There are Plane Spanners of Maximum Degree 4

Let E be the complete Euclidean graph on a set of points embedded in the plane. Given a constant t >= 1, a spanning subgraph G of E is said to be a t-spanner, or simply a spanner, if for any pair of vertices u,v in E the distance between u and v in G is at most t times their distance in E. A spanner is plane if its edges do not cross. This paper considers the question: "What is the smallest maximum degree that can always be achieved for a plane spanner of E?" Without the planarity constraint, it is known that the answer is 3 which is thus the best known lower bound on the degree of any plane spanner. With the planarity requirement, the best known upper bound on the maximum degree is 6, the last in a long sequence of results improving the upper bound. In this paper we show that the complete Euclidean graph always contains a plane spanner of maximum degree at most 4 and make a big step toward closing the question. Our construction leads to an efficient algorithm for obtaining the spanner from Chew's L1-Delaunay triangulation

Upper and Lower Bounds for Competitive Online Routing on Delaunay Triangulations

Consider a weighted graph G where vertices are points in the plane and edges are line segments. The weight of each edge is the Euclidean distance between its two endpoints. A routing algorithm on G has a competitive ratio of c if the length of the path produced by the algorithm from any vertex s to any vertex t is at most c times the length of the shortest path from s to t in G. If the length of the path is at most c times the Euclidean distance from s to t, we say that the routing algorithm on G has a routing ratio of c.We present an online routing algorithm on the Delaunay triangulation with competitive and routing ratios of 5.90. This improves upon the best known algorithm that has competitive and routing ratio 15.48. The algorithm is a generalization of the deterministic 1-local routing algorithm by Chew on the L1-Delaunay triangulation. When a message follows the routing path produced by our algorithm, its header need only contain the coordinates of s and t. This is an improvement over the currently known competitive routing algorithms on the Delaunay triangulation, for which the header of a message must additionally contain partial sums of distances along the routing path.We also show that the routing ratio of any deterministic k-local algorithm is at least 1.70 for the Delaunay triangulation and 2.70 for the L1-Delaunay triangulation. In the case of the L1-Delaunay triangulation, this implies that even though there exists a path between two points x and y whose length is at most 2.61|[xy]| (where |[xy]| denotes the length of the line segment [xy]), it is not always possible to route a message along a path of length less than 2.70|[xy]|. From these bounds on the routing ratio, we derive lower bounds on the competitive ratio of 1.23 for Delaunay triangulations and 1.12 for L1-Delaunay triangulations

Severe acute caffeine poisoning due to intradermal injections: Mesotherapy hazard

Introduction. Caffeine is indicated in the treatment of migraine headaches, as well as neonatal apnea and bradycardia syndrome. In mild poisoning, the most prevalent symptoms are nausea, vomiting, diarrhea, tremor, anxiety and headache. In more severe cases, symptoms consist of heart rythym abnormalities, myocardial infarction and seizures. Due to its common lipolytic effect, caffeine is used in mesotherapy, usually in combination with drugs of similar effect. We presented a patient with acute iatrogenic caffeine poisoning. Case report. A 51-year-old woman, with preexisting hypertension and hypertensive cardiomyopathy was subjected to cosmetic treatment in order to remove fat by intradermal caffeine injections. During the treatment the patient felt sickness, an urge to vomit, and a pronounced deterioration of general condition. Upon examination, the patient exhibited somnolence, hypotension and nonsustained ventricular tachycardia, which was sufficient enough evidence for further hospitalization. On admission to the intensive care unit the patient was anxious with increased heart rate, normotensive, with cold, damp skin, and visible traces of injection sites with surrounding hematomas on the anterior abdominal wall. Paroxysmal supraventricular tachycardia (PSVT) on electrocardiographic monitoring was found. The laboratory analysis determined a lowered potassium level of 2.1 mmol/L (normal range 3,5 - 5.2 mmol/L), and a toxicological analysis (liquid chromatography with ultraviolet detection) proved a toxic concentration of caffeine in plasma - 85.03 mg/L (toxic concentration over 25 mg/L). On application of intensive therapy, antiarrhythmics, and substitution of potassium, as well as both symptomatic and supportive therapy, there was a significant recovery. The patient was discharged without any sequele within four days. Conclusion. A presented rare iatrogenic acute caffeine poisoning occured due to massive absorption of caffeine from the subcutaneous adipose tissue into the circulation when injected directly into the tiny blood vessels, as evidenced by hematoma formation. Poisoning manifestations were registered in gastrointestinal, CNS (anxiety, somnolence) and cardiovascular (hypotension, ventricular tachycardia and nonsustained PSVT) system. In this era of mesotherapeutic treatment promotion, one should keep in mind toxic prevention, with application being carried out exclusively in a specialized institutio

Computing Branches Out: On Revitalizing Computing Education

Abstract Computing professionals are employed in so many different ways that it is difficult to even define the common foundation that all of them should master; however, a single degree program, computer science, is currently providing the training for most of these jobs. Computing education, following the model of engineering, may need to branch out into specialized degree programs to address the diversity of jobs that computing professionals currently hold. In this paper we describe the approach to computing education taken at DePaul University’s School of Computer Science, Telecommunications, and Information Systems, an approach that involves both specialized computing degrees and the integration of computing in general education requirements. Both of these efforts have had a positive effect on enrollments at DePaul CTI and hold promise for revitalizing computing education more broadly in the United States

An Optimal Localized Approximation Scheme for Euclidean MST

We consider the problem of locally constructing a spanning subgraph that approximates the Euclidean minimum spanning tree of a unit disk graph. We show that for any k ≥ 2 there exists a k-localized distributed algorithm that, given a unit disk graph U in the plane, constructs a planar subgraph of U containing a Euclidean MST on V (U), whose degree is at most 6, and whose total weight is at most 1+ 2 k−1 times the weight of the Euclidean MST on V (U). We prove that this approximation bound is tight by showing that, for any ɛ> 0, no k-localized algorithm can construct a spanning subgraph of U whose total weight is less than 1+ 2 k−1 −ɛ times the weight of a Euclidean MST on V (U). En route to our results, we prove a nice result about weighted planar graphs.