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

The optimal solution for Maximum Element-Sum of a Complete Subset of Indices runs in O(n * sqrt(max(nums))) time and uses O(n) space. This complexity arises from iterating through the nums array and calculating the prime product, which can take up to sqrt(max(nums)) time for each element.

What is the brute force solution for Maximum Element-Sum of a Complete Subset of Indices?

The brute force approach for Maximum Element-Sum of a Complete Subset of Indices is Brute Force, with O(n²) time complexity and O(1) space complexity. The brute force approach involves checking every possible pair of indices to see if their product is a perfect square. If it is, we can consider the corresponding values from the nums array for our subset sum. The optimal Optimal Solution approach improves this to O(n * sqrt(max(nums))).

What algorithm or data structure is used to solve Maximum Element-Sum of a Complete Subset of Indices?

Maximum Element-Sum of a Complete Subset of Indices uses the following concepts: Array, Math, Number Theory. The recommended patterns to study are: Hash Map, Array.

What are the interview tips for Maximum Element-Sum of a Complete Subset of Indices?

Always clarify the problem requirements and constraints before diving into coding. Think about how to optimize your approach after the brute force solution. Practice explaining your thought process clearly, as communication is key in interviews.

What are common mistakes when solving Maximum Element-Sum of a Complete Subset of Indices?

Failing to recognize that indices must be grouped based on their prime product. Not considering edge cases where no pairs can be formed.

#2862

Maximum Element-Sum of a Complete Subset of Indices

Hard

ArrayMathNumber TheoryHash MapArray

LeetCode ↗

Approaches

Brute ForceOptimal

Complexity Comparison

	Brute Force	Optimal Solution★
Time	O(n²)	O(n * sqrt(max(nums)))
Space	O(1)	O(n)

💡

Intuition

Time O(n * sqrt(max(nums)))Space O(n)

The optimal approach involves grouping indices based on their prime factorization characteristics. By using a HashMap to store sums of elements with the same prime product, we can efficiently calculate the maximum subset sum.

⚙️

Algorithm

4 steps

1Step 1: Create a function to compute the prime product P(x) for each index based on its number.
2Step 2: Use a HashMap to group the sums of nums based on their prime product.
3Step 3: Iterate through nums, calculate P(i) for each index, and update the HashMap with the sums.
4Step 4: The maximum value in the HashMap will be our result.

solution.py23 lines

1# Full working Python code
2import math
3from collections import defaultdict
4
5def prime_product(x):
6    factors = defaultdict(int)
7    for i in range(2, int(math.sqrt(x)) + 1):
8        while x % i == 0:
9            factors[i] += 1
10            x //= i
11    if x > 1:
12        factors[x] += 1
13    return tuple(p for p, exp in factors.items() if exp % 2 == 1)
14
15def max_subset_sum(nums):
16    sum_map = defaultdict(int)
17    for i in range(len(nums)):
18        p = prime_product(nums[i])
19        sum_map[p] += nums[i]
20    return max(sum_map.values())
21
22# Example usage
23print(max_subset_sum([8,7,3,5,7,2,4,9]))  # Output: 16

ℹ

Complexity note: This complexity arises from iterating through the nums array and calculating the prime product, which can take up to sqrt(max(nums)) time for each element.

1Understanding prime factorization is crucial for grouping indices effectively.
2Perfect squares have specific properties that can be exploited using prime factorization.

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