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Arboricity and Random Edge Queries Matter for Triangle Counting using Sublinear Queries

Arijit Bishnu, Debarshi Chanda, Gopinath Mishra

TL;DR

The paper advances triangle counting in sublinear query models by integrating arboricity with RandomEdge access. It develops a weight-function framework that separates light and heavy triangles, and introduces an oracle-based sampling scheme that uses heavy/light edge testing to achieve an $ ilde{O}\left(\frac{m\alpha\log(1/\delta)}{\varepsilon^3 T}\right)$ query upper bound for an $\varepsilon$-approximate count, while proving a near-matching lower bound of $\tilde{\Omega}\left(\frac{m\alpha\log(1/\delta)}{\varepsilon^2 T}\right)$. The method leverages carbono-like bucketing and Chernoff bounds to control estimation variance and extends prior work by explicitly incorporating arboricity in the presence of RandomEdge queries. This yields a tighter understanding of how graph density (via $\alpha$) and RandomEdge access jointly influence sublinear triangle counting, and suggests directions for counting other subgraphs under similar models.

Abstract

Given a simple, unweighted, undirected graph $G=(V,E)$ with $|V|=n$ and $|E|=m$, and parameters $0 < \varepsilon, δ<1$, along with \texttt{Degree}, \texttt{Neighbour}, \texttt{Edge} and \texttt{RandomEdge} query access to $G$, we provide a query based randomized algorithm to generate an estimate $\widehat{T}$ of the number of triangles $T$ in $G$, such that $\widehat{T} \in [(1-\varepsilon)T , (1+\varepsilon)T]$ with probability at least $1-δ$. The query complexity of our algorithm is $\widetilde{O}\left({m α\log(1/δ)}/{\varepsilon^3 T}\right)$, where $α$ is the arboricity of $G$. Our work can be seen as a continuation in the line of recent works [Eden et al., SIAM J Comp., 2017; Assadi et al., ITCS 2019; Eden et al. SODA 2020] that considered subgraph or triangle counting with or without the use of \texttt{RandomEdge} query. Of these works, Eden et al. [SODA 2020] considers the role of arboricity. Our work considers how \texttt{RandomEdge} query can leverage the notion of arboricity. Furthermore, continuing in the line of work of Assadi et al. [APPROX/RANDOM 2022], we also provide a lower bound of $\widetildeΩ\left({m α\log(1/δ)}/{\varepsilon^2 T}\right)$ that matches the upper bound exactly on arboricity and the parameter $δ$ and almost on $\varepsilon$.

Arboricity and Random Edge Queries Matter for Triangle Counting using Sublinear Queries

TL;DR

The paper advances triangle counting in sublinear query models by integrating arboricity with RandomEdge access. It develops a weight-function framework that separates light and heavy triangles, and introduces an oracle-based sampling scheme that uses heavy/light edge testing to achieve an query upper bound for an -approximate count, while proving a near-matching lower bound of . The method leverages carbono-like bucketing and Chernoff bounds to control estimation variance and extends prior work by explicitly incorporating arboricity in the presence of RandomEdge queries. This yields a tighter understanding of how graph density (via ) and RandomEdge access jointly influence sublinear triangle counting, and suggests directions for counting other subgraphs under similar models.

Abstract

Given a simple, unweighted, undirected graph with and , and parameters , along with \texttt{Degree}, \texttt{Neighbour}, \texttt{Edge} and \texttt{RandomEdge} query access to , we provide a query based randomized algorithm to generate an estimate of the number of triangles in , such that with probability at least . The query complexity of our algorithm is , where is the arboricity of . Our work can be seen as a continuation in the line of recent works [Eden et al., SIAM J Comp., 2017; Assadi et al., ITCS 2019; Eden et al. SODA 2020] that considered subgraph or triangle counting with or without the use of \texttt{RandomEdge} query. Of these works, Eden et al. [SODA 2020] considers the role of arboricity. Our work considers how \texttt{RandomEdge} query can leverage the notion of arboricity. Furthermore, continuing in the line of work of Assadi et al. [APPROX/RANDOM 2022], we also provide a lower bound of that matches the upper bound exactly on arboricity and the parameter and almost on .

Paper Structure

This paper contains 19 sections, 25 theorems, 28 equations, 1 table, 4 algorithms.

Table of Contents

  1. Introduction
  2. Our Results
  3. Related Work
  4. RAM model.
  5. Streaming model.
  6. Property Testing model.
  7. Paper organization
  8. Technical Overview
  9. Preliminaries
  10. Notations
  11. Arboricity and its properties
  12. Chernoff Bounds
  13. Algorithm
  14. Weight Function
  15. Oracle Based Algorithm
  16. ...and 4 more sections

Key Result

Theorem 2

Given a simple, unweighted and undirected graph $G = \left({V,E}\right)$ with $\left|{V}\right| = n$, $\left|{E}\right| = m$ and arboricity $\alpha$, and access to the graph via Degree, Neighbour, Edge and RandomEdge queries, an $\left({\varepsilon,\delta}\right)$ estimate $\widehat{T}$ of $T$ can b

Theorems & Definitions (45)

  • Definition 1: Arboricity$(\alpha)$
  • Theorem 2: Upper Bound(Simplified)
  • Theorem 3: Lower Bound(Simplified)
  • Definition 4: Arboricity$(\alpha)$
  • Lemma 5
  • Lemma 6: Triangle Upper Bound DBLP:conf/soda/EdenRS20
  • Lemma 7: Multiplicative Chernoff Bound
  • Definition 8: $\tau$-heavy and $\tau$-light edges
  • Definition 9: $\tau$-heavy and $\tau$-light triangles
  • Lemma 10: Upper Bound on $T^{\tau}_{H}$
  • ...and 35 more