Distributed Triangle Enumeration in Hypergraphs

TL;DR

This paper initiates the systematic study of distributed sub-hypergraph enumeration in hypergraphs and introduces several computational models for hypergraphs that generalize the CONGEST model for graphs and evaluates their relative computational power.

Abstract

In the last decade, subgraph detection and enumeration have emerged as a central problem in distributed graph algorithms. This is largely due to the theoretical challenges and practical applications of these problems. In this paper, we initiate the systematic study of distributed sub-hypergraph enumeration in hypergraphs. To this end, we (1)~introduce several computational models for hypergraphs that generalize the CONGEST model for graphs and evaluate their relative computational power, (2)~devise algorithms for distributed triangle enumeration in our computational models and prove their optimality in two such models, (3)~introduce classes of sparse and ``everywhere sparse'' hypergraphs and describe efficient distributed algorithms for triangle enumeration in these classes, and (4)~describe general techniques that we believe to be useful for designing efficient algorithms in our hypergraph models.

Figures (1)

• Figure 1: A figure demonstrating the relationships between models given in Proposition \ref{['prop:model-relationships']}. Directed edges correspond to simulation results, where the edge label gives the multiplicative overhead of the simulation. The vertical position of each model in the figure indicates the power of the model, with EC and EB being able to simulate all lower models without computational overhead.

Theorems & Definitions (60)

• theorem 1: Informal
• theorem 2: Informal
• proposition 1
• proof
• proposition 2
• corollary 1
• proof
• theorem 3
• proof : Proof of Theorem \ref{['thm:clique-ub']}
• theorem 4