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.

1. 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.