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.
Bravermanet al. REF gave a two-pass O(m/T 1/3 )-space algorithm to detect if the input graph is triangle free or it has at least T many triangles.
How Hard is Counting Triangles in the Streaming Model
ArXiv
ArXiv
Mathematics Computer Science