Computing the smallest k-enclosing circle and related problems

AbstractWe present an efficient algorithm for solving the “smallest k-enclosing circle” (kSC) problem: Given a set of n points in the plane and an integer k ⩽ n, find the smallest disk containing k of the points. We present two solutions. When using O(nk) storage, the problem can be solved in time O(nk log2 n). When only O(n log n) storage is allowed, the running time is O(nk log2 n log n/k). We also extend our technique to obtain efficient solutions of several related problems (with similar time and storage bounds). These related problems include: finding the smallest homothetic copy of a given convex polygon P which contains k points from a given planar set, and finding the smallest disk intersecting k segments from a given planar set of non-intersecting segments

From Cutting Planes Algorithms to Compression Schemes and Active Learning

Cutting-plane methods are well-studied localization(and optimization) algorithms. We show that they provide a natural framework to perform machinelearning ---and not just to solve optimization problems posed by machinelearning--- in addition to their intended optimization use. In particular, theyallow one to learn sparse classifiers and provide good compression schemes.Moreover, we show that very little effort is required to turn them intoeffective active learning methods. This last property provides a generic way todesign a whole family of active learning algorithms from existing passivemethods. We present numerical simulations testifying of the relevance ofcutting-plane methods for passive and active learning tasks.Comment: IJCNN 2015, Jul 2015, Killarney, Ireland. 2015, \<http://www.ijcnn.org/\&g

Witness (Delaunay) Graphs

Proximity graphs are used in several areas in which a neighborliness relationship for input data sets is a useful tool in their analysis, and have also received substantial attention from the graph drawing community, as they are a natural way of implicitly representing graphs. However, as a tool for graph representation, proximity graphs have some limitations that may be overcome with suitable generalizations. We introduce a generalization, witness graphs, that encompasses both the goal of more power and flexibility for graph drawing issues and a wider spectrum for neighborhood analysis. We study in detail two concrete examples, both related to Delaunay graphs, and consider as well some problems on stabbing geometric objects and point set discrimination, that can be naturally described in terms of witness graphs.Comment: 27 pages. JCCGG 200

Similarity between polygonal shapes

In this thesis we review existing algorithms for measuring shape similarity between polygons. We present a new approach to measure similarity based on the notion of annular profile. We also present the implementation of three shape measuring algorithms: signature function, turning function, and annular profile. The implementation is done by using the Visual C++ programming language. Finally, we discuss the comparative performances of the above three methods for capturing shape similarity. Measurement of shape similarity has applications in pattern recognition and artificial intelligence

Farthest-Polygon Voronoi Diagrams

Given a family of k disjoint connected polygonal sites in general position and of total complexity n, we consider the farthest-site Voronoi diagram of these sites, where the distance to a site is the distance to a closest point on it. We show that the complexity of this diagram is O(n), and give an O(n log^3 n) time algorithm to compute it. We also prove a number of structural properties of this diagram. In particular, a Voronoi region may consist of k-1 connected components, but if one component is bounded, then it is equal to the entire region