Maximum and Minimum Sums of at Most Size K Subsequences

This problem requires calculating the sum of maximum and minimum elements of subsequences of size up to k from an integer array. The solution involves using state transition dynamic programming, which optimizes the process of generating subsequences by leveraging sorting and dynamic state updates.

Problem Statement

You are given an integer array nums and a positive integer k. Your task is to compute the sum of the maximum and minimum elements of all subsequences of nums with at most k elements. The output should be returned modulo 10^9 + 7 due to potentially large sums.

The subsequences should consider sizes from 1 to k elements. Sorting the array first helps in efficiently generating the necessary subsequences for calculating both the minimum and maximum values across all possible subsequences.

Examples

Example 1

Input: nums = [1,2,3], k = 2

Output: 24

The subsequences of nums with at most 2 elements are: The output would be 24.

Example 2

Input: nums = [5,0,6], k = 1

Output: 2 2

For subsequences with exactly 1 element, the minimum and maximum values are the element itself. Therefore, the total is 5 + 5 + 0 + 0 + 6 + 6 = 22 .

Example 3

Input: nums = [1,1,1], k = 2

Output: 12

The subsequences [1, 1] and [1] each appear 3 times. For all of them, the minimum and maximum are both 1. Thus, the total is 12.

Constraints

• 1 <= nums.length <= 105
• 0 <= nums[i] <= 109
• 1 <= k <= min(70, nums.length)

Solution Approach

State Transition Dynamic Programming

The solution builds on state transition dynamic programming. First, sort the array, and then use dynamic programming to compute the sums for all subsequences of size up to k. This allows efficient computation of the sum of minimum and maximum elements for each subsequence.

Efficient Calculation with Sorting

By sorting the array, the problem simplifies into efficiently computing sums for subsequences. This sorting ensures that finding maximum and minimum values across subsequences becomes straightforward by leveraging the properties of sorted arrays.

Modulo Operation for Large Results

Since the result can be large, every intermediate result and the final answer is computed modulo 10^9 + 7 to avoid overflow and ensure correctness within the constraints.

Complexity Analysis

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

The time complexity of this solution depends on the sorting step, which is O(n log n), followed by dynamic programming, which operates in O(k). The space complexity is O(k) for storing intermediate results.

What Interviewers Usually Probe

• The candidate demonstrates understanding of dynamic programming and state transitions.
• The candidate suggests sorting the array as a means to simplify subsequence generation.
• The candidate efficiently handles the modulo operation to ensure correct results for large sums.

Common Pitfalls or Variants

Common pitfalls

• Forgetting to apply the modulo operation may lead to incorrect results.
• Not sorting the array, which leads to inefficient subsequence calculations.
• Overlooking edge cases, such as when k equals 1 or when nums contains duplicate values.

Follow-up variants

• Adjusting the value of k to examine subsequences of varying sizes.
• Changing the problem to find only the maximum or minimum sums.
• Modifying the problem to return only the sum for a specific subsequence size.

FAQ

What is the primary approach to solve 'Maximum and Minimum Sums of at Most Size K Subsequences'?

The primary approach is state transition dynamic programming, where you sort the array and calculate sums for subsequences using dynamic states.

How does sorting help in solving the problem?

Sorting the array simplifies finding the maximum and minimum values for subsequences, allowing dynamic programming to efficiently compute the required sums.

What are the constraints of the 'Maximum and Minimum Sums of at Most Size K Subsequences' problem?

The length of the array nums can be up to 10^5, the array elements can be as large as 10^9, and k is bounded by min(70, nums.length).

What is the role of modulo 10^9 + 7 in this problem?

The modulo operation ensures that the computed sums remain within bounds, as the sums can become extremely large.

Can I solve the 'Maximum and Minimum Sums of at Most Size K Subsequences' problem without dynamic programming?

While it is possible to approach this problem without dynamic programming, using dynamic programming optimizes the solution, especially for large arrays and subsequences.