Degree Four Plane Spanners: Simpler and Better

Let P be a set of n points embedded in the plane, and let C be the complete Euclidean graph whose point-set is P. Each edge in C between two points p, q is realized as the line segment [pq], and is assigned a weight equal to the Euclidean distance |pq|. In this paper, we show how to construct in O(nlg{n}) time a plane spanner of C of maximum degree at most 4 and of stretch factor at most 20. This improves a long sequence of results on the construction of bounded degree plane spanners of C. Our result matches the smallest known upper bound of 4 by Bonichon et al. on the maximum degree while significantly improving their stretch factor upper bound from 156.82 to 20. The construction of our spanner is based on Delaunay triangulations defined with respect to the equilateral-triangle distance, and uses a different approach than that used by Bonichon et al. Our approach leads to a simple and intuitive construction of a well-structured spanner, and reveals useful structural properties of the Delaunay triangulations defined with respect to the equilateral-triangle distance

Computational Thinking across the Curriculum: A Conceptual Framework

We describe a framework for implementing computational thinking in a broad variety of general education courses. The framework is designed to be used by faculty without formal training in information technology in order to understand and integrate computational thinking into their own general education courses. The framework includes examples of computational thinking in a variety of general education courses, as well as sample in-class activities, assignments, and other assessments for the courses. The examples in the different courses are related and differentiated using categories taken from Peter Denning’s Great Principles of Computing, so that similar types of computational thinking appearing in different contexts are brought together. This aids understanding of the computational thinking found in the courses and provides a template for future work on new course materials

The Stretch Factor of L1L_1- and L∞L_\infty-Delaunay Triangulations

In this paper we determine the stretch factor of the $L_1$-Delaunay and $L_\infty$-Delaunay triangulations, and we show that this stretch is $\sqrt{4+2\sqrt{2}} \approx 2.61$. Between any two points $x,y$ of such triangulations, we construct a path whose length is no more than $\sqrt{4+2\sqrt{2}}$ times the Euclidean distance between $x$ and $y$, and this bound is best possible. This definitively improves the 25-year old bound of $\sqrt{10}$ by Chew (SoCG '86). To the best of our knowledge, this is the first time the stretch factor of the well-studied $L_p$-Delaunay triangulations, for any real $p\ge 1$, is determined exactly

On Geometric Spanners of Euclidean and Unit Disk Graphs

We consider the problem of constructing bounded-degree planar geometric spanners of Euclidean and unit-disk graphs. It is well known that the Delaunay subgraph is a planar geometric spanner with stretch factor C_{del\approx 2.42; however, its degree may not be bounded. Our first result is a very simple linear time algorithm for constructing a subgraph of the Delaunay graph with stretch factor \rho =1+2\pi(k\cos{\frac{\pi{k)^{-1 and degree bounded by $k$, for any integer parameter $k\geq 14$. This result immediately implies an algorithm for constructing a planar geometric spanner of a Euclidean graph with stretch factor \rho \cdot C_{del and degree bounded by $k$, for any integer parameter $k\geq 14$. Moreover, the resulting spanner contains a Euclidean Minimum Spanning Tree (EMST) as a subgraph. Our second contribution lies in developing the structural results necessary to transfer our analysis and algorithm from Euclidean graphs to unit disk graphs, the usual model for wireless ad-hoc networks. We obtain a very simple distributed, {\em strictly-localized algorithm that, given a unit disk graph embedded in the plane, constructs a geometric spanner with the above stretch factor and degree bound, and also containing an EMST as a subgraph. The obtained results dramatically improve the previous results in all aspects, as shown in the paper

Computing Lightweight Spanning Subgraphs Locally

We consider the problem of computing bounded-degree lightweight plane spanning subgraphs of unit disk graphs in the local distributed model of computation. We are motivated by the hypothesis that such subgraphs can provide the underlying network topology for efficient unicasting and/or multicasting in wireless distributed systems. We start by showing that, for any integer $k \geq 2$, there exists a $k$-local distributed algorithm that, given a unit disk graph $U$ embedded in the plane, constructs a plane subgraph of $U$ containing a Euclidean Minimum Spanning Tree (EMST) of $V(U)$, whose degree is at most 6, and whose total weight is at most $(1 + \frac{2}{k-1})$ times the weight of an EMST of $V(U)$. We show that this bound is tight by proving that, for any $\epsilon \u3e 0$, there exists a unit disk graph $U$ such that no $k$-local distributed algorithm can construct a spanning subgraph of $U$ whose total weight is at most $(1 + \frac{2}{k-1} -\epsilon)$ times the weight of an EMST of $V(U)$. We then go further and present the first $k$-local distributed algorithm, where $k$ is a constant, that computes a bounded-degree plane lightweight {\em spanner} of a given unit disk graph. The upper bounds on the number of communication rounds of the algorithm, the degree, the stretch factor, and the weight of the spanner, are very small. For example, our results imply an $18$-local distributed algorithm that computes a plane spanner of a given unit disk graph $U$, whose degree is at most 14, stretch factor at most 8.81, and weight at most $8.81$ times the weight of an EMST of $V(U)$. All the obtained results rely on an elegant structural result that we develop for weighted planar graphs. We show a wider application of this result by giving an $O(n\lg{n})$ time centralized algorithm that constructs bounded-degree plane lightweight spanners of unit disk graphs (which include Euclidean graphs), with the best upper bounds on the spanner degree, stretch factor, and weight

Comment résumer le plan

International audienceCet article concerne les graphes de recouvrement d'un ensemble fini de points du plan Euclidien. Un graphe de recouvrement $H$ est de facteur d'étirement $t$ pour un ensemble de points $S$ si, entre deux points quelconques de $S$, le coût d'un plus court chemin dans $H$ est au plus $t$ fois leur distance Euclidenne. Les graphes de recouvrement d'étirement $t$ (ci-après nommés \emph{$t$-spanneurs}) sont à la base de nombreux algorithmes de routage et de navigation dans le plan. Le graphe (ou triangulation) de Delaunay, le graphe de Gabriel, le graphe de Yao ou le Theta-graphe sont des exemples bien connus de $t$-spanneurs. L'étirement $t$ et le degré maximum des spanneurs sont des paramètres important à minimiser pour l'optimisation des ressources. En même temps le caractère planaire des constructions se révèle essentiel dans les algorithmes de navigation. Nous présentons une série de résultats dans ce domaine, en particulier: \begin{itemize} \item Nous montrons que le graphe $\Theta_6$ (le Theta-graphe où $k=6$ cônes d'angle $\Theta_k = 2\pi/k$ par sommet sont utilisées) est l'union de deux spanneurs planaires d'étirement deux. En particulier, nous établissons que l'étirement maximum du graphe $\Theta_6$ est deux, ce qui est optimal. Des bornes supérieures sur l'étirement du graphe $\Theta_k$ n'étaient connues que lorsque $k > 6$. Pour $k=7$, la meilleure borne connue est d'environ $7.56$ et pour $k=6$ il était ouvert de savoir si le graphe était un $t$-spanneur pour une valeur constante de $t$. \item Nous montrons que le graphe $\Theta_6$ contient comme sous-graphe couvrant un $3$-spanneur planaire de degré maximum au plus~$9$. \item Finalement, en utilisant une variante du résultat précédant, nous montrons que le plan Euclidien possède un $6$-spanneur planaire de degré maximum au plus~$6$. \end{itemize} La dernière construction, non décrite ici par manque de place, améliore une longue série de résultats sur le problème largement ouvert de déterminer la plus petite valeur $\delta$ telle que tout ensemble du plan possède un spanneur planaire d'étirement constant et de degré maximum $\delta$. Le meilleur résultat en date montrait que $3 \le \delta\le 14$

Computational Thinking across the Curriculum: A Conceptual Framework

We describe a framework for implementing computational thinking in a broad variety of general education courses. The framework is designed to be used by faculty without formal training in information technology in order to understand and integrate computational thinking into their own general education courses. The framework includes examples of computational thinking in a variety of general education courses, as well as sample in-class activities, assignments, and other assessments for the courses. The examples in the different courses are related and differentiated using categories taken from Peter Denning’s Great Principles of Computing, so that similar types of computational thinking appearing in different contexts are brought together. This aids understanding of the computational thinking found in the courses and provides a template for future work on new course materials