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- H Gutfreund, Y Stein
- 2001

We study the optimal storage capacity of neural networks with discrete local constraints on the synaptic couplings J,,. Models with such constraints inlcude those with lJ8,1s 1 ('box confinement'). We find that the optimal storage capacity (Y (K) is best determined by the vanishing of a suitably defined 'entropy' as calculated in the replica symmetric… (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)

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, 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)

- 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 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)

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 S be a planar n-point set. Classically, one can find the Voronoi diagram VD(S) for S in O(n log n) time and O(n) space. We study the situation when the available workspace is limited: for s ∈ {1,. .. , n}, an s-workspace algorithm has read-only access to an input array with the points from S in arbitrary order, and it may use only O(s) additional words… (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)