Random k-out subgraph leaves only O(n/k) inter-component edges

Jacob Holm; Valerie King; Mikkel Thorup; Or Zamir; Uri Zwick

doi:10.1109/FOCS.2019.00058

Random k-out subgraph leaves only O(n/k) inter-component edges

Jacob Holm, Valerie King, Mikkel Thorup, Or Zamir, Uri Zwick

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

7 Scopus citations

Abstract

Each vertex of an arbitrary simple graph on n vertices chooses k random incident edges. What is the expected number of edges in the original graph that connect different connected components of the sampled subgraph? We prove that the answer is O(n/k), when k ≥ c log n, for some large enough c. We conjecture that the same holds for smaller values of k, possibly for any k ≥ 2. Such a result is best possible for any k ≥ 2. As an application, we use this sampling result to obtain a one-way communication protocol with private randomness for finding a spanning forest of a graph in which each vertex sends only O (√n log n) bits to a referee.

Original language	English
Title of host publication	Proceedings - 2019 IEEE 60th Annual Symposium on Foundations of Computer Science, FOCS 2019
Publisher	IEEE Computer Society
Pages	896-909
Number of pages	14
ISBN (Electronic)	9781728149523
DOIs	https://doi.org/10.1109/FOCS.2019.00058
State	Published - Nov 2019
Event	60th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2019 - Baltimore, United States Duration: 9 Nov 2019 → 12 Nov 2019

Publication series

Name	Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
Volume	2019-November
ISSN (Print)	0272-5428

Conference

Conference	60th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2019
Country/Territory	United States
City	Baltimore
Period	9/11/19 → 12/11/19

Funding

Funders	Funder number
Basic Algorithms Research Copenhagen
Blavatnik Computer Science Research Fund
VIL-LUM Foundation
Natural Sciences and Engineering Research Council of Canada

Keywords

Connected components
Random subgraph
communication complexity

Access to Document

10.1109/FOCS.2019.00058

Cite this

Holm, J., King, V., Thorup, M., Zamir, O., & Zwick, U. (2019). Random k-out subgraph leaves only O(n/k) inter-component edges. In Proceedings - 2019 IEEE 60th Annual Symposium on Foundations of Computer Science, FOCS 2019 (pp. 896-909). Article 8948658 (Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS; Vol. 2019-November). IEEE Computer Society. https://doi.org/10.1109/FOCS.2019.00058

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abstract = "Each vertex of an arbitrary simple graph on n vertices chooses k random incident edges. What is the expected number of edges in the original graph that connect different connected components of the sampled subgraph? We prove that the answer is O(n/k), when k ≥ c log n, for some large enough c. We conjecture that the same holds for smaller values of k, possibly for any k ≥ 2. Such a result is best possible for any k ≥ 2. As an application, we use this sampling result to obtain a one-way communication protocol with private randomness for finding a spanning forest of a graph in which each vertex sends only O (√n log n) bits to a referee.",

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author = "Jacob Holm and Valerie King and Mikkel Thorup and Or Zamir and Uri Zwick",

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Holm, J, King, V, Thorup, M, Zamir, O & Zwick, U 2019, Random k-out subgraph leaves only O(n/k) inter-component edges. in Proceedings - 2019 IEEE 60th Annual Symposium on Foundations of Computer Science, FOCS 2019., 8948658, Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS, vol. 2019-November, IEEE Computer Society, pp. 896-909, 60th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2019, Baltimore, United States, 9/11/19. https://doi.org/10.1109/FOCS.2019.00058

Random k-out subgraph leaves only O(n/k) inter-component edges. / Holm, Jacob; King, Valerie; Thorup, Mikkel et al.
Proceedings - 2019 IEEE 60th Annual Symposium on Foundations of Computer Science, FOCS 2019. IEEE Computer Society, 2019. p. 896-909 8948658 (Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS; Vol. 2019-November).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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Random k-out subgraph leaves only O(n/k) inter-component edges

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