Let <code>p[i]</code> be whether <code>nums[i]</code> is a peak in the original array. Namely <code>p[i] = nums[i] > nums[i - 1] && nums[i] > nums[i + 1]</code>.

Updating <code>nums[i]</code>, only affects <code>p[i]</code>, <code>p[i - 1]</code> and <code>p[i + 1]</code>. We can recalculate the 3 values in constant time.

The answer for <code>[l_i, r_i]</code> is <code>p[l_i + 1] + p[l_i + 2] + … + p[r_i - 1]</code> (note that <code>l_i</code> and <code>r_i</code> are not included).

Use some data structures (i.e. segment tree or binary indexed tree) to maintain the subarray sum efficiently.