Stretchability of Star-Like Pseudo-Visibility Graphs

We present advances on the open problem of characterizing vertex-edge visibility graphs (ve-graphs), reduced by results of O\u27Rourke and Streinu to a stretchability question for pseudo-polygons. We introduce star-like pseudo-polygons as a special subclass containing all the known instances of non-stretchable pseudo-polygons. We give a complete combinatorial characterization and a linear-time decision procedure for star-like pseudo-polygon stretchability and star-like ve-graph recognition. To the best of our knowledge, this is the first problem in computational geometry for which a combinatorial characterization was found by first isolating the oriented matroid substructure and then separately solving the stretchability question. It is also the first class (as opposed to isolated examples) of oriented matroids for which an efficient stretchability decision procedure based on combinatorial criteria is given. The difficulty of the general stretchability problem implied by Mnev\u27s Universality Theorem makes this a result of independent interest in the theory of oriented matroids

Stretchability of Star-Like Pseudo-Visibility Graphs

We present advances on the open problem of characterizing vertex-edge visibility graphs (ve-graphs), reduced by results of O\u27Rourke and Streinu to a stretchability question for pseudo-polygons. We introduce star-like pseudo-polygons as a special subclass containing all the known instances of non-stretchable pseudo-polygons. We give a complete combinatorial characterization and a linear-time decision procedure for star-like pseudo-polygon stretchability and star-like ve-graph recognition. To the best of our knowledge, this is the first problem in computational geometry for which a combinatorial characterization was found by first isolating the oriented matroid substructure and then separately solving the stretchability question. It is also the first class (as opposed to isolated examples) of oriented matroids for which an efficient stretchability decision procedure based on combinatorial criteria is given. The difficulty of the general stretchability problem implied by Mnev\u27s Universality Theorem makes this a result of independent interest in the theory of oriented matroids

Non-Stretchable Pseudo-Visibility Graphs

We exhibit a family of graphs which can be realized as pseudo-visibility graphs of pseudo-polygons, but not of straight-line polygons. The example is based on the characterization of vertex-edge pseudo-visibility graphs of O\u27Rourke and Streinu [Proc. ACM Symp. Comput. Geometry, Nice, France, 1997, pp. 119-128] and extends a recent result of the author [Proc. ACM Symp. Comput. Geometry, Miami Beach, 1999, pp. 274-280] on non-stretchable vertex-edge visibility graphs. We construct a pseudo-visibility graph for which there exists a unique compatible vertex-edge visibility graph, which is then shown to be non-stretchable. The construction is then extended to an infinite family. © 2004 Elsevier B.V

Clusters of Stars

We solve two open problems posed by Goodman and Pollack[GP84] about sets of signed circular permutations (clusters of stars) arising from generalized configurations of points: recognition and efficient reconstruction (drawing). As a biproduct we get an O(n2) space data structure constructible in O(n2) time, representing the order type of a (generalized) configuration of points and from which the orientation of each triple can be found in constant time, a problem posed in [EHN]

Sign rank versus VC dimension

This work studies the maximum possible sign rank of $N \times N$ sign matrices with a given VC dimension $d$. For $d=1$, this maximum is {three}. For $d=2$, this maximum is $\tilde{\Theta}(N^{1/2})$. For $d >2$, similar but slightly less accurate statements hold. {The lower bounds improve over previous ones by Ben-David et al., and the upper bounds are novel.} The lower bounds are obtained by probabilistic constructions, using a theorem of Warren in real algebraic topology. The upper bounds are obtained using a result of Welzl about spanning trees with low stabbing number, and using the moment curve. The upper bound technique is also used to: (i) provide estimates on the number of classes of a given VC dimension, and the number of maximum classes of a given VC dimension -- answering a question of Frankl from '89, and (ii) design an efficient algorithm that provides an $O(N/\log(N))$ multiplicative approximation for the sign rank. We also observe a general connection between sign rank and spectral gaps which is based on Forster's argument. Consider the $N \times N$ adjacency matrix of a $\Delta$ regular graph with a second eigenvalue of absolute value $\lambda$ and $\Delta \leq N/2$. We show that the sign rank of the signed version of this matrix is at least $\Delta/\lambda$. We use this connection to prove the existence of a maximum class $C\subseteq\{\pm 1\}^N$ with VC dimension $2$ and sign rank $\tilde{\Theta}(N^{1/2})$. This answers a question of Ben-David et al.~regarding the sign rank of large VC classes. We also describe limitations of this approach, in the spirit of the Alon-Boppana theorem. We further describe connections to communication complexity, geometry, learning theory, and combinatorics.Comment: 33 pages. This is a revised version of the paper "Sign rank versus VC dimension". Additional results in this version: (i) Estimates on the number of maximum VC classes (answering a question of Frankl from '89). (ii) Estimates on the sign rank of large VC classes (answering a question of Ben-David et al. from '03). (iii) A discussion on the computational complexity of computing the sign-ran

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum

Discrete Geometry

[no abstract available

From Radio Channel Modeling to a System Level Perspective in Body-Centric Communications

Body-centric communications are emerging as a new paradigm in the panorama of personal communications. Being concerned with human behaviour, they are suitable for a wide variety of applications. The advances in the miniaturization of portable devices to be placed on or around the body, foster the diffusion of these systems, where the human body is the key element defining communication characteristics. This thesis investigates the human impact on body-centric communications under its distinctive aspects. First of all, the unique propagation environment defined by the body is described through a scenario-based channel modeling approach, according to the communication scenario considered, i.e., on- or on- to off-body. The novelty introduced pertains to the description of radio channel features accounting for multiple sources of variability at the same time. Secondly, the importance of a proper channel characterisation is shown integrating the on-body channel model in a system level simulator, allowing a more realistic comparison of different Physical and Medium Access Control layer solutions. Finally, the structure of a comprehensive simulation framework for system performance evaluation is proposed. It aims at merging in one tool, mobility and social features typical of the human being, together with the propagation aspects, in a scenario where multiple users interact sharing space and resources