one 11). Result: 3 groups (10, 10, 11). If frequencies were [2, 3]: Try S-1 = 2 (S = 3).
• 2: one group of 2.
• 3: one group of 3. Total: 2 groups.

5. Common mistakes candidates make

A common error is not checking all possible values of S. Candidates often assume S must be the minimum frequency, but it could be smaller. Another mistake is the math for the "can it be partitioned" check. Some try to use nested loops to find the combination of S and S-1, but simple division and remainder checks are much faster.

6. Interview preparation tip

When a problem involves grouping by frequency, always start by building a frequency map (Hash Table). Then, think about the constraints on the group size. Often, the smallest frequency limits the possible range of the answer, which helps you narrow down your search space.