2026-08-07T10:23:55Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:20066 2024-05-31T13:11:11Z ddc:004 open_access

No-Dimensional Tverberg Partitions Revisited Har-Peled, Sariel Robson, Eliot W. Points partitions convex hull high dimension Given a set P ⊂ ℝ^d of n points, with diameter Δ, and a parameter δ ∈ (0,1), it is known that there is a partition of P into sets P_1, …, P_t, each of size O(1/δ²), such that their convex hulls all intersect a common ball of radius δΔ. We prove that a random partition, with a simple alteration step, yields the desired partition, resulting in a (randomized) linear time algorithm (i.e., O(dn)). We also provide a deterministic algorithm with running time O(dn log n). Previous proofs were either existential (i.e., at least exponential time), or required much bigger sets. In addition, the algorithm and its proof of correctness are significantly simpler than previous work, and the constants are slightly better. We also include a number of applications and extensions using the same central ideas. For example, we provide a linear time algorithm for computing a "fuzzy" centerpoint, and prove a no-dimensional weak ε-net theorem with an improved constant. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Sariel Har-Peled and Eliot W. Robson 2024 Is Part Of LIPIcs, Volume 294, 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.SWAT.2024.26 urn:nbn:de:0030-drops-200664 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.26 eng https://creativecommons.org/licenses/by/4.0/legalcode