3509 - Maximum Product of Subsequences With an Alternating Sum Equal to K.

Hard

You are given an integer array nums and two integers, k and limit. Your task is to find a non-empty subsequences of nums that:

• Has an alternating sum equal to k.

• Maximizes the product of all its numbers without the product exceeding limit.

Return the product of the numbers in such a subsequence. If no subsequence satisfies the requirements, return -1.

The alternating sum of a 0-indexed array is defined as the sum of the elements at even indices minus the sum of the elements at odd indices.

Example 1:

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

Output: 6

Explanation:

The subsequences with an alternating sum of 2 are:

• [1, 2, 3]

• [2]

The maximum product within the limit is 6.

Example 2:

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

Output: \-1

Explanation:

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

Explanation:

The subsequences with an alternating sum of 0 are:

• [2, 2]

• [3, 3]

• [2, 2, 3, 3]

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

• <code>-10⁵<= k <= 10⁵</code>

• 1 <= limit <= 5000