Set-Valued Recursions Arising from Vantage-Point Trees.
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| Title: | Set-Valued Recursions Arising from Vantage-Point Trees. |
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| 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] |
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| Database: | Engineering Source |
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