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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, (opti- mal) 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, where the complexity of the algorithm is the sum of two terms. One term equals the complexity of the known algorithms by Assadi, Kapralov, and Khanna (ITCS 2019) and Fichtenberger and Peng (ICALP 2020) in the (strictly stronger) augmented model and the other is an additional, necessary, additive overhead. Our second contribution is a variant of our algorithm that en- ables nearly uniform sampling of these motifs, a capability pre- viously 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)). 

Constructing a spanning tree of a graph is one of the most basic tasks in graph theory. We consider a relaxed version of this problem in the setting of local algorithms. The relaxation is that the constructed subgraph is a sparse spanning subgraph containing at most (1+ϵ)n edges (where n is the number of vertices and ϵ is a given approximation/sparsity parameter). In the local setting, the goal is to quickly determine whether a given edge e belongs to such a subgraph, without constructing the whole subgraph, but rather by inspecting (querying) the local neighborhood of e. The challenge is to maintain consistency. That is, to provide answers concerning different edges according to the same spanning subgraph. We first show that for general bounded-degree graphs, the query complexity of any such algorithm must be Ω(n−−√). This lower bound holds for constant-degree graphs that have high expansion. Next we design an algorithm for (bounded-degree) graphs with high expansion, obtaining a result that roughly matches the lower bound. We then turn to study graphs that exclude a fixed minor (and are hence non-expanding). We design an algorithm for such graphs, which may have an unbounded maximum degree. The query complexity of this algorithm is poly(1/ϵ,h) (independent of n and the maximum degree), where h is the number of vertices in the excluded minor. Though our two algorithms are designed for very different types of graphs (and have very different complexities), on a high-level there are several similarities, and we highlight both the similarities and the differences. 

Constructing a spanning tree of a graph is one of the most basic tasks in graph theory. Motivated by several recent studies of local graph algorithms, we consider the following variant of this problem. Let G be a connected bounded-degree graph. Given an edge e in G we would like to decide whether e belongs to a connected subgraph math formula consisting of math formula edges (for a prespecified constant math formula), where the decision for different edges should be consistent with the same subgraph math formula. Can this task be performed by inspecting only a constant number of edges in G? Our main results are: We show that if every t-vertex subgraph of G has expansion math formula then one can (deterministically) construct a sparse spanning subgraph math formula of G using few inspections. To this end we analyze a “local” version of a famous minimum-weight spanning tree algorithm. We show that the above expansion requirement is sharp even when allowing randomization. To this end we construct a family of 3-regular graphs of high girth, in which every t-vertex subgraph has expansion math formula. We prove that for this family of graphs, any local algorithm for the sparse spanning graph problem requires inspecting a number of edges which is proportional to the girth.