Is Maximum Element-Sum of a Complete Subset of Indices easy or hard?

Maximum Element-Sum of a Complete Subset of Indices is classified as Hard on LeetCode. The challenge comes from recognizing the number theory property that converts the pairwise perfect-square condition into grouping by square-free index factors.

Maximum Element-Sum of a Complete Subset of Indices Python/Java solution

Python and Java implementations typically compute the square-free kernel for each index, store cumulative sums in a hash map, and track the maximum group sum. The logic is identical across languages and runs in O(n log n) time with O(n) space.

How to solve Maximum Element-Sum of a Complete Subset of Indices in O(n)?

A near-linear solution is possible by precomputing smallest prime factors using a sieve and deriving the square-free kernel of each index in constant amortized time. Group indices by this kernel and accumulate sums, effectively achieving O(n) to O(n log log n) preprocessing with O(n) traversal.

What is the best approach for Maximum Element-Sum of a Complete Subset of Indices?

The optimal approach groups indices by their square-free kernel using prime factorization. If two indices have the same square-free representation, their product is guaranteed to be a perfect square. Summing array values for each group and returning the maximum produces the result in O(n log n) time and O(n) space.

Is Maximum Element-Sum of a Complete Subset of Indices asked at Google/Amazon/Meta?

This problem is categorized as Hard and involves number theory and mathematical transformations, patterns commonly seen in interviews at companies like Google, Amazon, and Meta. Variations involving perfect squares, factorization, or index grouping appear in algorithmic interview rounds.

What data structure is used in Maximum Element-Sum of a Complete Subset of Indices?

The main data structure is a hash map that groups indices by their square-free kernel. Additional tools include prime factorization or sieve arrays for number theory preprocessing, along with basic array traversal.

What is the time complexity of Maximum Element-Sum of a Complete Subset of Indices?

The optimal factorization-based grouping runs in O(n log n) time with O(n) space. Each index is reduced to its square-free form using prime factorization or precomputed smallest prime factors, and values are aggregated using a hash map.

Description Hints 3 Solution Video 11:50

2862. Maximum Element-Sum of a Complete Subset of Indices

Hard42.9% acceptance1 company

You are given a 1-indexed array nums. Your task is to select a complete subset from nums where every pair of selected indices multiplied is a perfect square,. i. e. if you select a_i and a_j, i * j must be a perfect square.

Return the sum of the complete subset with the maximum sum.

Example 1:

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

Output: 16

Explanation:

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

Explanation:

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

Asked at 1 company

Purplle

Topics

Hints open one at a time

Hint 1

Define P(x) as the product of primes p with odd exponents in x's factorization. Examples: For x = 18, factorization 2¹ × 3², P(18) = 2; for x = 45, factorization 3² × 5¹, P(45) = 5; for x = 50, factorization 2¹ × 5², P(50) = 2; for x = 210, factorization 2¹ × 3¹ × 5¹ × 7¹, P(210) = 210.

Hint 2

If P(i) = P(j), nums[i] and nums[j] can be grouped together.

Hint 3

Pick the group with the largest sum.

11:50

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