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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 binary couplings J,, = i l or J,, = 0, 1, quantised couplings with larger synaptic range, e.g. J,, = *l/L, *2/ L, . . . , *1 and, in the limit, continuous couplings confined to the hypercube… (More)

The average eigenvalue distribution p(k) of N xN real random asymmetric matrices J;, (Ji,cJ„)is calculated in the limit of N ~. It is found that p(X) is uniform in an ellipse, in the complex plane, whose real and imaginary axes are 1+ r and 1 —i, respectively. The parameter r is given by r=N[J;, J,;]J and N[J;, lJ is normalized to l. In the r=1 limit,… (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… (More)

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

Let P1, . . . , Pd+1 ⊂ R be d-dimensional point sets such that the convex hull of each Pi contains the origin. We call the sets Pi color classes, and we think of the points in Pi 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 convex… (More)

- Bahareh Banyassady, Matias Korman, +4 authors Yannik Stein
- STACS
- 2017

Let P be a planar n-point set in general position. For k ∈ {1, . . . , n − 1}, the Voronoi diagram of order k is obtained by subdividing the plane into regions such that points in the same cell have the same set of nearest k neighbors in P . The (nearest point) Voronoi diagram (NVD) and the farthest point Voronoi diagram (FVD) are the particular cases of k… (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 logn) 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)

- 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 dn/(d + 1)(t + 1)e… (More)

Let P1, . . . , Pd+1 ⊂ R 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)

<lb>Let C1, ..., Cd+1 be d+ 1 point sets in R, each containing the origin in its convex hull. A subset C of ⋃d+1<lb>i=1 Ci<lb>is called a colorful choice (or rainbow) for C1, . . . ,<lb>Cd+1, if it contains exactly one point from each set Ci. The<lb>colorful Carathéodory theorem states that there always exists a colorful choice for C1, . . . , Cd+1 that has… (More)