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. Specifically, 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. 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 Θ(m/ρ(G)). We match this by a second algorithm that solves the distinguishing problem using O(m/ρ(G))-memory.
The full version of this paper is available at http://arxiv.org/abs/1304.1458
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Braverman, V., Ostrovsky, R., Vilenchik, D. (2013). How Hard Is Counting Triangles in the Streaming Model?. In: Fomin, F.V., Freivalds, R., Kwiatkowska, M., Peleg, D. (eds) Automata, Languages, and Programming. ICALP 2013. Lecture Notes in Computer Science, vol 7965. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39206-1_21
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DOI: https://doi.org/10.1007/978-3-642-39206-1_21
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