Floyd State Cycle Detection: When to Use It

Start with the idea

A problem is a good match for Floyd State Cycle Detection when it asks for repeated state machines, linked transitions, duplicate-by-cycle reductions. Here the concrete task is: Given next indexes, return the first value in the cycle reached from index 0. The visible input next=[1,2,3,1] is an example, not a definition of the whole problem. Ask what operations the solution must support and what the input guarantees. Ask whether one pass is enough, whether order matters, and whether a later query will need information you already discarded. The right choice follows from those guarantees, not from a familiar noun in the prompt.

The decisive invariant is: Advance one state once and another twice; their meeting proves a cycle without a visited set. 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? A scan turns a sequence into a small piece of state. Say what the state means after the first i elements, before you describe a loop body. 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(mu + lambda) time · O(1) space; identify the operation responsible for it and the extra memory it needs.