For several well-studied triangle detection problems, it is known that trivial enumeration algorithms are optimal (up to the exponent) under popular fine-grained conjectures. For instance, in All-Edges Sparse Triangle and Sparse Monochromatic Triangle, each node has a degree of $n^{\beta}$ for some $\beta > \log^2 n$.
Along the way, we show how to sort $n$ items over a universe of size $2^u$ using only $AC0$ word operations in $O(n u \log n)/w$ time.
Blogger's Review: This article sheds light on the complexity of triangle detection, showcasing significant advancements in theoretical computer science. The approach to sparse graphs opens new avenues for future research in this area.