Exploiting Air-Pressure to Map Floorplans on Point Sets

We prove a conjecture of Ackerman, Barequet and Pinter. Every floorplan with n segments can be embedded on every set of n points in generic position. The construction makes use of area universal floorplans also known as area universal rectangular layouts. The notion of area used in our context depends on a nonuniform density function. We, therefore, have to generalize the theory of area universal floorplans to this situation. The method is then used to prove a result about accommodating points in floorplans that is slightly more general than the conjecture of Ackerman et al

An Upper Bound for the Number of Rectangulations of a Planar Point Set

We prove that every set of n points in the plane has at most $17^n$ rectangulations. This improves upon a long-standing bound of Ackerman. Our proof is based on the cross-graph charging-scheme technique.Comment: 8 pages, 5 figure

Colored anchored visibility representations in 2D and 3D space

© 2020. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/In a visibility representation of a graph G, the vertices are represented by nonoverlapping geometric objects, while the edges are represented as segments that only intersect the geometric objects associated with their end-vertices. Given a set P of n points, an Anchored Visibility Representation of a graph G with n vertices is a visibility representation such that for each vertex v of G, the geometric object representing v contains a point of P. We prove positive and negative results about the existence of anchored visibility representations under various models, both in 2D and in 3D space. We consider the case when the mapping between the vertices and the points is not given and the case when it is only partially given.Peer ReviewedPostprint (author's final draft

Fixed-energy harmonic functions

Fixed-energy harmonic functions, Discrete Analysis 2017:18, 21 pp. The classical Dirichlet problem asks for a harmonic function in the interior of a region that takes specified values on the boundary. One can formulate a discrete version of the problem as follows. Let $G$ be a finite graph. Associate with each edge $e$ a positive real number $c_e$ called its conductance, and define a subset $B$ of the vertices, of size at least 2, to be the boundary. Given a function $f$ defined on the vertices, we say that it is _harmonic_ if $\sum_{y\sim x}c_{xy}(f(x)-f(y))=0$ for every vertex $x\notin B$, where we have written $\sim$ for the relation "is a neighbour of". That is, the value at each non-boundary vertex $x$ is an average, weighted by the conductances, of the values at its neighbours. One of the methods of solving the classical Dirichlet problem is an energy-minimization argument, and that works for the discrete version as well. First, one defines the energy of $f$ to be $\sum_{x\sim y}c_{xy}(f(x)-f(y))^2$. To understand why this weighting is appropriate, one should think of $f$ as a voltage. Then if $c_{xy}$ is large, meaning that the resistance of the edge is small, there should be a tendency for the values $f(x)$ and $f(y)$ not to be too different, whereas if $c_{xy}$ is small and the resistance is large, then this tendency should be reduced. An easy variational argument (just differentiate with respect to the value at $x$) shows that if this energy is minimized and $x\notin B$, then $f$ is indeed harmonic at $x$. This paper turns the usual discrete Dirichlet problem on its head, in the following sense. We can regard the usual problem as providing for us a function that takes as its input a set of conductances and outputs the energies of each of the edges. Here this is reversed: one is given the edge energies and also directions on the edges (which have to satisfy some simple compatibility conditions), and one finds conductances such that the solution $f$ to the corresponding Dirichlet problem gives rise to those energies and such that if the edge $xy$ is directed from $x$ to $y$, then $f(x)>f(y)$. The main theorem of the paper is that the solution to this problem is unique, from which it follows that the number of solutions if one just wishes to obtain the given energies is obtained by enumerating the compatible sets of directions on the edges. These solutions turn out to be the local maxima of the expression $\prod_{x\sim y}|f(x)-f(y)|^{\mathcal E_{xy}}$, where $\mathcal E_{xy}$ is the energy associated with the edge $xy$, and if $f$ is such a solution, then it satisfies the equation $\sum_{y\sim x}\frac{\mathcal E_{xy}}{f(x)-f(y))}=0$ for every interior vertex $x$. The authors call such functions _enharmonic_. The results of this study have some interesting consequences. For example, as is well known, there are close connections between electrical networks and rectangle tilings: one of the applications in this paper is that that certain polygons cannot be tiled by rectangles with rational areas. The authors also define an enharmonic conjugate function and obtain results analogous to the Cauchy-Riemann equations and the Riemann mapping theorem for the resulting "analytic functions". </div

Distributed-in/ distributed-out sensor networks : a new framework to analyze distributed phenomena

Thesis (Ph. D.)--Massachusetts Institute of Technology, School of Architecture and Planning, Program in Media Arts and Sciences, 2006.Includes bibliographical references (p. 155-165).With a new way of thinking about organizing sensor networks, we demonstrate that we can more easily deploy and program these networks to solve a variety of different problems. We describe sensor networks that can analyze and actuate distributed phenomena without a central coordinator. Previous implementations of sensor networks have approached the problem from the perspective of centralized reporting of distributed events. By contrast, we create a system that allows users to infer the global state from within the sensor network itself, rather than by accessing an outside, central middleware layer. This is accomplished via dynamic creation of clusters of nodes based on application or intent, rather than proximity. The data collected and returned by these clusters is returned directly to the inquirer at his current location. By creating this Distributed-in/Distributed-out (DiDo) system that bypasses a middleware layer, our networks have the principal advantage of being easily configurable and deployable. We show that a system with this structure can solve path problems in a random graph. These graph problems are directly applicable to real-life applications such as discovering escape routes for people in a building with changing pathways. We show that the system is scalable, as reconfiguration requires only local communication.(cont.) To test our assumptions, we build a suite of applications to create different deployment scenarios that model the physical world and set up simulations that allow us to measure performance. Finally, we create a set of simple primitives that serve as a high-level organizing protocol. These primitives can be used to solve different problems with distributed sensors, regardless of the underlying network protocols. The instructions provided by the sensors result in tangible performance improvements when the sensors' instructions are directed to agents within a simulated physical world.by Constantine Kleomenis Christakos.Ph.D

The IPIN 2019 Indoor Localisation Competition - Description and Results

IPIN 2019 Competition, sixth in a series of IPIN competitions, was held at the CNR Research Area of Pisa (IT), integrated into the program of the IPIN 2019 Conference. It included two on-site real-time Tracks and three off-site Tracks. The four Tracks presented in this paper were set in the same environment, made of two buildings close together for a total usable area of 1000 m 2 outdoors and and 6000 m 2 indoors over three floors, with a total path length exceeding 500 m. IPIN competitions, based on the EvAAL framework, have aimed at comparing the accuracy performance of personal positioning systems in fair and realistic conditions: past editions of the competition were carried in big conference settings, university campuses and a shopping mall. Positioning accuracy is computed while the person carrying the system under test walks at normal walking speed, uses lifts and goes up and down stairs or briefly stops at given points. Results presented here are a showcase of state-of-the-art systems tested side by side in real-world settings as part of the on-site real-time competition Tracks. Results for off-site Tracks allow a detailed and reproducible comparison of the most recent positioning and tracking algorithms in the same environment as the on-site Tracks