How Hard Is Counting Triangles in the Streaming Model?

Abstract

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 Ω(m) for graphs G with m edges on n vertices. If a constant number of passes is allowed, we show a lower bound of Ω(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 ρ(G) -the triangle density, and conjecture that the space complexity of the triangles problem is Ω(m/ρ(G)). We match this by a second algorithm that solves the distinguishing problem using O(m/ρ(G))-memory.

A second look at counting triangles in graph streams (corrected)

“…In this paper, while we refute such a possibility, we show that a more modest bound is possible. Specifically here we show that the sampling strategy of [BOV13], namely uniform sampling of the edges at a rate of 1 √ T in the first pass and counting detected triangles in the second pass gives a O(1) approximation of the number of triangles. To bring down the approximation precision to 1 + ε, we use a simple summary structure for identifying heavy edges (edges shared by many triangles which introduce large variance in the estimator) in order to deal with them separately from the rest of the graph.…”

“…In a recent work by Braverman et al [BOV13], it has been shown that at the expense of an extra pass over stream, a straightforward sampling strategy gives a sublinear bound that depends only on m (number of edges) and T (a lower bound on the number of triangles 1 ). More precisely [BOV13] have shown that one extra pass yields an algorithm that distinguishes between triangle-free graphs from graphs with at least T triangles using O( m T 1/3 ) words of space.…”

“…More precisely [BOV13] have shown that one extra pass yields an algorithm that distinguishes between triangle-free graphs from graphs with at least T triangles using O( m T 1/3 ) words of space. Although their algorithm does not give an estimate of the number of triangles and more important is not clearly superior to the O( m∆ T ) one pass algorithm by [PT12,PTTW13] (especially for graphs with small maximum degree ∆), it creates some hope that perhaps with the expense of extra passes one could get improved and cleaner space 1 In this and prior works, some assumption on the number of triangles is required.…”

“…The construction essentially encodes Ω(n 2 ) bits of information, and uses the presence or absence of a single triangle to recover a single bit. Braverman et al [BOV13] show a lower bound of Ω(m) by demonstrating a family of graphs with m chosen between n and n 2 . Their construction encodes m bits in a graph, then adds τ edges such that there are either τ triangles or 0 triangles, which reveal the value of an encoded bit.…”

“…Specifically, they create a graph that encodes two binary strings, so that the resulting graph has T triangles if the strings are disjoint, and 2T if they have an intersection. In a similar way, Braverman et al [BOV13] encode binary strings into a graph, so that it either has no triangles (disjoint strings) or at least T triangles (intersecting strings). This implies that Ω(m/T ) space is required to distinguish the two cases.…”