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A note on approximating the average degree of bounded arboricity graphs

Talya Eden, C. Seshadhri

TL;DR

An algorithm is described that gives a $(1+\varepsilon)-approximation to d degree using $O(\varepsilon^{-2}\alpha/d)$ queries and modified to get a $O(\varepsilon^{-2} \sqrt{n/d})$ query.

Abstract

Estimating the average degree of graph is a classic problem in sublinear graph algorithm. Eden, Ron, and Seshadhri (ICALP 2017, SIDMA 2019) gave a simple algorithm for this problem whose running time depended on the graph arboricity, but the underlying simplicity and associated analysis were buried inside the main result. Moreover, the description there loses logarithmic factors because of parameter search. The aim of this note is to give a full presentation of this algorithm, without these losses. Consider standard access (vertex samples, degree queries, and neighbor queries) to a graph $G = (V,E)$ of arboricity at most $α$. Let $d$ denote the average degree of $G$. We describe an algorithm that gives a $(1+\varepsilon)$-approximation to $d$ degree using $O(\varepsilon^{-2}α/d)$ queries. For completeness, we modify the algorithm to get a $O(\varepsilon^{-2} \sqrt{n/d})$ query

A note on approximating the average degree of bounded arboricity graphs

TL;DR

An algorithm is described that gives a O(\varepsilon^{-2}\alpha/d)O(\varepsilon^{-2} \sqrt{n/d})$ query.

Abstract

Estimating the average degree of graph is a classic problem in sublinear graph algorithm. Eden, Ron, and Seshadhri (ICALP 2017, SIDMA 2019) gave a simple algorithm for this problem whose running time depended on the graph arboricity, but the underlying simplicity and associated analysis were buried inside the main result. Moreover, the description there loses logarithmic factors because of parameter search. The aim of this note is to give a full presentation of this algorithm, without these losses. Consider standard access (vertex samples, degree queries, and neighbor queries) to a graph of arboricity at most . Let denote the average degree of . We describe an algorithm that gives a -approximation to degree using queries. For completeness, we modify the algorithm to get a query
Paper Structure (4 sections, 4 theorems, 7 equations, 2 algorithms)

This paper contains 4 sections, 4 theorems, 7 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Setup
  3. The ERS algorithm
  4. The general case, for unknown arboricity

Key Result

Lemma 1.2

$\sum_{(u,v) \in E} \min(d_u, d_v) \leq 2m\alpha(G)$.

Theorems & Definitions (14)

  • Definition 1.1: Arboricity
  • Lemma 1.2: Chiba-Nishizeki, Lemma 2 of ChNi85
  • proof
  • Definition 1.3: The degree ordering
  • Lemma 1.4
  • proof
  • Theorem 1.5
  • Claim 2.1
  • proof
  • Claim 2.2
  • ...and 4 more