1,450 research outputs found

    Approximate Euclidean shortest paths in polygonal domains

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    Given a set P\mathcal{P} of hh pairwise disjoint simple polygonal obstacles in R2\mathbb{R}^2 defined with nn vertices, we compute a sketch Ω\Omega of P\mathcal{P} whose size is independent of nn, depending only on hh and the input parameter ϵ\epsilon. We utilize Ω\Omega to compute a (1+ϵ)(1+\epsilon)-approximate geodesic shortest path between the two given points in O(n+h((lgn)+(lgh)1+δ+(1ϵlghϵ)))O(n + h((\lg{n}) + (\lg{h})^{1+\delta} + (\frac{1}{\epsilon}\lg{\frac{h}{\epsilon}}))) time. Here, ϵ\epsilon is a user parameter, and δ\delta is a small positive constant (resulting from the time for triangulating the free space of P\cal P using the algorithm in \cite{journals/ijcga/Bar-YehudaC94}). Moreover, we devise a (2+ϵ)(2+\epsilon)-approximation algorithm to answer two-point Euclidean distance queries for the case of convex polygonal obstacles.Comment: a few updates; accepted to ISAAC 201

    Query processing of spatial objects: Complexity versus Redundancy

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    The management of complex spatial objects in applications, such as geography and cartography, imposes stringent new requirements on spatial database systems, in particular on efficient query processing. As shown before, the performance of spatial query processing can be improved by decomposing complex spatial objects into simple components. Up to now, only decomposition techniques generating a linear number of very simple components, e.g. triangles or trapezoids, have been considered. In this paper, we will investigate the natural trade-off between the complexity of the components and the redundancy, i.e. the number of components, with respect to its effect on efficient query processing. In particular, we present two new decomposition methods generating a better balance between the complexity and the number of components than previously known techniques. We compare these new decomposition methods to the traditional undecomposed representation as well as to the well-known decomposition into convex polygons with respect to their performance in spatial query processing. This comparison points out that for a wide range of query selectivity the new decomposition techniques clearly outperform both the undecomposed representation and the convex decomposition method. More important than the absolute gain in performance by a factor of up to an order of magnitude is the robust performance of our new decomposition techniques over the whole range of query selectivity

    Query-points visibility constraint minimum link paths in simple polygons

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    We study the query version of constrained minimum link paths between two points inside a simple polygon PP with nn vertices such that there is at least one point on the path, visible from a query point. The method is based on partitioning PP into a number of faces of equal link distance from a point, called a link-based shortest path map (SPM). Initially, we solve this problem for two given points ss, tt and a query point qq. Then, the proposed solution is extended to a general case for three arbitrary query points ss, tt and qq. In the former, we propose an algorithm with O(n)O(n) preprocessing time. Extending this approach for the latter case, we develop an algorithm with O(n3)O(n^3) preprocessing time. The link distance of a qq-visiblevisible path between ss, tt as well as the path are provided in time O(logn)O(\log n) and O(m+logn)O(m+\log n), respectively, for the above two cases, where mm is the number of links

    Geodesic-Preserving Polygon Simplification

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    Polygons are a paramount data structure in computational geometry. While the complexity of many algorithms on simple polygons or polygons with holes depends on the size of the input polygon, the intrinsic complexity of the problems these algorithms solve is often related to the reflex vertices of the polygon. In this paper, we give an easy-to-describe linear-time method to replace an input polygon P\mathcal{P} by a polygon P\mathcal{P}' such that (1) P\mathcal{P}' contains P\mathcal{P}, (2) P\mathcal{P}' has its reflex vertices at the same positions as P\mathcal{P}, and (3) the number of vertices of P\mathcal{P}' is linear in the number of reflex vertices. Since the solutions of numerous problems on polygons (including shortest paths, geodesic hulls, separating point sets, and Voronoi diagrams) are equivalent for both P\mathcal{P} and P\mathcal{P}', our algorithm can be used as a preprocessing step for several algorithms and makes their running time dependent on the number of reflex vertices rather than on the size of P\mathcal{P}
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