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1. On the Distributed Complexity of Large-Scale Graph Computations

   https://doi.org/10.1145/3460900  

   Pandurangan, Gopal; Robinson, Peter; Scquizzato, Michele (June 2021, ACM Transactions on Parallel Computing)

   null (Ed.)

   Motivated by the increasing need to understand the distributed algorithmic foundations of large-scale graph computations, we study some fundamental graph problems in a message-passing model for distributed computing where k ≥ 2 machines jointly perform computations on graphs with n nodes (typically, n >> k). The input graph is assumed to be initially randomly partitioned among the k machines, a common implementation in many real-world systems. Communication is point-to-point, and the goal is to minimize the number of communication rounds of the computation. Our main contribution is the General Lower Bound Theorem , a theorem that can be used to show non-trivial lower bounds on the round complexity of distributed large-scale data computations. This result is established via an information-theoretic approach that relates the round complexity to the minimal amount of information required by machines to solve the problem. Our approach is generic, and this theorem can be used in a “cookbook” fashion to show distributed lower bounds for several problems, including non-graph problems. We present two applications by showing (almost) tight lower bounds on the round complexity of two fundamental graph problems, namely, PageRank computation and triangle enumeration . These applications show that our approach can yield lower bounds for problems where the application of communication complexity techniques seems not obvious or gives weak bounds, including and especially under a stochastic partition of the input. We then present distributed algorithms for PageRank and triangle enumeration with a round complexity that (almost) matches the respective lower bounds; these algorithms exhibit a round complexity that scales superlinearly in k , improving significantly over previous results [Klauck et al., SODA 2015]. Specifically, we show the following results: PageRank: We show a lower bound of Ὼ(n/k 2 ) rounds and present a distributed algorithm that computes an approximation of the PageRank of all the nodes of a graph in Õ(n/k 2 ) rounds. Triangle enumeration: We show that there exist graphs with m edges where any distributed algorithm requires Ὼ(m/k 5/3 ) rounds. This result also implies the first non-trivial lower bound of Ὼ(n 1/3 ) rounds for the congested clique model, which is tight up to logarithmic factors. We then present a distributed algorithm that enumerates all the triangles of a graph in Õ(m/k 5/3 + n/k 4/3 ) rounds. 

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2. Cover Edge-Based Novel Triangle Counting

   https://doi.org/10.3390/a18110685  

   Bader, David A; Li, Fuhuan; Du, Zhihui; Pauliuchenka, Palina; Rodriguez, Oliver Alvarado; Gupta, Anant; Minnal, Sai_Sri Vastav; Nahata, Valmik; Ganeshan, Anya; Gundogdu, Ahmet Cemal; et al (November 2025, Algorithms)

   Counting and listing triangles in graphs is a fundamental task in network analysis, supporting applications such as community detection, clustering coefficient computation, k-truss decomposition, and triangle centrality. We introduce the cover-edge set, a novel concept that eliminates unnecessary edges during triangle enumeration, thereby improving efficiency. This compact cover-edge set is rapidly constructed using a breadth-first search (BFS) strategy. Using this concept, we develop both sequential and parallel triangle-counting algorithms and conduct comprehensive comparisons with state-of-the-art methods. We also design a benchmarking framework to evaluate our sequential and parallel algorithms in a systematic and reproducible manner. Extensive experiments on the latest Intel Xeon 8480+ processor reveal clear performance differences among algorithms, demonstrate the benefits of various optimization strategies, and show how graph characteristics, such as diameter and degree distribution, affect algorithm performance. Our source code is available on GitHub. 

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3. Multi-dimensional balanced graph partitioning via projected gradient descent

   https://doi.org/10.14778/3324301.3324307  

   Avdiukhin, Dmitrii; Pupyrev, Sergey; Yaroslavtsev, Grigory (April 2019, Proceedings of the VLDB Endowment)

   Motivated by performance optimization of large-scale graph processing systems that distribute the graph across multiple machines, we consider the balanced graph partitioning problem. Compared to most of the previous work, we study the multi-dimensional variant in which balance according to multiple weight functions is required. As we demonstrate by experimental evaluation, such multi-dimensional balance is essential for achieving performance improvements for typical distributed graph processing workloads. We propose a new scalable technique for the multidimensional balanced graph partitioning problem. It is based on applying randomized projected gradient descent to a non-convex continuous relaxation of the objective. We show how to implement the new algorithm efficiently in both theory and practice utilizing various approaches for the projection step. Experiments with large-scale graphs containing up to hundreds of billions of edges indicate that our algorithm has superior performance compared to the state of the art. 

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4. Batch Updates of Distributed Streaming Graphs using Linear Algebra

   https://doi.org/10.1109/SCW63240.2024.00089  

   Hassani, Elaheh; Hussain, Md Taufique; Azad, Ariful (November 2024, IEEE)

   We develop a distributed-memory parallel algorithm for performing batch updates on streaming graphs, where vertices and edges are continuously added or removed. Our algorithm leverages distributed sparse matrices as the core data structures, utilizing equivalent sparse matrix operations to execute graph updates. By reducing unnecessary communication among processes and employing shared-memory parallelism, we accelerate updates of distributed graphs. Additionally, we maintain a balanced load in the output matrix by permuting the resultant matrix during the update process. We demonstrate that our streaming update algorithm is at least 25 times faster than alternative linear-algebraic methods and scales linearly up to 4,096 cores (32 nodes) on a Cray EX supercomputer. 

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5. Wedge-Parallel Triangle Counting for GPUs

   https://doi.org/10.1007/978-3-031-99872-0_1  

   Spaan, Jeffrey; Chen, Kuan-Hsun; Bader, David A; Varbanescu, Ana-Lucia (August 2025, Springer Nature Switzerland)

   For fast processing of increasingly large graphs, triangle counting - a common building block of graph processing algorithms, is often performed on GPUs. However, applying massive parallelism to triangle counting is challenging due to the algorithm’s inherent irregular access patterns and workload imbalance. In this work, we propose WeTriC, a novel wedge-parallel triangle counting algorithm for GPUs, which, using fine(r)-grained parallelism through a lightweight static mapping of wedges to threads, improves load balancing and efficiency. Our theoretical analysis compares different parallelization granularities, while optimizations enhance caching, reduce work-per-intersection, and minimize overhead. Performance experiments indicate that WeTriC yields 5.63x and 4.69x speedup over optimized vertex-parallel and edge-parallel binary search triangle counting algorithms, respectively. Furthermore, we show that WeTriC consistently outperforms the state-of-the-art (i.e., on avg. 2.86x faster than Trust and 2.32x faster than GroupTC). 

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