Maximum Product of Subsequences With an Alternating Sum Equal to K

To solve this problem, first identify subsequences that match the alternating sum. Then compute the maximum product for valid subsequences, checking against the limit. Use dynamic programming and hash lookup for efficient computation.

Problem Statement

You are given an integer array nums and two integers, k and limit. Your task is to find a non-empty subsequence of nums such that the alternating sum equals k. The alternating sum of a 0-indexed array is the sum of the elements at even indices minus the sum of the elements at odd indices.

Once you find such a subsequence, return the product of its elements. If no such subsequence exists or the product exceeds the given limit, return -1. Make sure to optimize your solution by leveraging dynamic programming and hash lookup techniques.

Examples

Example 1

Input: nums = [1,2,3], k = 2, limit = 10

Output: 6

The subsequences with an alternating sum of 2 are: The maximum product within the limit is 6.

Example 2

Input: nums = [0,2,3], k = -5, limit = 12

Output: -1

A subsequence with an alternating sum of exactly -5 does not exist.

Example 3

Input: nums = [2,2,3,3], k = 0, limit = 9

Output: 9

The subsequences with an alternating sum of 0 are: The subsequence [2, 2, 3, 3] has the greatest product with an alternating sum equal to k , but 36 > 9 . The next greatest product is 9, which is within the limit.

Constraints

• 1 <= nums.length <= 150
• 0 <= nums[i] <= 12
• -105 <= k <= 105
• 1 <= limit <= 5000

Solution Approach

Dynamic Programming with Hash Lookup

Use dynamic programming (DP) to maintain the states of possible alternating sums as you iterate over the array. Hash lookup can help track the maximum product for each valid sum, ensuring efficiency in computing the result.

Subsequence Generation and Alternating Sum Calculation

Generate subsequences by scanning the array. For each subsequence, calculate its alternating sum and check if it equals k. If valid, track its product while ensuring it does not exceed the limit.

Efficient Product Calculation within Limit

For subsequences that match the alternating sum, calculate the product of their elements. Use dynamic programming to efficiently check each subsequence's product and ensure it remains within the specified limit.

Complexity Analysis

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

The time complexity depends on the approach used for dynamic programming and hashing. Typically, it involves scanning the array and checking sums with hash lookups, which could lead to a time complexity of O(n * k) where n is the array length and k is the range of sums. The space complexity is also determined by the size of the DP table and hash map, usually O(n * k).

What Interviewers Usually Probe

• Ability to handle dynamic programming efficiently for array-based problems.
• Understanding of hash lookups for optimizing subproblem storage.
• Skills in balancing time and space complexity for non-trivial problem constraints.

Common Pitfalls or Variants

Common pitfalls

• Not properly maintaining the DP states for alternating sums.
• Failing to limit the product calculation within the given limit.
• Incorrect handling of edge cases where no valid subsequences exist.

Follow-up variants

• Consider cases where the alternating sum can be zero.
• Alter the problem by changing the range of k or the product limit.
• Introduce additional constraints on the subsequence length.

FAQ

What is the pattern for solving the Maximum Product of Subsequences With an Alternating Sum Equal to K?

The main pattern involves scanning the array while dynamically tracking alternating sums and their associated products using dynamic programming and hash lookups.

What should I focus on when solving this problem?

Focus on efficiently managing subsequences and their alternating sums using dynamic programming. Pay attention to the product limit and ensure it’s respected.

What happens if no subsequence matches the alternating sum?

If no valid subsequence matches the alternating sum of k, return -1 as specified in the problem statement.

How does dynamic programming apply to this problem?

Dynamic programming is used to track the possible sums and their corresponding products, which allows for efficient checking and updating as you process the array.

Can this problem be solved using brute force?

While brute force could solve the problem by generating all subsequences, it would be inefficient for larger inputs. A dynamic programming approach significantly improves performance.