This work focuses on the problem of learning an unknown $3$-uniform hypergraph using edge-detecting queries. Our goal is to design a querying strategy that recovers the hyperedge set using as few queries as possible. We restrict our attention to random hypergraphs under the Erdős--Rényi (ER) model, in which each potential hyperedge appears independently with probability $q = Θ(n^{-3(1-θ)})$ for $θ\in (0;1)$. Prior work [Austhof-Reyzin-Tani, ISIT 2025] presents a testing-decoding scheme that uses $O(\bar{m}\log n)$ tests but requires a decoding time of $Ω(n^3)$, where $\bar{m} = q\binom{n}{3}$ denotes the expected number of hyperedges. In this work, we extend the binary splitting framework and adapt it to the $3$-uniform hypergraph setting. We obtain a testing-decoding scheme that recovers the hyperedge set with high probability using $O(\bar{m} \log n)$ tests and achieves decoding time $O(\bar{m}^{5/3}\log n)$ for the case $θ> \dfrac{2}{3}$ and $O(\bar{m}^{5/3}\log^2{\bar{m}}\log n)$ for the case $θ\leq \dfrac{2}{3}$. Thus, compared with prior work, our result significantly improves the decoding complexity while maintaining optimal query complexity.

翻译：本文关注于使用边检测查询学习未知$3$-均匀超图的问题。我们的目标是设计一种查询策略，以尽可能少的查询恢复超边集。我们将注意力限制在埃尔德什—雷尼（ER）模型下的随机超图上，其中每条潜在超边以概率$q = Θ(n^{-3(1-θ)})$独立出现，$θ\in (0;1)$。先前的工作[Austhof-Reyzin-Tani, ISIT 2025]提出了一种测试-解码方案，使用$O(\bar{m}\log n)$次测试，但需要$Ω(n^3)$的解码时间，其中$\bar{m} = q\binom{n}{3}$表示期望的超边数。在本文中，我们扩展了二分分裂框架，并将其适配到$3$-均匀超图场景。我们获得了一种测试-解码方案，该方案能以高概率恢复超边集，使用$O(\bar{m} \log n)$次测试，并实现解码时间：对于$θ> \dfrac{2}{3}$的情况为$O(\bar{m}^{5/3}\log n)$，对于$θ\leq \dfrac{2}{3}$的情况为$O(\bar{m}^{5/3}\log^2{\bar{m}}\log n)$。因此，与先前工作相比，我们的结果在保持最优查询复杂度的同时，显著降低了解码复杂度。