Maximum Element-Sum of a Complete Subset of Indices

Maximum Element-Sum of a Complete Subset of Indices

1. What is this problem about?

This Hard-level problem involves finding a 'complete' subset of indices from a 1-indexed array such that the sum of elements at these indices is maximized. A subset of indices is 'complete' if the product of any two indices in the subset is a perfect square. This requires a deep dive into Number Theory and understanding the properties of square-free parts of numbers.

2. Why is this asked in interviews?

The Maximum Element-Sum of a Complete Subset of Indices interview question is a test of mathematical maturity. It's less about traditional data structures and more about Number Theory and efficient categorization. It assesses if a candidate can break down a complex mathematical condition into a grouping problem that can be solved with a Hash Map or frequency array.

3. Algorithmic pattern used

The core pattern is Math and Number Theory. The condition "product of any two indices is a perfect square" implies that all indices in the subset must share the same 'square-free' part. For example, index 8 (2 * 2²) and index 18 (2 * 3²) both have a square-free part of 2. Their product 8 * 18 = 144 (12²) is a perfect square. We can iterate through the array, calculate the square-free part of each index, and sum the values of indices that share the same square-free part.

4. Example explanation

Array (1-indexed): [5, 10, 15, 20, 25]

• Index 1: Square-free part 1. Sum = 5.
• Index 2: Square-free part 2. Sum = 10.
• Index 3: Square-free part 3. Sum = 15.
• Index 4: 2², Square-free part 1. Index 1 and 4 can be in a subset (1*4=4). Sum = 5 + 20 = 25.
• Index 5: Square-free part 5. Sum = 25. The maximum sum is 25 (from indices {1, 4} or just index {5}).

5. Common mistakes candidates make

The biggest challenge is identifying the square-free property. Without it, candidates might try to check all pairs or use graph-based approaches, which are too slow. Another mistake is incorrect 1-indexing vs 0-indexing management. Calculating the square-free part inefficiently can also lead to TLE (Time Limit Exceeded).

6. Interview preparation tip

For Hard Math problems, look for properties of squares, primes, and divisors. Practice reducing numbers to their core components (like the square-free part). This 'grouping by property' is a powerful technique for many advanced competitive programming challenges.