Topological Stability of Kinetic kk-Centers

We study the $k$-center problem in a kinetic setting: given a set of continuously moving points $P$ in the plane, determine a set of $k$ (moving) disks that cover $P$ at every time step, such that the disks are as small as possible at any point in time. Whereas the optimal solution over time may exhibit discontinuous changes, many practical applications require the solution to be stable: the disks must move smoothly over time. Existing results on this problem require the disks to move with a bounded speed, but this model is very hard to work with. Hence, the results are limited and offer little theoretical insight. Instead, we study the topological stability of $k$-centers. Topological stability was recently introduced and simply requires the solution to change continuously, but may do so arbitrarily fast. We prove upper and lower bounds on the ratio between the radii of an optimal but unstable solution and the radii of a topologically stable solution---the topological stability ratio---considering various metrics and various optimization criteria. For $k = 2$ we provide tight bounds, and for small $k > 2$ we can obtain nontrivial lower and upper bounds. Finally, we provide an algorithm to compute the topological stability ratio in polynomial time for constant $k$

Guest Editor's Foreword (Special Issue with Selected Papers from the 19th International Symposium on Graph Drawing, GD 2011)

This issue of the Journal of Graph Algorithms and Applications is devoted to the nineteenth International Symposium on Graph Drawing, held September 19-21, 2011, in Eindhoven, the Netherlands

Higher order Delaunay triangulations

For a set P of points in the plane, we introduce a class of triangulations that is an extension of the Delaunay triangulation. Instead of requiring that for each triangle the circle through its vertices contains no points of P inside, we require that at most k points are inside the circle. Since there are many different higher-order Delaunay triangulations for a point set, other useful criteria for triangulations can be incorporated without sacrificing the well-shapedness too much. Applications include realistic terrain modelling and mesh generation

Spatial information retrieval and geographical ontologies: an overview of the SPIRIT project

A large proportion of the resources available on the world-wide web refer to information that may be regarded as geographically located. Thus most activities and enterprises take place in one or more places on the Earth's surface and there is a wealth of survey data, images, maps and reports that relate to specific places or regions. Despite the prevalence of geographical context, existing web search facilities are poorly adapted to help people find information that relates to a particular location. When the name of a place is typed into a typical search engine, web pages that include that name in their text will be retrieved, but it is likely that many resources that are also associated with the place may not be retrieved. Thus resources relating to places that are inside the specified place may not be found, nor may be places that are nearby or that are equivalent but referred to by another name. Specification of geographical context frequently requires the use of spatial relationships concerning distance or containment for example, yet such terminology cannot be understood by existing search engines. Here we provide a brief survey of existing facilities for geographical information retrieval on the web, before describing a set of tools and techniques that are being developed in the project SPIRIT : Spatially-Aware Information Retrieval on the Internet (funded by European Commission Framework V Project IST-2001-35047)