The Number of Beautiful Subsets

The Number of Beautiful Subsets

What is this problem about?

A "beautiful" subset is defined by a specific constraint on the relationships between its elements. In "The Number of Beautiful Subsets," you are given an array of positive integers and a target difference 'k'. A subset is considered beautiful if it does not contain any two integers such that their absolute difference is exactly 'k'. Your task is to count how many non-empty subsets of the given array satisfy this condition.

Why is this asked in interviews?

This the Number of Beautiful Subsets interview question is asked by companies like Infosys, Amazon, and Bloomberg. It tests a wide range of skills, including backtracking for simple constraints and dynamic programming/combinatorics for more optimized solutions. It's an excellent problem for gauging a candidate's ability to simplify a complex constraint into independent subproblems.

Algorithmic pattern used

This problem can be approached via the Array, Math, Hash Table, Sorting, Backtracking, Combinatorics, Dynamic Programming interview pattern.

1. The simple approach is backtracking: for each element, decide whether to include it in the subset. Only include it if its "forbidden partners" (x-k and x+k) are not already in the subset.
2. A more optimized approach involves grouping numbers into "chains" based on the difference 'k'. For example, if k=2, the numbers 1, 3, 5 form one chain, and 2, 4, 6 form another.
3. Elements in different chains never conflict with each other.
4. For each chain, you can use DP to count valid subsets: dp[i] is the number of valid subsets using the first i elements of the chain. This is similar to the "House Robber" problem.
5. Multiply the results from all chains to get the total number of beautiful subsets.

Example explanation

Array [1, 2, 3], k = 1.

• Chains: (1, 2, 3) - All conflict.
• Valid subsets: {1}, {2}, {3}, {1, 3}.
• Total = 4. Using the chain logic:
• Chain [1, 2, 3] with counts [1, 1, 1].
• For a chain like this, we can't pick adjacent elements.
• Subsets: None(1), {1}, {2}, {3}, {1,3}. Total 5 including empty.
• Total valid non-empty = 5 - 1 = 4.

Common mistakes candidates make

In "The Number of Beautiful Subsets coding problem," many candidates forget to subtract the empty set from their final count. Another mistake is failing to account for duplicate numbers in the input array, which changes the number of ways to pick a specific value. Finally, some might attempt a pure backtracking approach which is too slow (O(2^n)) for large arrays, missing the chain-based DP optimization.

Interview preparation tip

When a problem involves constraints between specific pairs of numbers (like "don't pick x and x+k"), try to represent those relationships as a graph. Often, the graph will be a collection of simple paths or components, allowing you to solve the problem for each component independently and combine the results. This is a very powerful technique in combinatorics.