Equivalence testing, a fundamental problem in the field of distribution testing, seeks to infer if two unknown distributions on $[n]$ are the same or far apart in the total variation distance. Conditional sampling has emerged as a powerful query model and has been investigated by theoreticians and practitioners alike, leading to the design of optimal algorithms albeit in a sequential setting (also referred to as adaptive tester). Given the profound impact of parallel computing over the past decades, there has been a strong desire to design algorithms that enable high parallelization. Despite significant algorithmic advancements over the last decade, parallelizable techniques (also termed non-adaptive testers) have $\tilde{O}(\log^{12}n)$ query complexity, a prohibitively large complexity to be of practical usage. Therefore, the primary challenge is whether it is possible to design algorithms that enable high parallelization while achieving efficient query complexity. Our work provides an affirmative answer to the aforementioned challenge: we present a highly parallelizable tester with a query complexity of $\tilde{O}(\log n)$, achieved through a single round of adaptivity, marking a significant stride towards harmonizing parallelizability and efficiency in equivalence testing.

翻译：等价性测试是分布测试领域的一个基本问题，旨在推断$[n]$上的两个未知分布在总变差距离下是否相同或相距甚远。条件采样已成为一种强大的查询模型，并受到理论家和实践者的广泛研究，尽管是在顺序设置（也称为适应性测试器）下，仍催生了最优算法的设计。鉴于过去几十年并行计算的深远影响，设计支持高度并行化的算法已成为强烈需求。尽管过去十年间算法取得了显著进展，可并行化的技术（也称为非适应性测试器）的查询复杂度为$\tilde{O}(\log^{12}n)$，这一复杂度过高，难以实际应用。因此，核心挑战在于是否可能设计出既支持高度并行化又实现高效查询复杂度的算法。我们的工作对上述挑战给出了肯定答案：我们提出了一种高度可并行化的测试器，其查询复杂度为$\tilde{O}(\log n)$，仅通过单轮适应性即可实现，这标志着在等价性测试中协调并行性与效率方面迈出了重要一步。