Alternating Groups III - Solution & Explanation

Problem Overview: You are given a circular array of colors and a sequence of queries. Some queries update the color at an index, while others ask how many alternating groups of a given size exist in the array. An alternating group means adjacent elements strictly alternate (e.g., 0,1,0,1). Because updates modify the structure dynamically, the solution must efficiently maintain alternating segments while answering queries quickly.

Approach 1: Recursive Approach with Memoization (Brute-Force Simulation) (Time: O(n * q), Space: O(n))

The most direct method recomputes alternating segments whenever a query asks for the number of valid groups. Traverse the array and recursively check whether consecutive elements alternate, storing intermediate results with memoization to avoid recomputing overlapping segments. For each query, scan the array and count segments whose length satisfies the requested size. This works for small inputs but becomes slow when updates and queries are frequent because each query may require scanning most of the array.

Approach 2: Dynamic Programming with Binary Indexed Tree (Optimal) (Time: O((n + q) log n), Space: O(n))

The efficient strategy tracks maximal alternating segments instead of recomputing them from scratch. First, determine whether each adjacent pair alternates. This transforms the problem into maintaining runs of valid alternating edges. When a color update occurs, only the two neighboring relationships change, so the affected segments can be split or merged locally. Use a Binary Indexed Tree to store counts of segment lengths and support prefix queries for "how many segments have length ≥ k".

Dynamic programming ideas help maintain the length of alternating runs starting at each index. When an update breaks or forms alternation, adjust the boundaries of the affected segments and update their lengths in the Fenwick tree. Each modification touches only a constant number of segments, keeping updates efficient. Query operations then become simple range queries over segment lengths using the BIT.

This approach works well because it converts a structural property of the array into a frequency problem that a Fenwick tree handles efficiently. The array itself is still accessed directly for updates, while the BIT maintains aggregate statistics about segment sizes.

Recommended for interviews: The Binary Indexed Tree solution is what interviewers expect for a hard problem with dynamic updates. The brute-force or recursive simulation demonstrates you understand how alternating groups are formed, but the optimized structure shows you can maintain derived properties under updates. Strong candidates recognize that only local relationships change after an update and use data structures like a array plus Fenwick tree or segment tree to maintain counts efficiently. Concepts from dynamic programming help track alternating run lengths, while the BIT enables fast aggregated queries.