#2862

Hard

LeetCode Problem Workspace

Maximum Element-Sum of a Complete Subset of Indices

Given a 1-indexed array, select a subset where indices' product is a perfect square, then return the maximum sum.

Problem Statement

You are given a 1-indexed array nums. Your task is to find a complete subset of indices such that every pair of selected indices, when multiplied, results in a perfect square. In other words, for any two selected indices i and j, i * j must be a perfect square.

Return the sum of the subset with the highest sum. For example, given nums = [8,7,3,5,7,2,4,9], the subset with indices 2 and 8 gives a sum of 16 as 2 * 8 = 16, which is a perfect square.

Examples

Example 1

Input: nums = [8,7,3,5,7,2,4,9]

Output: 16

We select elements at indices 2 and 8 and 2 * 8 is a perfect square.

Example 2

Input: nums = [8,10,3,8,1,13,7,9,4]

Output: 20

We select elements at indices 1, 4, and 9. 1 * 4 , 1 * 9 , 4 * 9 are perfect squares.

Constraints

1 <= n == nums.length <= 104
1 <= nums[i] <= 109

Solution Approach

Prime Factorization Approach

Using prime factorization, define the product of primes with odd exponents in a number's factorization. This can help determine the pairs of indices where their product is a perfect square.

Efficient Search for Subsets

You can use dynamic programming or memoization techniques to efficiently search for subsets that satisfy the condition, ensuring the solution works within the time constraints.

Optimization through Number Theory

Optimize the approach by recognizing mathematical properties of numbers, specifically leveraging perfect squares' divisibility rules to eliminate unnecessary computations.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time and space complexity depend on the approach used, specifically in terms of prime factorization, subset checking, and dynamic programming. A well-optimized solution can bring the time complexity down to approximately O(n^2) or better for large inputs.

What Interviewers Usually Probe

Look for understanding of prime factorization and perfect square properties.
Check if the candidate can apply number theory efficiently in subset problems.
Evaluate the ability to optimize solutions with dynamic programming or other techniques.

Common Pitfalls or Variants

Common pitfalls

Overlooking the importance of prime factorization for identifying perfect square pairs.
Failing to optimize the solution for large input sizes (n up to 10^4).
Choosing a brute force approach without considering number theory for efficiency.

Follow-up variants

Extend the problem by finding the subset with the minimum sum instead of the maximum sum.
Consider a variant where the perfect square condition is replaced with a different mathematical property (e.g., perfect cube).
Modify the problem to allow subsets that are not necessarily contiguous but still respect the perfect square condition.

FAQ

What is the key to solving the 'Maximum Element-Sum of a Complete Subset of Indices' problem?

The key is understanding prime factorization and identifying subsets where every pair of indices' product results in a perfect square.

How can I optimize my solution for large inputs in the 'Maximum Element-Sum of a Complete Subset of Indices' problem?

Use dynamic programming or memoization to efficiently find valid subsets, and apply number theory to avoid unnecessary computations.

What is the mathematical principle that makes two indices' product a perfect square?

Two indices' product is a perfect square if the product of their prime factors has even exponents for every prime factor.

What are the common pitfalls when solving the 'Maximum Element-Sum of a Complete Subset of Indices' problem?

Common pitfalls include not utilizing prime factorization efficiently, failing to optimize the solution, and using brute force methods for large inputs.

Can I solve the 'Maximum Element-Sum of a Complete Subset of Indices' problem without using number theory?

Number theory techniques, especially prime factorization, are critical for solving this problem efficiently and correctly, so avoiding them is not recommended.

terminal

Solution

Solution 1: Enumeration

We note that if a number can be expressed in the form of $k \times j^2$, then all numbers of this form have the same $k$.

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class Solution:
    def maximumSum(self, nums: List[int]) -> int:
        n = len(nums)
        ans = 0
        for k in range(1, n + 1):
            t = 0
            j = 1
            while k * j * j <= n:
                t += nums[k * j * j - 1]
                j += 1
            ans = max(ans, t)
        return ans

Continue Practicing

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