On the Complexity of Triangle Counting Using Emptiness Queries
Beame et al. [ITCS'18 & TALG'20] introduced and used the Bipartite Independent Set (BIS) and Independent Set (IS) oracle access to an unknown, simple, unweighted and undirected graph and solved the edge estimation problem. The introduction of this oracle set forth a series of works in a short time that either solved open questions mentioned by Beame et al. or were generalizations of their work as in Dell and Lapinskas [STOC'18 and TOCT'21], Dell, Lapinskas, and Meeks [SODA'20 and SICOMP'22], Bhattacharya et al. [ISAAC'19 & TOCS'21], and Chen et al. [SODA'20]. Edge estimation using BIS can be done using polylogarithmic queries, while IS queries need sub-linear but more than polylogarithmic queries. Chen et al. improved Beame et al.’s upper bound result for edge estimation using IS and also showed an almost matching lower bound. Beame et al. in their introductory work asked a few open questions out of which one was on estimating structures of higher order than edges, like triangles and cliques, using BIS queries.
In this work, we almost resolve the query complexity of estimating triangles using BIS oracle. While doing so, we prove a lower bound for an even stronger query oracle called Edge Emptiness (EE) oracle, recently introduced by Assadi, Chakrabarty, and Khanna [ESA'21] to test graph connectivity.
Triangle Counting
Emptiness Queries
Bipartite Independent Set Query
Theory of computation~Streaming, sublinear and near linear time algorithms
Theory of computation~Lower bounds and information complexity
48:1-48:22
RANDOM
https://arxiv.org/abs/2110.03836
Arijit
Bishnu
Arijit Bishnu
Indian Statistical Institute, Kolkata, India
https://www.isical.ac.in/~arijit/
Arijit
Ghosh
Arijit Ghosh
Indian Statistical Institute, Kolkata, India
https://sites.google.com/site/homepagearijitghosh/
Gopinath
Mishra
Gopinath Mishra
University of Warwick, Coventry, UK
https://sites.google.com/view/gopinathmishra
https://orcid.org/0000-0003-0540-0292
Research supported in part by the Centre for Discrete Mathematics and its Applications (DIMAP), by EPSRC award EP/V01305X/13.
10.4230/LIPIcs.APPROX/RANDOM.2023.48
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Arijit Bishnu, Arijit Ghosh, and Gopinath Mishra
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