Core Concepts
The author presents a novel algorithm for distributed conductance testing in the CONGEST model, utilizing random walks and spectral graph theory to determine if a graph is a good conductor or far from it.
Abstract
The content discusses a new algorithm for property testing graphs' conductance in the CONGEST model. It introduces the concept of sticky vertices and leverages spectral graph theory to analyze the process efficiently. The algorithm aims to provide an optimal solution without requiring global information collection, offering insights into distributed decision problems and property testing.
The proposed algorithm involves running multiple random walks from random sources to determine if a network is a good conductor. By leveraging sticky vertices and spectral graph theory, the algorithm can make local decisions without aggregating data centrally. This approach aims to improve efficiency and robustness in distributed computing models like CONGEST.
The content highlights the importance of conducting property testing in distributed networks and explores the challenges faced in verifying global properties efficiently. By introducing innovative techniques like sticky vertices and parallel random walks, the algorithm offers a time-optimal solution for conductance testing without relying on global aggregation.
Stats
Our main result is the algorithm presented in Section 3 (Pseudocodes 1 and 2).
The algorithm uses Oϵ,α(log n) communication rounds.
The algorithm runs 2m2 walks of length 32 α2 log n from each of θ(1/ϵ) starting vertices.
The rejection threshold for vertex v ∈ V , τv = m · deg(v) · (1 + 2n−1/4).