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Near Uniform Triangle Sampling Over Adjacency List Graph Streams

Arijit Bishnu, Arijit Ghosh, Gopinath Mishra, Sayantan Sen

TL;DR

The paper addresses near-uniform triangle sampling from graph streams, formalizing the objective as outputting a triangle from a distribution within $\varepsilon$ in $\ell_1$ distance of uniform over all triangles. It introduces a heavy-light edge framework and develops both multi-pass and one-pass algorithms in the Adjacency List model, achieving space that matches the corresponding triangle-counting bounds, namely $\tilde{\Theta}(m/T^{2/3})$ for 3-pass and $\tilde{\Theta}(m/\sqrt{T})$ for 1-pass; it also extends sampling results to Ea/Va models with $\tilde{O}(\min\{m, m^{2}/T\})$ and $\tilde{O}(m^{3/2}/T)$ bounds, respectively, along with lower bounds. The key techniques include charging triangles to edges, distinguishing heavy and light edges/triangles, and running parallel subroutines (SampLightHelper and SampHeavyHelper) to enable single-pass sampling. The findings have significant implications for streaming subgraph queries and sampling-with-guarantees, with potential applications in databases, social networks, and biology, and open avenues for sampling other substructures in different streaming models.

Abstract

Triangle counting and sampling are two fundamental problems for streaming algorithms. Arguably, designing sampling algorithms is more challenging than their counting variants. It may be noted that triangle counting has received far greater attention in the literature than the sampling variant. In this work, we consider the problem of approximately sampling triangles in different models of streaming with the focus being on the adjacency list model. In this problem, the edges of a graph $G$ will arrive over a data stream. The goal is to design efficient streaming algorithms that can sample and output a triangle from a distribution, over the triangles in $G$, that is close to the uniform distribution over the triangles in $G$. The distance between distributions is measured in terms of $\ell_1$-distance. The main technical contribution of this paper is to design algorithms for this triangle sampling problem in the adjacency list model with the space complexities matching their counting variants. For the sake of completeness, we also show results on the vertex and edge arrival models.

Near Uniform Triangle Sampling Over Adjacency List Graph Streams

TL;DR

The paper addresses near-uniform triangle sampling from graph streams, formalizing the objective as outputting a triangle from a distribution within in distance of uniform over all triangles. It introduces a heavy-light edge framework and develops both multi-pass and one-pass algorithms in the Adjacency List model, achieving space that matches the corresponding triangle-counting bounds, namely for 3-pass and for 1-pass; it also extends sampling results to Ea/Va models with and bounds, respectively, along with lower bounds. The key techniques include charging triangles to edges, distinguishing heavy and light edges/triangles, and running parallel subroutines (SampLightHelper and SampHeavyHelper) to enable single-pass sampling. The findings have significant implications for streaming subgraph queries and sampling-with-guarantees, with potential applications in databases, social networks, and biology, and open avenues for sampling other substructures in different streaming models.

Abstract

Triangle counting and sampling are two fundamental problems for streaming algorithms. Arguably, designing sampling algorithms is more challenging than their counting variants. It may be noted that triangle counting has received far greater attention in the literature than the sampling variant. In this work, we consider the problem of approximately sampling triangles in different models of streaming with the focus being on the adjacency list model. In this problem, the edges of a graph will arrive over a data stream. The goal is to design efficient streaming algorithms that can sample and output a triangle from a distribution, over the triangles in , that is close to the uniform distribution over the triangles in . The distance between distributions is measured in terms of -distance. The main technical contribution of this paper is to design algorithms for this triangle sampling problem in the adjacency list model with the space complexities matching their counting variants. For the sake of completeness, we also show results on the vertex and edge arrival models.
Paper Structure (26 sections, 24 theorems, 24 equations, 8 algorithms)

This paper contains 26 sections, 24 theorems, 24 equations, 8 algorithms.

Table of Contents

  1. Introduction
  2. The complexity of sampling may be more than that of counting.
  3. Our Results
  4. Notations.
  5. Related Works
  6. Overview of our algorithms
  7. Multi-pass triangle sampling algorithm in the Al model
  8. Single pass triangle sampling algorithm in the Al model
  9. Intuition of sampling a light triangle:
  10. Intuition of sampling a heavy triangle:
  11. Multi-pass Triangle Sampling in Al model
  12. One-Pass Triangle Sampling in Al model
  13. Descriptions of SampLightHelper and SampHeavyHelper
  14. Notions of heavy and light edges and triangles
  15. Descriptions of Sample-Light-Triangle and Sample-heavy-Triangle
  16. ...and 11 more sections

Key Result

Theorem 1.1

(Main Result: Informal Statement). Triangle Sampling can be solved in Al model using: (i) $\widetilde{\Theta}\left(m/T^{2/3}\right)$ space in $3$-passes and (ii) $\widetilde{\Theta}\left(m/\sqrt{T}\right)$ space in $1$-pass. Here $T$ is a promised lower bound on the number of triangles in $G$. The p

Theorems & Definitions (43)

  • Theorem 1.1
  • Theorem 3.1: Upper Bound of Theorem \ref{['theo:two']}(i)
  • Definition 3.2: Heavy and light edge
  • Definition 3.3: Heavy and light triangle
  • Lemma 3.4: RIVIN2002647
  • proof
  • Lemma 3.6
  • proof
  • proof : Proof of Theorem \ref{['theo:tsamp-al-p-proof']}
  • Theorem 4.1: Upper bound of Theorem \ref{['theo:two']}(ii)
  • ...and 33 more