The Complexity of Symmetry Breaking in Massive Graphs

05/05/2021
by   Christian Konrad, et al.
0

The goal of this paper is to understand the complexity of symmetry breaking problems, specifically maximal independent set (MIS) and the closely related β-ruling set problem, in two computational models suited for large-scale graph processing, namely the k-machine model and the graph streaming model. We present a number of results. For MIS in the k-machine model, we improve the Õ(m/k^2 + Δ/k)-round upper bound of Klauck et al. (SODA 2015) by presenting an Õ(m/k^2)-round algorithm. We also present an Ω̃(n/k^2) round lower bound for MIS, the first lower bound for a symmetry breaking problem in the k-machine model. For β-ruling sets, we use hierarchical sampling to obtain more efficient algorithms in the k-machine model and also in the graph streaming model. More specifically, we obtain a k-machine algorithm that runs in Õ(β nΔ^1/β/k^2) rounds and, by using a similar hierarchical sampling technique, we obtain one-pass algorithms for both insertion-only and insertion-deletion streams that use O(β· n^1+1/2^β-1) space. The latter result establishes a clear separation between MIS, which is known to require Ω(n^2) space (Cormode et al., ICALP 2019), and β-ruling sets, even for β = 2. Finally, we present an even faster 2-ruling set algorithm in the k-machine model, one that runs in Õ(n/k^2-ϵ + k^1-ϵ) rounds for any ϵ, 0 ≤ϵ≤ 1.

READ FULL TEXT

Authors

page 1

page 2

page 3

page 4

01/11/2019

Exponentially Faster Massively Parallel Maximal Matching

The study of graph problems in the Massively Parallel Computations (MPC)...
07/17/2018

Massively Parallel Symmetry Breaking on Sparse Graphs: MIS and Maximal Matching

The success of massively parallel computation (MPC) paradigms such as Ma...
04/18/2022

Sleeping is Superefficient: MIS in Exponentially Better Awake Complexity

Maximal Independent Set (MIS) is one of the central and most well-studie...
02/19/2018

Breaking the Linear-Memory Barrier in MPC: Fast MIS on Trees with n^ Memory per Machine

Recently, studying fundamental graph problems in the Massive Parallel Co...
03/29/2020

How the Degeneracy Helps for Triangle Counting in Graph Streams

We revisit the well-studied problem of triangle count estimation in grap...
02/14/2020

A Breezing Proof of the KMW Bound

In their seminal paper from 2004, Kuhn, Moscibroda, and Wattenhofer (KMW...
11/25/2017

Optimal Gossip Algorithms for Exact and Approximate Quantile Computations

This paper gives drastically faster gossip algorithms to compute exact a...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.