[Paper Review] Distributed triangle detection via expander decomposition

Yi‐Jun Chang, Seth Pettie+1 more|arXiv (Cornell University)|Jan 6, 2019

Algorithms and Data Compression25 citations

TL;DR

This paper presents a novel distributed graph partitioning technique that enables triangle detection, counting, and enumeration in O(n^{1/2}) rounds in the CONGEST model—improving upon prior bounds of O(n^{2/3}) and O(n^{3/4}). The approach partitions edges into high-conductance, low-arboricity, and residual components, leveraging specialized algorithms for each and simulating clique algorithms on high-conductance subgraphs via a routing scheme.

ABSTRACT

We present improved distributed algorithms for triangle detection and its variants in the CONGEST model. We show that Triangle Detection, Counting, and Enumeration can be solved in O(n1/2) rounds. In contrast, the previous state-of-the-art bounds for Triangle Detection and Enumeration were O(n2/3) and O(n3/4), respectively, due to Izumi and LeGall (PODC 2017).The main technical novelty in this work is a distributed graph partitioning algorithm. We show that in O(n1 − δ) rounds we can partition the edge set of the network G = (V, E) into three parts E = Em ∩ Es ∩ Er such that• Each connected component induced by Em has minimum degree Ω(nδ) and conductance Ω(1/polylog(n)). As a consequence the mixing time of a random walk within the component is O(polylog(n)).• The subgraph induced by Es has arboricity at most nδ.• |Er| ≤ |E|/6.All of our algorithms are based on the following generic framework, which we believe is of interest beyond this work. Roughly, we deal with the set Es by an algorithm that is efficient for low-arboricity graphs, and deal with the set Er using recursive calls. For each connected component induced by Em, we are able to simulate CONGESTED-CLIQUE algorithms with small overhead by applying a routing algorithm due to Ghaffari, Kuhn, and Su (PODC 2017) for high conductance graphs.

Motivation & Objective

• To improve the round complexity of distributed triangle detection, counting, and enumeration in the CONGEST model.
• To design a distributed graph partitioning scheme that separates edges into components with distinct structural properties for targeted algorithmic treatment.
• To enable efficient simulation of CONGESTED-CLIQUE algorithms on high-conductance subgraphs using a routing protocol for high-conductance graphs.
• To reduce the overall round complexity by recursively handling residual edges and exploiting low-arboricity structures.

Proposed method

• The algorithm partitions the edge set E into three parts: Em (high-conductance, high-min-degree components), Es (low-arboricity), and Er (small residual set).
• Each component in Em has minimum degree Ω(n^δ) and conductance Ω(1/polylog(n)), ensuring fast mixing times for random walks.
• The subgraph induced by Es has arboricity at most n^δ, enabling efficient processing via low-arboricity algorithms.
• The residual set Er has size at most |E|/6, allowing recursive handling with diminishing size.
• For each Em component, the method simulates CONGESTED-CLIQUE algorithms with small overhead using a routing protocol from Ghaffari, Kuhn, and Su (PODC 2017).
• The overall framework combines specialized algorithms for Es and Er with simulation on Em, achieving improved round complexity.

Experimental results

Research questions

• RQ1 Can triangle detection in the CONGEST model be solved in o(n^{2/3}) rounds?
• RQ2 Can a distributed graph partitioning scheme be designed to enable efficient processing of triangle-related problems via component-specific algorithms?
• RQ3 Can high-conductance subgraphs be leveraged to simulate faster algorithms, such as those for the CONGESTED-CLIQUE model?
• RQ4 What is the trade-off between component conductance, arboricity, and residual edge size in achieving improved round complexity?

Key findings

• Triangle Detection, Counting, and Enumeration can all be solved in O(n^{1/2}) rounds in the CONGEST model, improving over prior O(n^{2/3}) and O(n^{3/4}) bounds.
• The proposed distributed expander decomposition partitions edges into three parts with distinct structural guarantees: high conductance, low arboricity, and small residual size.
• Each connected component in Em has minimum degree Ω(n^δ) and conductance Ω(1/polylog(n)), ensuring O(polylog(n)) mixing time for random walks.
• The subgraph induced by Es has arboricity at most n^δ, enabling efficient processing with low-arboricity algorithms.
• The residual set Er contains at most |E|/6 edges, allowing recursive handling with diminishing size across rounds.
• The framework enables simulation of CONGESTED-CLIQUE algorithms on high-conductance components with small overhead, leveraging a routing protocol from Ghaffari, Kuhn, and Su (PODC 2017).