NDLI: Approximately Counting Triangles in Sublinear Time

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Approximately Counting Triangles in Sublinear Time

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Ron, Dana Seshadhri, C. Levi, Amit Eden, Talya
Copyright Year	2017
Abstract	We consider the problem of estimating the number of triangles in a graph. This problem has been extensively studied in both theory and practice, but all existing algorithms read the entire graph. In this work we design a sublinear-time algorithm for approximating the number of triangles in a graph, where the algorithm is given query access to the graph. The allowed queries are degree queries, vertex-pair queries, and neighbor queries. We show that for any given approximation parameter $0<\epsilon<1$, the algorithm provides an estimate $\widehat{t}$ such that, with high constant probability, $(1-\epsilon)\cdot t< \widehat{t}<(1+\epsilon)\cdot t$, where $t$ is the number of triangles in the graph $G$. The expected query complexity of the algorithm is $(\frac{n}{t^{1/3}} + \min\{m, \frac{m^{3/2}}{t}\})\cdot {poly}(\log n, \frac{1}{\epsilon})$, where $n$ is the number of vertices in the graph and $m$ is the number of edges. The expected running time of the algorithm is $(\frac{n}{t^{1/3}} + \frac{m^{3/2}}{t})\cdot {poly}(\log n, \frac{1}{\epsilon})$. We also prove that $\Omega(\frac{n}{t^{1/3}} + \min\{m, \frac{m^{3/2}}{t}\})$ queries are necessary, thus establishing that the query complexity of this algorithm is optimal up to the dependence on ${poly}(\log n, \frac{1}{\epsilon})$.
Sponsorship	Blavatnik Family Foundation. Israel Science Foundation
Starting Page	1603
Ending Page	1646
Page Count	44
File Format	PDF
ISSN	00975397
DOI	10.1137/15M1054389
e-ISSN	10957111
Journal	SIAM Journal on Computing (SMJCAT)
Issue Number	5
Volume Number	46
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2017-10-10
Access Restriction	Subscribed
Subject Keyword	sublinear algorithms Analysis of algorithms and problem complexity Randomized algorithms Approximation algorithms approximation algorithms triangles counting
Content Type	Text
Resource Type	Article
Subject	Mathematics Computer Science

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