Edge Estimation with Independent Set Oracles

Beame, Paul; Har-Peled, Sariel; Ramamoorthy, Sivaramakrishnan Natarajan; Rashtchian, Cyrus; Sinha, Makrand

doi:10.1145/3404867

Cited by 12 publications

(41 citation statements)

References 34 publications

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“…We feel that the analysis in [6] does not go through in our case because of a subtle difference between counting the number of triangles and the number of edges in a graph. An edge is an explicit structure, whereas, a triangle is an implicit structure -triangles (a, b, x), (b, c, y) and (a, c, z) in G imply the existence of the triangle (a, b, c) in G. In Figure 1, we give a flowchart of the algorithm and show the corresponding lemmas that support the steps of the algorithm.…”

Section: Overview Of the Algorithmmentioning

confidence: 98%

“…This result would almost have closed the edge estimation problem but for having a relook at the problem with stronger query models and hoping for polylogarithmic number of queries. Beame et al [6] precisely did that by estimating the number of edges in a graph using O −4 log 14 n bipartite independent set (BIS) queries. Motivated by this result, we explore whether triangle estimation can be solved using only polylogarithmic queries to TIS.…”

Section: Prior Workmentioning

confidence: 99%

“…Our algorithmic framework is inspired by [6] but the detailed analysis is markedly different using color coding due to Alon et al [3] and a relatively new concentration inequality, due to Janson [23], for handling sums of random variables with bounded dependency. Apart from Lemmas 5 and 13, all other proofs require different ideas.…”

Section: Overview Of the Algorithmmentioning

confidence: 99%

“…After estimating the number of triangles in each subgraph coarsely, we generate a bounded number of samples of the set of subgraphs using a sampling technique given by Beame et al [6]. The sampling maintains the triangle count approximately.…”

Section: Lemma 7 (Coarse Estimation)mentioning

confidence: 99%

“…Notations, the query model, the problem and the result query oracle is its worth 1 . Beame et al [6] had given a subjective justification in favor of BIS to establish it as a query oracle. It is easy to verify that TIS, being in the same class of group queries, have the interesting connections to group testing and computational geometry as BIS.…”

Section: Introductionmentioning

confidence: 99%

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On Triangle Estimation Using Tripartite Independent Set Queries

et al. 2021

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Estimating the number of triangles in a graph is one of the most fundamental problems in sublinear algorithms. In this work, we provide an approximate triangle counting algorithm using only polylogarithmic queries when the number of triangles on any edge in the graph is polylogarithmically bounded. Our query oracle Tripartite Independent Set (TIS) takes three disjoint sets of vertices A, B and C as input, and answers whether there exists a triangle having one endpoint in each of these three sets. Our query model generally belongs to the class of group queries (Ron and Tsur, ACM ToCT, 2016; Dell and Lapinskas, STOC 2018) and in particular is inspired by the Bipartite Independent Set (BIS) query oracle of Beame et al. (ITCS 2018). We extend the algorithmic framework of Beame et al., with TIS replacing BIS, for triangle counting using ideas from color coding due to Alon et al. (J. ACM, 1995) and a concentration inequality for sums of random variables with bounded dependency (Janson, Rand. Struct. Alg., 2004). ACM Subject ClassificationTheory of computation → Models of computation; Theory of computation → Streaming, sublinear and near linear time algorithms; Mathematics of computing → Probabilistic algorithms

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Section: Overview Of the Algorithmmentioning

confidence: 98%

Section: Prior Workmentioning

confidence: 99%

Section: Overview Of the Algorithmmentioning

confidence: 99%

Section: Lemma 7 (Coarse Estimation)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On Triangle Estimation Using Tripartite Independent Set Queries

et al. 2021

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Algorithms that Access the Input via Queries

Erlebach

2021

SOFSEM 2021: Theory and Practice of Computer Science

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Problems where an algorithm cannot simply access the whole input but needs to obtain information about it using queries arise naturally in many settings. We discuss different aspects of models where an algorithm needs to query the input, and of how the performance of algorithms for such models can be measured. After that, we give some concrete examples of algorithmic settings and results for scenarios where algorithms access the input via queries. Finally, we discuss recent results for the setting of computing with explorable uncertainty with parallel queries and with untrusted predictions.

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