How Hard is Counting Triangles in the Streaming Model

TL;DR

This paper investigates the memory requirements for counting triangles in streaming graph algorithms, establishing lower bounds and proposing algorithms that approach these bounds, with a focus on the number of passes and graph parameters.

Contribution

It introduces new lower bounds for triangle counting in streaming models and proposes algorithms that nearly match these bounds, including a novel graph parameter called triangle density.

Findings

Lower bound of Ω(m) memory for one-pass algorithms

Lower bound of Ω(m/T) memory for multi-pass algorithms

A 2-pass algorithm with O(m/T^{1/3}) memory that distinguishes triangle-rich graphs

Abstract

The problem of (approximately) counting the number of triangles in a graph is one of the basic problems in graph theory. In this paper we study the problem in the streaming model. We study the amount of memory required by a randomized algorithm to solve this problem. In case the algorithm is allowed one pass over the stream, we present a best possible lower bound of $\Omega(m)$ for graphs $G$ with $m$ edges on $n$ vertices. If a constant number of passes is allowed, we show a lower bound of $\Omega(m/T)$, $T$ the number of triangles. We match, in some sense, this lower bound with a 2-pass $O(m/T^{1/3})$-memory algorithm that solves the problem of distinguishing graphs with no triangles from graphs with at least $T$ triangles. We present a new graph parameter $\rho(G)$ -- the triangle density, and conjecture that the space complexity of the triangles problem is $\Omega(m/\rho(G))$. We match this by a second algorithm that solves the distinguishing problem using $O(m/\rho(G))$-memory.