Implement several OA simulation problems

Reconstruct and solve the following coding problems from an online assessment. 1. **Case-insensitive adjacent differences** Given a string `s`, treat uppercase and lowercase letters as the same. Count how many indices `i` with `0 <= i < len(s) - 1` satisfy `lower(s[i]) != lower(s[i + 1])`. 2. **Memory allocator with 8-aligned starts** You are given an initial binary array `memory[0..n-1]`, where `0` means free and `1` means occupied. Support two operations: - `alloc(x)`: find the leftmost start index `s` such that `s % 8 == 0`, `s + x <= n`, and every value in `memory[s..s+x-1]` is `0`. Allocate that entire block to a newly created allocation ID, and mark all those cells as occupied by the same ID. If no such block exists, report failure. - `erase(id)`: free every cell currently assigned to `id` by setting it back to `0`. If `id` does not exist or has already been erased, report failure. Implement a simulator for these operations. 3. **Online counting of same-color contiguous segments** Points lie on integer positions of a line. Each update assigns or changes the color of one point. After every update, return the current number of contiguous segments, where a segment is a maximal run of adjacent positions with the same color. Only colored positions participate; a missing position breaks continuity. Design an efficient online method to maintain the segment count as points are inserted or recolored.