A filling subgroup of a finitely generated free group F (X) is a subgroup which does not fix a point in any very small action free action on an R-tree. For the free group of rank two, we construct a combinatorial algorithm to determine whether or not a given finitely generated subgroup is filling. In higher ranks, we discuss two types of non-filling subgroups: those contained in loop vertex sub...

Since its introduction in 1984, R-tree has been proven to be one of the most practical and well-behaved data structures for accommodating dynamic massive sets of geometric objects and conducting a very diverse set of queries on such datasets in real-world applications. This success has led to a variety of versions, each one trying to tune the performance parameters of the original proposal. Amo...

ÐSelection and join queries are fundamental operations in Data Base Management Systems (DBMS). Support for nontraditional data, including spatial objects, in an efficient manner is of ongoing interest in database research. Toward this goal, access methods and cost models for spatial queries are necessary tools for spatial query processing and optimization. In this paper, we present analytical m...

We propose a new Rtree structure that outperforms all the older ones. The heart of the idea is to facilitate the deferred splitting ap preach in R-trees. This is done by proposing an ordering on the R-tree nodes. This ordering has to be ‘good’, in the sense that it should group ‘similar’ data rectangles to gether, to minimize the area and perimeter of the resulting minimum bounding rectangles (...

A geodesic lamination L on a closed hyperbolic surface S, when provided with a transverse measure μ, gives rise to a “dual R-tree” Tμ, together with an action of G = π1S on Tμ by isometries. A point of Tμ corresponds precisely to a leaf of the lift L̃ of L to the universal covering S̃ of S, or to a complementary component of L̃ in S̃. The G-action on T is induced by the G-action on S̃ as deck transf...

Content-based Image Retrieval (CBIR) is a flexible approach to access modern image database with large data size, whereby the actual images are described by multi-dimensional feature vectors. These are then indexed in the database to provide low retrieval latency. Currently, many different structures have been proposed, among them, the structures of the R-tree family, especially the R-tree, R*-...