Counting small subgraphs, referred to as motifs, in large graphs is a fundamental task in graph analysis, extensively studied across various contexts and computational models. In the sublinear-time regime, the relaxed problem of approximate counting has been explored within two prominent query frameworks: the standard model, which permits degree, neighbor, and pair queries, and the strictly more powerful augmented model, which additionally allows for uniform edge sampling. Currently, in the standard model, (optimal) results have been established only for approximately counting edges, stars, and cliques, all of which have a radius of one. This contrasts sharply with the state of affairs in the augmented model, where algorithmic results (some of which are optimal) are known for any input motif, leading to a disparity which we term the ``scope gap" between the two models. In this work, we make significant progress in bridging this gap. Our approach draws inspiration from recent advancements in the augmented model and utilizes a framework centered on counting by uniform sampling, thus allowing us to establish new results in the standard model and simplify on previous results. In particular, our first, and main, contribution is a new algorithm in the standard model for approximately counting any Hamiltonian motif in sublinear time. Our second contribution is a variant of our algorithm that enables nearly uniform sampling of these motifs, a capability previously limited in the standard model to edges and cliques. Our third contribution is to introduce even simpler algorithms for stars and cliques by exploiting their radius-one property. As a result, we simplify all previously known algorithms in the standard model for stars (Gonen, Ron, Shavitt (SODA 2010)), triangles (Eden, Levi, Ron Seshadhri (FOCS 2015)) and cliques (Eden, Ron, Seshadri (STOC 2018)).

翻译：在大图中计数被称为“子图模式”的小规模子图是图分析中的一项基本任务，已在多种背景和计算模型中得到广泛研究。在亚线性时间范畴内，近似计数这一宽松问题已在两种主流的查询框架下得到探索：标准模型（允许度查询、邻居查询和节点对查询）以及严格更强大的增强模型（额外允许均匀边采样）。目前，在标准模型中，（最优）结果仅针对半径为一的子图（如边、星形和团）的近似计数得以建立。这与增强模型中的现状形成鲜明对比——增强模型中对任意输入子图模式均已存在算法结果（部分为最优），我们称这种差异为两种模型间的“范围鸿沟”。本工作在此鸿沟的弥合上取得重要进展。我们的方法受增强模型最新进展的启发，采用以均匀采样为核心的计数框架，从而在标准模型中建立新结果并简化先前结果。具体而言，我们首要且主要的贡献是提出一种标准模型中的新算法，可在亚线性时间内近似计数任意哈密顿子图模式。第二项贡献是该算法的变体，能够实现对这些子图模式的近似均匀采样——此前在标准模型中该能力仅适用于边和团。第三项贡献是通过利用半径为一的特性，为星形和团设计了更为简洁的算法。由此，我们简化了标准模型中所有已知算法：星形（Gonen, Ron, Shavitt (SODA 2010)）、三角形（Eden, Levi, Ron Seshadhri (FOCS 2015)）以及团（Eden, Ron, Seshadri (STOC 2018)）。