We study the parallel (adaptive) complexity of the classic problem of finding a basis in an $n$-element matroid, given access via an \emph{independence oracle}. In this model, the algorithm may submit polynomially many independence queries in each round, and the central question is: how many rounds are necessary and sufficient to find a basis? Karp, Upfal, and Wigderson (FOCS~1985, JCSS~1988; hereafter KUW) initiated this study, showing that $O(\sqrt{n})$ adaptive rounds suffice for any matroid, and that $\widetildeΩ(n^{1/3})$ rounds are necessary even for partition matroids. This left a substantial gap that persisted for nearly four decades, until Khanna, Putterman, and Song (FOCS~2025; hereafter KPS) achieved $\widetilde O(n^{7/15})$ rounds, the first improvement since~KUW. In this work, we make another conceptual advance beyond KPS, giving a new algorithm that finds a matroid basis in $\widetilde O(n^{3/7})$ rounds. We develop a structural and algorithmic framework that brings a new lens to the analysis of random circuits, moving from reasoning about individual elements to understanding how dependencies span multiple elements simultaneously.

翻译：我们研究在经典问题——在 $n$ 元拟阵中寻找一个基——的并行（自适应）复杂度，其中通过一个*独立性预言机*访问。在该模型中，算法可在每一轮提交多项式数量的独立性查询，核心问题是：需要多少轮自适应查询才能找到基？Karp、Upfal 和 Wigderson（FOCS~1985, JCSS~1988；以下简称 KUW）开创了这一研究，证明 $O(\sqrt{n})$ 轮自适应查询足以适用于任何拟阵，且即使对于分区拟阵也需要 $\widetildeΩ(n^{1/3})$ 轮。这一巨大差距持续了近四十年，直到 Khanna、Putterman 和 Song（FOCS~2025；以下简称 KPS）实现了 $\widetilde O(n^{7/15})$ 轮，这是自 KUW 以来的首次改进。在本工作中，我们做出了超越 KPS 的另一概念性进展，提出了一种新算法，可在 $\widetilde O(n^{3/7})$ 轮内找到拟阵基。我们开发了一个结构性和算法性框架，为随机电路分析提供了新视角，从关注单个元素转向理解依赖关系如何同时跨越多个元素。