Antoine Vigneron, studies the problem of determining a rigid motion (or pure translation) of a convex set P in the plane in order to maximize its area of overlap with another convex set Q. The problem is motivated by applications in shape matching. The authors give efficient approximation algorithms that compute a (1 − ε)-approximation to an optimal solution for any fixed ε > 0. For convex polygons having n vertices, the algorithm runs in time O((1/ε) log n + (1/ε 2 ) log(1/ε)) for rigid motions and time O((1/ε) log n + (1/ε) log(1/ε)) for translational motions. Prior results computed an exact solution in high degree polynomial time.The second paper, "Kinetic Sorting and Kinetic Convex Hull", by Mohammad Ali Abam and Mark de Berg, studies query and maintenance cost tradeoffs in certain kinetic data structures. First, they consider the problem of maintaining a data structure for a set S of moving points on a line; the structure should support generation of the sorted list of points at any given time. The authors give tight lower bounds, based on explicitly constructing a family of instances, showing that with sub-quadratic maintenance cost one cannot obtain a significant improvement over the trivial time (O(n log n)) to generate the sorted list, even for linear motions of the points. Second, they give a kinetic data structure that trades off query time and maintenance cost for performing gift-wrapping and extreme-point queries for a set of points moving in the plane on trajectories of bounded algebraic degree.The third paper, "Minimum Dilation Stars", by David Eppstein and Kevin Wortman, considers an optimization problem that arises in Euclidean spanners. They efficiently compute, in time O(n log n), the dilation (worst-case ratio of distance in the graph to Euclidean distance) of a given star graph on n points. They also give a randomized O(n log n) expected time algorithm for finding the best position of the center of the star, in any fixed dimension, in order to minimize the dilation. In dimension two, they also give an efficient randomized algorithm for computing the best star center from among the input point set.The fourth paper, "Learning Smooth Shapes by Probing", by Jean-Daniel Boissonnat, Leonidas Guibas, and Steve Oudot, considers a problem in sensing and reconstructing a surface in three dimensions by carefully generating a sequence of probes, each of which gives a sample point on an unknown surface S where a point probing device first comes in contact with S. Their strategy gives a rich enough set of samples that S can be approximated to any degree of accuracy. They bound the number of probes and the number of movements of the probing device, and they give a method of maintaining the approximating surface mesh, which has provable properties with respect to topological type, normal estimation, etc.Finally, we wish to thank the authors for submitting their papers to the special issue and keeping to a tight time line. We are especially grateful to the referees for their careful and timely reviews. It is ...
