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No, you can't beat $\Theta(\sqrt{n})$ queries. I will explain how to formalize exfret's proof sketch of this, in a way that works for adaptive algorithms. This is all anticipated in exfret's answer; I am just filling in some of the details. Consider any (possibly adaptive) algorithm that issues a sequence of queries, where each query is either "fetch the $...


Let’s assume we can only query the $i$th edge of a given vertex’s adjacency list (which I am assuming is not sorted) or whether two given vertices are adjacent. In this case it should take $\sqrt n$ queries to even find a cycle. This is because there is a $1-o(1)$ chance that all our queries of the first type return different vertices and that all of our ...

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