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1. Sampling Multiple Edges Efficiently

   Eden, T.; Mossel, S.; Rubinfeld, R. (January 2021, Random)

   We present a sublinear time algorithm that allows one to sample multiple edges from a distribution that is pointwise ϵ-close to the uniform distribution, in an amortized-efficient fashion. We consider the adjacency list query model, where access to a graph G is given via degree and neighbor queries. The problem of sampling a single edge in this model has been raised by Eden and Rosenbaum (SOSA 18). Let n and m denote the number of vertices and edges of G, respectively. Eden and Rosenbaum provided upper and lower bounds of Θ∗(n/ √ m) for sampling a single edge in general graphs (where O ∗(·) suppresses poly(1/ϵ) and poly(log n) dependencies). We ask whether the query complexity lower bound for sampling a single edge can be circumvented when multiple samples are required. That is, can we get an improved amortized per-sample cost if we allow a preprocessing phase? We answer in the affirmative. We present an algorithm that, if one knows the number of required samples q in advance, has an overall cost that is sublinear in q, namely, O∗(√ q · (n/ √ m)), which is strictly preferable to O∗(q · (n/ √ m)) cost resulting from q invocations of the algorithm by Eden and Rosenbaum. Subsequent to a preliminary version of this work, Tětek and Thorup (arXiv, preprint) proved that this bound is essentially optimal. 

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2. Talya Eden, Saleet Mossel, Ronitt Rubinfeld, Sampling Multiple Edges Efficiently

   Eden, T.; Mossel, S.; Rubinfeld, R. (January 2021, RANDOM 2021)

   We present a sublinear time algorithm that allows one to sample multiple edges from a distribution that is pointwise ε-close to the uniform distribution, in an amortized-efficient fashion. We consider the adjacency list query model, where access to a graph G is given via degree and neighbor queries. The problem of sampling a single edge in this model has been raised by Eden and Rosenbaum (SOSA 18). Let n and m denote the number of vertices and edges of G, respectively. Eden and Rosenbaum provided upper and lower bounds of Θ^*(n/√ m) for sampling a single edge in general graphs (where O^*(⋅) suppresses poly(1/ε) and poly(log n) dependencies). We ask whether the query complexity lower bound for sampling a single edge can be circumvented when multiple samples are required. That is, can we get an improved amortized per-sample cost if we allow a preprocessing phase? We answer in the affirmative. We present an algorithm that, if one knows the number of required samples q in advance, has an overall cost that is sublinear in q, namely, O^*(√ q ⋅(n/√ m)), which is strictly preferable to O^*(q⋅ (n/√ m)) cost resulting from q invocations of the algorithm by Eden and Rosenbaum. Subsequent to a preliminary version of this work, Tětek and Thorup (arXiv, preprint) proved that this bound is essentially optimal. 

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3. Approximately counting and sampling Hamiltonian motifs in sublinear time

   Levi, Reut; Eden, Talya; Ron, Dana; Rubinfeld, Ronitt (June 2025, 57th Annual ACM Symposium on Theory of Computing (STOC 2025))

   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)). 

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4. Locally Computing Edge Orientations

   Mitrovic, Slobodan; Rubinfeld, Ronitt; Singhal, Mihir (September 2024, 32nd Annual European Symposium on Algorithms (ESA 2024))

   We consider the question of orienting the edges in a graph G such that every vertex has bounded out-degree. For graphs of arboricity α, there is an orientation in which every vertex has out-degree at most α and, moreover, the best possible maximum out-degree of an orientation is at least α - 1. We are thus interested in algorithms that can achieve a maximum out-degree of close to α. A widely studied approach for this problem in the distributed algorithms setting is a "peeling algorithm" that provides an orientation with maximum out-degree α(2+ε) in a logarithmic number of iterations. We consider this problem in the local computation algorithm (LCA) model, which quickly answers queries of the form "What is the orientation of edge (u,v)?" by probing the input graph. When the peeling algorithm is executed in the LCA setting by applying standard techniques, e.g., the Parnas-Ron paradigm, it requires Ω(n) probes per query on an n-vertex graph. In the case where G has unbounded degree, we show that any LCA that orients its edges to yield maximum out-degree r must use Ω(√ n/r) probes to G per query in the worst case, even if G is known to be a forest (that is, α = 1). We also show several algorithms with sublinear probe complexity when G has unbounded degree. When G is a tree such that the maximum degree Δ of G is bounded, we demonstrate an algorithm that uses Δ n^{1-log_Δ r + o(1)} probes to G per query. To obtain this result, we develop an edge-coloring approach that ultimately yields a graph-shattering-like result. We also use this shattering-like approach to demonstrate an LCA which 4-colors any tree using sublinear probes per query. 

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5. Sublinear Time Hypergraph Sparsification via Cut and Edge Sampling Queries

   https://doi.org/10.4230/LIPIcs.ICALP.2021.53  

   Chen, Yu; Khanna, Sanjeev; Nagda, Sanjeev (July 2021, 48th International Colloquium on Automata, Languages, and Programming)

   null (Ed.)

   The problem of sparsifying a graph or a hypergraph while approximately preserving its cut structure has been extensively studied and has many applications. In a seminal work, Benczúr and Karger (1996) showed that given any n-vertex undirected weighted graph G and a parameter ε ∈ (0,1), there is a near-linear time algorithm that outputs a weighted subgraph G' of G of size Õ(n/ε²) such that the weight of every cut in G is preserved to within a (1 ± ε)-factor in G'. The graph G' is referred to as a (1 ± ε)-approximate cut sparsifier of G. Subsequent recent work has obtained a similar result for the more general problem of hypergraph cut sparsifiers. However, all known sparsification algorithms require Ω(n + m) time where n denotes the number of vertices and m denotes the number of hyperedges in the hypergraph. Since m can be exponentially large in n, a natural question is if it is possible to create a hypergraph cut sparsifier in time polynomial in n, independent of the number of edges. We resolve this question in the affirmative, giving the first sublinear time algorithm for this problem, given appropriate query access to the hypergraph. Specifically, we design an algorithm that constructs a (1 ± ε)-approximate cut sparsifier of a hypergraph H(V,E) in polynomial time in n, independent of the number of hyperedges, when given access to the hypergraph using the following two queries: 1) given any cut (S, ̄S), return the size |δ_E(S)| (cut value queries); and 2) given any cut (S, ̄S), return a uniformly at random edge crossing the cut (cut edge sample queries). Our algorithm outputs a sparsifier with Õ(n/ε²) edges, which is essentially optimal. We then extend our results to show that cut value and cut edge sample queries can also be used to construct hypergraph spectral sparsifiers in poly(n) time, independent of the number of hyperedges. We complement the algorithmic results above by showing that any algorithm that has access to only one of the above two types of queries can not give a hypergraph cut sparsifier in time that is polynomial in n. Finally, we show that our algorithmic results also hold if we replace the cut edge sample queries with a pair neighbor sample query that for any pair of vertices, returns a random edge incident on them. In contrast, we show that having access only to cut value queries and queries that return a random edge incident on a given single vertex, is not sufficient. 

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