Core Concepts

A novel algorithm for counting the exact number of triangles in bounded arboricity graphs in O(1) rounds, O(nδ) space per machine, and O(mα) total space, where α is the arboricity of the graph.

Abstract

The paper presents a simple and efficient algorithm for counting the exact number of triangles in bounded arboricity graphs in the Massively Parallel Computation (MPC) model. The key insights are:
The algorithm enumerates the wedges adjacent to the lower degree endpoint of every edge, leveraging the Chiba-Nishizeki lemma which bounds the sum of the minimum degrees of edge endpoints.
The algorithm partitions the adjacency list of the lower degree endpoint of each edge into chunks of size nδ, where δ is a constant, and sends each edge and its corresponding partition to a separate machine.
On each machine, the algorithm forms wedges between the given edge and the edges in its partition, and checks if the third edge completing the wedge into a triangle exists in the graph.
The algorithm uses various MPC primitives like sorting, counting, filtering, etc. to efficiently implement this approach in O(1/δ) rounds, O(nδ) space per machine, and O(mα) total space, where α is the arboricity of the graph.
The paper also shows a lower bound of Ω(1/δ) rounds for triangle counting in the worst-case setting of disjoint subgraphs partitioned across machines, proving the optimality of the algorithm's round complexity.

Stats

Í(u,v)∈E min(deg(u), deg(v)) ≤ 2mα

Quotes

"Counting triangles in O(1) rounds (exactly) is listed as one of the interesting remaining open problems in the recent survey of Im et al. [IKL+23]."
"Our new algorithm is very simple, achieves the optimal O(1) rounds without increasing the space per machine and the total space, and has the potential of being easily implementable in practice."

Key Insights Distilled From

by Quanquan C. ... at **arxiv.org** 05-02-2024

Deeper Inquiries

The algorithm presented for triangle counting in the MPC model can indeed be extended to count other types of subgraphs beyond triangles. By modifying the process of forming queries and filtering based on the presence of specific subgraph structures, such as squares or cliques, the algorithm can be adapted to enumerate and count these subgraphs efficiently. The key lies in defining the appropriate queries and filtering mechanisms tailored to the subgraph of interest, similar to how triangles are identified and counted in the current algorithm.

To handle dynamic graph updates in the MPC setting, the algorithm needs to incorporate mechanisms for efficiently updating the counts of subgraphs as the graph evolves. One approach is to maintain incremental counters for each subgraph type, adjusting them as edges are added or removed from the graph. By tracking the changes in the adjacency structure and updating the subgraph counts accordingly, the algorithm can adapt to dynamic updates while maintaining efficiency in terms of rounds and space per machine.

The development of an efficient triangle counting algorithm in the MPC model has significant implications for various graph problems and applications. Firstly, the algorithm's optimal complexity in terms of rounds and space per machine sets a precedent for tackling other subgraph counting tasks with similar efficiency. This can lead to advancements in community detection, link recommendation, and other graph analysis tasks that rely on subgraph enumeration.
Furthermore, the algorithm's simplicity and optimality make it a promising candidate for implementation in real-world distributed systems and platforms like MapReduce, Hadoop, and Spark. Its potential for practical implementation opens up opportunities for scaling graph analytics on massive datasets efficiently.
Overall, the efficient triangle counting algorithm serves as a foundational building block for addressing a wide range of graph-related challenges in the MPC model, paving the way for advancements in distributed graph processing and analytics.

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