Cyclic sort

Start by clarifying the exact contract for Cyclic sort: input representation, mutation rules, duplicate behavior, and what should happen on invalid or empty input. The core idea is index placement under the explicit 1..n value-domain assumption. Work a tiny example aloud and name the invariant after each step, because that is the proof that the approach stays correct as the input grows.

A dependable implementation plan is to map x to index x - 1; swap until the current value is in its target index; use the settled layout to identify missing or duplicate values. Duplicates can cause a non-progressing swap, so detect nums[i] === nums[target]. Keep the code aligned with that invariant instead of adding special cases that silently change the contract. Test the smallest input, a normal case, duplicates or ties where relevant, an adversarial boundary, and a case that exercises the failure path. The expected complexity is O(n) time and O(1) extra space. Explain both time and auxiliary memory, including recursion or helper structures; do not hide sorting, copying, or repeated scans. If the title omits critical constraints, say what you need to know before committing to a more specialized solution.