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24
Parameterized Complexity of Geometric Problems
, 2007
"... This paper surveys parameterized complexity results for hard geometric algorithmic problems. It includes fixedparameter tractable problems in graph drawing, geometric graphs, geometric covering and several other areas, together with an overview of the algorithmic techniques used. Fixedparameter in ..."
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Cited by 11 (2 self)
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This paper surveys parameterized complexity results for hard geometric algorithmic problems. It includes fixedparameter tractable problems in graph drawing, geometric graphs, geometric covering and several other areas, together with an overview of the algorithmic techniques used. Fixedparameter intractability results are surveyed as well. Finally, we give some directions for future research.
Optimization for first order Delaunay triangulations
 COMPUTATIONAL GEOMETRY: THEORY AND APPLICATIONS
, 2009
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Multiresolution Amalgamation: Dynamic Spatial Data Cube
"... Aggregating spatial objects is a necessary step in generating spatial data cubes to support rollup/drilldown operations. Current approaches face performance bottleneck issues when attempting to dynamically aggregate geometries for a large set of spatial data. We observe that changing the resolutio ..."
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Cited by 5 (1 self)
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Aggregating spatial objects is a necessary step in generating spatial data cubes to support rollup/drilldown operations. Current approaches face performance bottleneck issues when attempting to dynamically aggregate geometries for a large set of spatial data. We observe that changing the resolution of a region is reflective of the fact that the precision of spatial data can be changed to certain extent without compromising its usefulness. Moreover most spatial datasets are stored at much higher resolutions than are necessary for some applications. The existing approaches, which aggregate objects at a base resolution, often results in a processing bottleneck due to extraneous I/O. In this paper, we develop a new aggregation methodology that can significantly reduce retrieval (I/O) costs and improve overall performance by utilising multiresolution data storage and retrieval techniques. Topological inconsistencies that may arise during resolution change, which are not handled by current amalgamation techniques, are identified. By factoring these issues into the amalgamation query processing, the retrieval loads can be further reduced with guaranteed topological correctness. Experimental results illustrate significant savings in data retrieval and overall processing time of dynamic aggregation.
Sparse Terrain Pyramids
 IN PROC. OF THE 16TH ACM SIGSPATIAL INT. CONFERENCE ON ADVANCES IN GEOGRAPHIC INFORMATION SYSTEMS
, 2008
"... Bintrees based on longest edge bisection and hierarchies of diamonds are popular multiresolution techniques on regularly sampled terrain datasets. In this work, we consider sparse terrain pyramids as a compact multiresolution representation for terrain datasets whose samples are a subset of those ly ..."
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Cited by 3 (3 self)
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Bintrees based on longest edge bisection and hierarchies of diamonds are popular multiresolution techniques on regularly sampled terrain datasets. In this work, we consider sparse terrain pyramids as a compact multiresolution representation for terrain datasets whose samples are a subset of those lying on a regular grid. While previous diamondbased approaches can efficiently represent meshes built on a complete grid of resolution (2 k + 1) 2, this is not suitable when the field values are uniform in large areas or simply nonexistent. We explore properties of diamonds to simplify an encoding of the implicit dependency relationship between diamonds. Additionally, we introduce a diamond clustering technique to further reduce the geometric and topological overhead of such representations. We demonstrate the coherence of our clustering technique as well as the compactness of our representation.
Computing the Independence Number of Intersection Graphs
"... Computing the maximum number of disjoint elements in a collection C of geometric objects is a classical problem in computational geometry with applications ranging from frequency assignment in cellular networks to map labeling in computational cartography. The problem is equivalent to finding the in ..."
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Cited by 3 (0 self)
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Computing the maximum number of disjoint elements in a collection C of geometric objects is a classical problem in computational geometry with applications ranging from frequency assignment in cellular networks to map labeling in computational cartography. The problem is equivalent to finding the independence number, α(GC), of the intersection graph GC of C, obtained by connecting two elements of C with an edge if and only if their intersection is nonempty. This is known to be an NPhard task even for systems of segments in the plane with at most two different slopes. The best known polynomial time approximation algorithm for systems of arbitrary segments is due to Agarwal and Mustafa, and returns in the worst case an n 1/2+o(1)approximation for α. Using extensions of the LiptonTarjan separator theorem, we improve this result and present, for every ɛ> 0, a polynomial time algorithm for computing α(GC) with approximation ratio at most n ɛ. In contrast, for general graphs, for any ɛ> 0 it is NPhard to approximate the independence number within a factor of n 1−ɛ. We also give a subexponential time exact algorithm for computing the independence number of intersection graphs of arcwise connected sets in the plane. 1
Towards a definition of higher order constrained Delaunay triangulations
 In Proc. 19th Annual Canadian Conference on Computational Geometry
, 2007
"... When a triangulation of a set of points and edges is required, the constrained Delaunay triangulation is often the preferred choice because of its wellshaped triangles. However, in applications like terrain modeling, it is sometimes necessary to have flexibility to optimize some other aspect of the ..."
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Cited by 2 (1 self)
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When a triangulation of a set of points and edges is required, the constrained Delaunay triangulation is often the preferred choice because of its wellshaped triangles. However, in applications like terrain modeling, it is sometimes necessary to have flexibility to optimize some other aspect of the triangulation, while still having nicelyshaped triangles and including a set of constraints. Higher order Delaunay triangulations were introduced to provide a class of wellshaped triangulations, flexible enough to allow the optimization of some extra criterion. But they are not able to handle constraints: a single constraining edge may cause that all triangulations with that edge have high order, allowing illshaped triangles at any part of the triangulation. In this paper we generalize the concept of the constrained Delaunay triangulation to higher order constrained Delaunay triangulations. We study several possible definitions that assure that an orderk constrained Delaunay triangulation exists for any k ≥ 0, while maintaining the character of higher order Delaunay triangulations of point sets. Several properties of these definitions are studied, and efficient algorithms to support computations with orderk constrained Delaunay triangulations are also discussed. For the special case of k = 1, we show that many measures can be optimized efficiently in the presence of constraints. 1
A Fast Triangle to Triangle Intersection Test for Collision Detection
"... The triangletotriangle intersection test is a basic component of all collision detection data structures and algorithms. This paper presents a fast method for testing whether two triangles embedded in three dimensions intersect. Our technique solves the basic sets of linear equations associated wi ..."
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The triangletotriangle intersection test is a basic component of all collision detection data structures and algorithms. This paper presents a fast method for testing whether two triangles embedded in three dimensions intersect. Our technique solves the basic sets of linear equations associated with the problem and exploits the strong relations between these sets to speed up their solution. Moreover, unlike previous techniques, with very little additional cost, the exact intersection coordinates can be determined. Finally, our technique uses general principles that can be applied to similar problems such as rectangletorectangle intersection tests, and generally to problems where several equation sets are strongly related. We show that our algorithm saves about 20 % of the mathematical operations used by the best previous triangletotriangle intersection algorithm. Our experiments also show that it runs 18.9 % faster than the fastest previous algorithm on average for typical scenarios of collision detection (on Pentium 4).