What is the brute force solution for Maximum and Minimum Sums of at Most Size K Subsequences?

The brute force approach for Maximum and Minimum Sums of at Most Size K Subsequences is Brute Force, with O(n²) time complexity and O(1) space complexity. The brute-force approach generates all possible subsequences of the array and calculates the sum of the minimum and maximum elements for each subsequence with at most k elements. This method is straightforward but inefficient due to the exponential number of subsequences. The optimal Optimal Solution approach improves this to O(n log n).

What algorithm or data structure is used to solve Maximum and Minimum Sums of at Most Size K Subsequences?

Maximum and Minimum Sums of at Most Size K Subsequences uses the following concepts: Array, Math, Dynamic Programming, Sorting, Combinatorics. The recommended patterns to study are: Combinatorics, Sorting.

What are the interview tips for Maximum and Minimum Sums of at Most Size K Subsequences?

Always clarify the constraints and edge cases with the interviewer. Discuss your thought process and potential optimizations as you work through the problem. Practice combinatorial problems to become comfortable with counting techniques.

What are common mistakes when solving Maximum and Minimum Sums of at Most Size K Subsequences?

Forgetting to consider subsequences of size 1. Not handling large numbers properly due to overflow.

#3428

Maximum and Minimum Sums of at Most Size K Subsequences

Medium

ArrayMathDynamic ProgrammingSortingCombinatoricsCombinatoricsSorting

LeetCode ↗

Approaches

Brute ForceOptimal

Complexity Comparison

	Brute Force	Optimal Solution★
Time	O(n²)	O(n log n)
Space	O(1)	O(1)

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Intuition

Time O(n log n)Space O(1)

The optimal solution leverages sorting and combinatorial mathematics to efficiently calculate the contributions of each element as both minimum and maximum in valid subsequences. By sorting the array, we can directly compute how many times each element contributes to the total sum without generating all subsequences.

⚙️

Algorithm

3 steps

1Step 1: Sort the array nums.
2Step 2: For each element in the sorted array, calculate its contribution as a minimum and maximum in all valid subsequences.
3Step 3: Use combinatorial counting to determine how many subsequences of size up to k include the current element as min or max.

solution.py15 lines

1# Full working Python code
2from math import comb
3
4def max_min_sum(nums, k):
5    MOD = 10**9 + 7
6    nums.sort()
7    n = len(nums)
8    total_sum = 0
9
10    for i in range(n):
11        min_contrib = nums[i] * (comb(i, 0) + sum(comb(i, j) for j in range(1, k + 1))) % MOD
12        max_contrib = nums[i] * (comb(n - i - 1, 0) + sum(comb(n - i - 1, j) for j in range(1, k + 1))) % MOD
13        total_sum = (total_sum + min_contrib + max_contrib) % MOD
14
15    return total_sum

ℹ

Complexity note: The optimal solution sorts the array in O(n log n) time, and then iterates through the array in O(n) time to calculate contributions, leading to an overall time complexity of O(n log n). The space complexity is O(1) since we only use a few variables.

1Sorting the array allows for efficient calculation of contributions.
2Using combinatorial mathematics reduces the need to generate all subsequences.

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