Maximum Sum of Subsequence With Non-adjacent Elements

After each point update to an array, compute the maximum sum of a subsequence with no two adjacent elements (i.e., the maximum-weight independent set on a path) and return the sum of all query answers modulo 1e9+7. The challenge is to support up to 5·10^4 updates efficiently by maintaining range DP states (often with a segment tree / 2×2 max-plus matrices) to combine non-adjacent-selection choices in O(log n) per update.