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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. Join Size Bounds using l _p -Norms on Degree Sequences

   https://doi.org/10.1145/3651597  

   Abo_Khamis, Mahmoud; Nakos, Vasileios; Olteanu, Dan; Suciu, Dan (May 2024, Proceedings of the ACM on Management of Data)

   Estimating the output size of a query is a fundamental yet longstanding problem in database query processing. Traditional cardinality estimators used by database systems can routinely underestimate the true output size by orders of magnitude, which leads to significant system performance penalty. Recently, upper bounds have been proposed that are based on information inequalities and incorporate sizes and max-degrees from input relations, yet their main benefit is limited to cyclic queries, because they degenerate to rather trivial formulas on acyclic queries. We introduce a significant extension of the upper bounds, by incorporating l_p-norms of the degree sequences of join attributes. Our bounds are significantly lower than previously known bounds, even when applied to acyclic queries. These bounds are also based on information theory, they come with a matching query evaluation algorithm, are computable in exponential time in the query size, and are provably tight when all degrees are ''simple''. 

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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. A Scalable Randomized Algorithm for Triangle Enumeration on Graphs Based on SQL Queries

   https://doi.org/10.1007/978-3-030-59065-9_12  

   Farouzi, Abir; Bellatreche, Ladjel; Ordonez, Carlos; Pandurangan, Gopal; Malki, Mimoun (January 2020, International Conference on Big Data Analytics and Knowledge Discovery (DaWaK))

   null (Ed.)

   Triangle enumeration is a fundamental problem in large-scale graph analysis. For instance, triangles are used to solve practical problems like community detection and spam filtering. On the other hand, there is a large amount of data stored on database management systems (DBMSs), which can be modeled and analyzed as graphs. Alternatively, graph data can be quickly loaded into a DBMS. Our paper shows how to adapt and optimize a randomized distributed triangle enumeration algorithm with SQL queries, which is a significantly different approach from programming graph algorithms in traditional languages such as Python or C++. We choose a parallel columnar DBMS given its fast query processing, but our solution should work for a row DBMS as well. Our randomized solution provides a balanced workload for parallel query processing, being robust to the existence of skewed degree vertices. We experimentally prove our solution ensures a balanced data distribution, and hence workload, among machines. The key idea behind the algorithm is to evenly partition all possible triplets of vertices among machines, sending edges that may form a triangle to a proxy machine; this edge redistribution eliminates shuffling edges during join computation and therefore triangle enumeration becomes local and fully parallel. In summary, our algorithm exhibits linear speedup with large graphs, including graphs that have high skewness in vertex degree distributions. 

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