Find the Earliest Contiguous Removal That Balances Index Sums

Problem Statement

Given an integer array nums and an integer k, you need to remove a contiguous subarray of length k. After the removal, the remaining elements are concatenated, and their indices are recalculated (from 0 to n-k-1). Your goal is to find the starting index of the earliest contiguous subarray of length k such that the sum of elements at even indices equals the sum of elements at odd indices in the resulting array.

Return the smallest starting index i (0 int:`

Examples

Example 1: Input: nums = [2, 1, 6, 4], k = 1

• Remove index 0: [1, 6, 4]. Even sums (index 0+2): 1+4=5. Odd sums (index 1): 6. (5 != 6)
• Remove index 1: [2, 6, 4]. Even sums: 2+4=6. Odd sums: 6. (6 == 6). Return 1.

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

• Remove index 0: [1, 1]. Even: 1. Odd: 1. Return 0.

Constraints

• n = len(nums)
• 1 <= k < n <= 10^5
• -10^4 <= nums[i] <= 10^4

Edge Cases

• Array length equals k+1.
• No solution exists.
• Removing the subarray leaves an empty list or a list of size 1.