We present the first constant-factor approximation algorithm for a non-trivial instance of the optimal guarding (coverage) problem in polygons. In particular, we give an O(1)-approximation algorithm for placing the fewest point guards on a 1.5D terrain, so that every point of the terrain is seen by at least one guard. While polylogarithmic-factor approximations follow from set cover results, our new results exploit geometric structure of terrains to obtain a substantially improved approximation algorithm.
The Indoor Positioning and Indoor Navigation (IPIN) conference holds an annual competition in which indoor localization systems from different research groups worldwide are evaluated empirically. The objective of this competition is to establish a systematic evaluation methodology with rigorous metrics both for real-time (on-site) and post-processing (off-site) situations, in a realistic environment unfamiliar to the prototype developers. For the IPIN 2018 conference, this competition was held on September 22nd, 2018, in Atlantis, a large shopping mall in Nantes (France). Four competition tracks (two on-site and two off-site) were designed. They consisted of several 1 km routes traversing several floors of the mall. Along these paths, 180 points were topographically surveyed with a 10 cm accuracy, to serve as ground truth landmarks, combining theodolite measurements, differential global navigation satellite system (GNSS) and 3D scanner systems. 34 teams effectively competed. The accuracy score corresponds to the third quartile (75 th percentile) of an error metric that combines the horizontal positioning error and the floor detection. The best results for the on-site tracks showed an accuracy score of 11.70 m (Track 1) and 5.50 m (Track 2), while the best results for the off-site tracks showed an accuracy score of 0.90 m (Track 3) and 1.30 m (Track 4). These results showed that it is possible to obtain high accuracy indoor positioning solutions in large, realistic environments using wearable light-weight sensors without deploying any beacon. This paper describes the organization work of the tracks, analyzes the methodology used to quantify the results, reviews the lessons learned from the competition and discusses its future.
We study the problem of computing the visibility graph defined by a set P of n points inside a polygon Q: two points p, q ∈ P are joined by an edge if the segment pq ⊂ Q. Efficient output-sensitive algorithms are known for the case in which P is the set of all vertices of Q. We examine the general case in which P is an arbitrary set of points, interior or on the boundary of Q and study a variety of algorithmic questions. We give an output-sensitive algorithm, which is nearly optimal, when Q is a simple polygon. We introduce a notion of "fat" or "robust" visibility, and give a nearly optimal algorithm for computing visibility graphs according to it, in polygons Q that may have holes. Other results include an algorithm to detect if there are any visible pairs among P, and algorithms for output-sensitive computation of visibility graphs with distance restrictions, invisibility graphs, and rectangle visibility graphs.
Given a terrain and a point p on or above it, we wish to compute the region R p that is visible from p. We present a generic radar-like algorithm for computing an approximation of R p . The algorithm interpolates the visible region between two consecutive rays (emanating from p) whenever the rays are close enough; that is, whenever the difference between the sets of visible segments along the cross sections in the directions specified by the rays is below some threshold. Thus the density of the sampling by rays is sensitive to the shape of the visible region. We suggest a specific way to measure the resemblance (difference) and to interpolate the visible region between two consecutive rays. We also present an alternative algorithm, which uses circles of increasing radii centered at p instead of rays emanating from p. Both algorithms compute a representation of the (approximated) visible region that is especially suitable for is-visible-from-p queries, i.e., given a query point q on the terrain determine whether q is visible from p. Finally, we report on the experiments that we performed with these algorithms and with their corresponding fixed versions, using a natural error measure. Our main conclusion is that the radar-like algorithm is significantly better than the others.
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