Set-Valued Recursions Arising from Vantage-Point Trees.
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| Title: | Set-Valued Recursions Arising from Vantage-Point Trees. |
|---|---|
| Authors: | Dong, Congzao1 (AUTHOR) czdong@xidian.edu.cn, Marynych, Alexander2,3 (AUTHOR) marynych@knu.ua, Molchanov, Ilya4 (AUTHOR) ilya.molchanov@unibe.ch |
| Source: | Discrete & Computational Geometry. Jun2026, Vol. 75 Issue 4, p1134-1150. 17p. |
| Subjects: | Convex bodies, Data structures, Polyhedra, Limit theorems, Markov processes, Vector spaces |
| Abstract: | We study vantage-point trees constructed using an independent sample from the uniform distribution on a fixed convex body K in (R d , ‖ · ‖) , where ‖ · ‖ is an arbitrary norm on R d . We prove that a sequence of sets, associated with the left boundary of a vantage-point tree, forms a recurrent Harris chain on the space of convex bodies in (R d , ‖ · ‖) . The limiting object is a ball polyhedron, that is, an a.s. finite intersection of closed balls in (R d , ‖ · ‖) of possibly different radii. As a consequence, we derive a limit theorem for the length of the leftmost path of a vantage-point tree. [ABSTRACT FROM AUTHOR] |
| Copyright of Discrete & Computational Geometry is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) | |
| Database: | Engineering Source |
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| Header | DbId: egs DbLabel: Engineering Source An: 194162859 AccessLevel: 6 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 0 |
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| Items | – Name: Title Label: Title Group: Ti Data: Set-Valued Recursions Arising from Vantage-Point Trees. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Dong%2C+Congzao%22">Dong, Congzao</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> czdong@xidian.edu.cn</i><br /><searchLink fieldCode="AR" term="%22Marynych%2C+Alexander%22">Marynych, Alexander</searchLink><relatesTo>2,3</relatesTo> (AUTHOR)<i> marynych@knu.ua</i><br /><searchLink fieldCode="AR" term="%22Molchanov%2C+Ilya%22">Molchanov, Ilya</searchLink><relatesTo>4</relatesTo> (AUTHOR)<i> ilya.molchanov@unibe.ch</i> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Discrete+%26+Computational+Geometry%22">Discrete & Computational Geometry</searchLink>. Jun2026, Vol. 75 Issue 4, p1134-1150. 17p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Convex+bodies%22">Convex bodies</searchLink><br /><searchLink fieldCode="DE" term="%22Data+structures%22">Data structures</searchLink><br /><searchLink fieldCode="DE" term="%22Polyhedra%22">Polyhedra</searchLink><br /><searchLink fieldCode="DE" term="%22Limit+theorems%22">Limit theorems</searchLink><br /><searchLink fieldCode="DE" term="%22Markov+processes%22">Markov processes</searchLink><br /><searchLink fieldCode="DE" term="%22Vector+spaces%22">Vector spaces</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: We study vantage-point trees constructed using an independent sample from the uniform distribution on a fixed convex body K in (R d , ‖ · ‖) , where ‖ · ‖ is an arbitrary norm on R d . We prove that a sequence of sets, associated with the left boundary of a vantage-point tree, forms a recurrent Harris chain on the space of convex bodies in (R d , ‖ · ‖) . The limiting object is a ball polyhedron, that is, an a.s. finite intersection of closed balls in (R d , ‖ · ‖) of possibly different radii. As a consequence, we derive a limit theorem for the length of the leftmost path of a vantage-point tree. [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of Discrete & Computational Geometry is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.) |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1007/s00454-024-00714-1 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 17 StartPage: 1134 Subjects: – SubjectFull: Convex bodies Type: general – SubjectFull: Data structures Type: general – SubjectFull: Polyhedra Type: general – SubjectFull: Limit theorems Type: general – SubjectFull: Markov processes Type: general – SubjectFull: Vector spaces Type: general Titles: – TitleFull: Set-Valued Recursions Arising from Vantage-Point Trees. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Dong, Congzao – PersonEntity: Name: NameFull: Marynych, Alexander – PersonEntity: Name: NameFull: Molchanov, Ilya IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 06 Text: Jun2026 Type: published Y: 2026 Identifiers: – Type: issn-print Value: 01795376 Numbering: – Type: volume Value: 75 – Type: issue Value: 4 Titles: – TitleFull: Discrete & Computational Geometry Type: main |
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