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- H J Sommers, A Crisanti, H Sompolinsky, Y Stein
- 2011

Let S be a planar n-point set. A triangulation for S is a maximal plane straight-line graph with vertex set S. The Voronoi diagram for S is the subdivision of the plane into cells such that each cell has the same nearest neighbors in S. Classically, both structures can be computed in O(n log n) time and O(n) space. We study the situation when the available… (More)

- Wolfgang Mulzer, Huy L. Nguyen, Paul Seiferth, Yannik Stein
- STOC
- 2015

Let k ≥ 0 be an integer. In the <i>approximate k-flat nearest neighbor</i> (k-ANN) problem, we are given a set P ⊂ R<sup>d</sup> of n points in d-dimensional space and a fixed approximation factor c > 1. Our goal is to preprocess P so that we can efficiently answer <i>approximate k-flat nearest neighbor queries</i>: given a k-flat F, find a… (More)

Let P 1 ,. .. , P d+1 ⊂ R d be point sets whose convex hulls each contain the origin. Each set represents a color class. The Colorful Carathéodory theorem guarantees the existence of a colorful choice, i.e., a set that contains exactly one point from each color class, whose convex hull also contains the origin. The computational complexity of finding such a… (More)

Let C 1 , ..., C d+1 be d + 1 point sets in R d , each containing the origin in its convex hull. A subset C of d+1 i=1 C i is called a colorful choice (or rainbow) for C 1 ,. .. , C d+1 , if it contains exactly one point from each set C i. The colorful Carathéodory theorem states that there always exists a colorful choice for C 1 ,. .. , C d+1 that has the… (More)

- Wolfgang Mulzer, Yannik Stein
- Int. J. Comput. Geometry Appl.
- 2014

Let P be a d-dimensional n-point set. A partition T of P is called a Tverberg partition if the convex hulls of all sets in T intersect in at least one point. We say T is t-tolerated if it remains a Tverberg partition after deleting any t points from P. Soberón and Strausz proved that there is always a t-tolerated Tverberg partition with n/(d + 1)(t + 1)… (More)

- Wolfgang Mulzer, Yannik Stein
- Symposium on Computational Geometry
- 2015

Let P 1 ,. .. , P d+1 ⊂ R d be d-dimensional point sets such that the convex hull of each P i contains the origin. We call the sets P i color classes, and we think of the points in P i as having color i. A colorful choice is a set with at most one point of each color. The colorful Carathéodory theorem guarantees the existence of a colorful choice whose… (More)

- Wolfgang Mulzer, Yannik Stein
- ISAAC
- 2013

Let P be a d-dimensional n-point set. A partition T of P is called a Tverberg partition if the convex hulls of all sets in T intersect in at least one point. We say T is t-tolerated if it remains a Tverberg partition after deleting any t points from P. Soberón and Strausz proved that there is always a t-tolerated Tverberg partition with n/(d + 1)(t + 1)… (More)

- Bahareh Banyassady, Matias Korman, Wolfgang Mulzer, André van Renssen, Marcel Roeloffzen, Paul Seiferth +1 other
- STACS
- 2017

Let P 1 ,. .. , P d+1 ⊂ R d be d-dimensional point sets such that the convex hull of each P i contains the origin. We call the sets P i color classes, and we think of the points in P i as having color i. A colorful choice is a set with at most one point from each color class. The colorful Carathéodory theorem guarantees the existence of a colorful choice… (More)