Mo's Algorithm: When to Use It

Start with the idea

A problem is a good match for Mo's Algorithm when it asks for many immutable range queries with cheap add/remove. Here the concrete task is: Return distinct counts for ranges [0,3], [2,5], [0,5]. The visible input [1,2,1,3,2,4] is an example, not a definition of the whole problem. Ask what operations the solution must support and what the input guarantees. Choose it when the required operation sequence, query volume, or substring structure exceeds a simple scan or prefix summary; compare the real time and memory limits first. The right choice follows from those guarantees, not from a familiar noun in the prompt.

The decisive invariant is: Order offline range queries so adjacent queries move their boundaries only a small distance. Try to say why it applies to the task without quoting the implementation. Which fact becomes known after one update, and which possible answers remain? An advanced structure or offline query order packages an invariant across operations. Name what each node, block, or reordered query state stores. If that claim cannot be maintained on a legal input, the pattern is wrong or one of its preconditions is missing. A counterexample is more useful than a vague warning: change one property of the sample, trace the first two steps, and identify the exact claim that fails.

Compare costs before committing. The advertised bound is O((n+q)*sqrt(n)) operations · O(state) space; identify the operation responsible for it and the extra memory it needs.