A Second Look at Counting Triangles in Graph Streams (revised)

TL;DR

This paper introduces an improved two-pass streaming algorithm for counting triangles in graphs, achieving better space efficiency than previous methods and establishing limits on the benefits of additional passes.

Contribution

It presents a new two-pass streaming algorithm with reduced space complexity for approximate triangle counting and proves the optimality of this approach regarding the number of passes.

Findings

New two-pass algorithm with $O(rac{m}{ ext{poly}(rac{1}{ ext{epsilon}}) ext{sqrt}(T)} ext{polylog}(m))$ space

Improved bounds over previous triangle detection algorithms

Proves no constant-pass algorithm can significantly improve space bounds for triangle detection

Abstract

In this paper we present improved results on the problem of counting triangles in edge streamed graphs. For graphs with $m$ edges and at least $T$ triangles, we show that an extra look over the stream yields a two-pass treaming algorithm that uses $O(\frac{m}{\eps^{2.5}\sqrt{T}}\polylog(m))$ space and outputs a $(1+\eps)$ approximation of the number of triangles in the graph. This improves upon the two-pass streaming tester of Braverman, Ostrovsky and Vilenchik, ICALP 2013, which distinguishes between triangle-free graphs and graphs with at least $T$ triangle using $O(\frac{m}{T^{1/3}})$ space. Also, in terms of dependence on $T$, we show that more passes would not lead to a better space bound. In other words, we prove there is no constant pass streaming algorithm that distinguishes between triangle-free graphs from graphs with at least $T$ triangles using $O(\frac{m}{T^{1/2+\rho}})$ space for any constant $\rho \ge 0$.