Quadtrees as an Abstract Domain

Quadtrees have proved popular in computer graphics and spatial databases as a way of representing regions in two dimensional space. This hierarchical data-structure is flexible enough to support non-convex and even disconnected regions, therefore it is natural to ask whether this datastructure can form the basis of an abstract domain. This paper explores this question and suggests that quadtrees offer a new approach to weakly relational domains whilst their hierarchical structure naturally lends itself to representation with boolean functions

A limit field for orthogonal range searches in two-dimensional random point search trees

We consider the cost of general orthogonal range queries in random quadtrees. The cost of a given query is encoded into a (random) function of four variables which characterize the coordinates of two opposite corners of the query rectangle. We prove that, when suitably shifted and rescaled, the random cost function converges uniformly in probability towards a random field that is characterized as the unique solution to a distributional fixed-point equation. We also state similar results for $2$-d trees. Our results imply for instance that the worst case query satisfies the same asymptotic estimates as a typical query, and thereby resolve an old question of Chanzy, Devroye and Zamora-Cura [\emph{Acta Inf.}, 37:355--383, 2000]Comment: 24 pages, 8 figure

Singularity analysis, Hadamard products, and tree recurrences

We present a toolbox for extracting asymptotic information on the coefficients of combinatorial generating functions. This toolbox notably includes a treatment of the effect of Hadamard products on singularities in the context of the complex Tauberian technique known as singularity analysis. As a consequence, it becomes possible to unify the analysis of a number of divide-and-conquer algorithms, or equivalently random tree models, including several classical methods for sorting, searching, and dynamically managing equivalence relationsComment: 47 pages. Submitted for publicatio

Quadtrees as an Abstract Domain

Quadtrees have proved popular in computer graphics and spatial databases as a way of representing regions in two dimensional space. This hierarchical data-structure is flexible enough to support nonconvex and even disconnected regions, therefore it is natural to ask whether this data-structure can the form the basis of an abstract domain. This paper explores this question and suggests that quadtrees offer a new approach to weakly relation domains whilst their hierarchical structure naturally lends itself to representation with boolean functions

The K-framed quadtrees approach for path planning through a known environment

One of the most important tasks for a mobile robot is to navigate in an environment. The path planning is required to design the trajectory that generates useful motions from the original to the desired position. There are several methodologies to perform the path planning. In this paper, a new method of approximate cells decomposition, called K-Framed Quadtrees is present, to which the algorithm A ⋆ is applied to determine trajectories between two points. To validate the new approach, we made a comparative analysis between the present method, the grid decomposition, quadtree decomposition and framed quadtree decomposition. Results and implementation specifications of the four methods are presented.Project ”TEC4Growth - Pervasive Intelligence, Enhancers and Proofs of Concept with Industrial Impact/NORTE-01-0145-FEDER-000020” is financed by the North Portugal Regional Operational. Programme (NORTE 2020), under the PORTUGAL 2020 Partnership Agreement, and through the European Regional Development Fund (ERDF). This work is also financed by the ERDF – European Regional Development Fund through the Operational Programme for Competitiveness and Internationalisation - COMPETE 2020 Programme within project POCI-01-0145-FEDER-006961, and by National Funds through the FCT – Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) as part of project UID/EEA/50014/2013.info:eu-repo/semantics/publishedVersio

KBGIS-2: A knowledge-based geographic information system

The architecture and working of a recently implemented knowledge-based geographic information system (KBGIS-2) that was designed to satisfy several general criteria for the geographic information system are described. The system has four major functions that include query-answering, learning, and editing. The main query finds constrained locations for spatial objects that are describable in a predicate-calculus based spatial objects language. The main search procedures include a family of constraint-satisfaction procedures that use a spatial object knowledge base to search efficiently for complex spatial objects in large, multilayered spatial data bases. These data bases are represented in quadtree form. The search strategy is designed to reduce the computational cost of search in the average case. The learning capabilities of the system include the addition of new locations of complex spatial objects to the knowledge base as queries are answered, and the ability to learn inductively definitions of new spatial objects from examples. The new definitions are added to the knowledge base by the system. The system is currently performing all its designated tasks successfully, although currently implemented on inadequate hardware. Future reports will detail the performance characteristics of the system, and various new extensions are planned in order to enhance the power of KBGIS-2