We investigate ways in which an algorithm can improve its expected performance by finetuning itself automatically with respect to an arbitrary, unknown input distribution. We give such self-improving algorithms for sorting and computing Delaunay triangulations. The highlights of this work: (i) an algorithm to sort a list of numbers with optimal expected limiting complexity; and (ii) an algorithm to compute the Delaunay triangulation of a set of points with optimal expected limiting complexity. In both cases, the algorithm begins with a training phase during which it adjusts itself to the input distribution, followed by a stationary regime in which the algorithm settles to its optimized incarnation. 1

pefully make them curious about further powerful treasures to be discovered in the area of computational geometry. In order to achieve these goals in an engaging yet sound manner, the general concept throughout the course is to present each geometric data structure in the following way: first, the data strucure will be defined and described in detail; then, some of its fundamental properties will be highlighted; after that, one or more computational geometry algorithms based on the data structure will be presented; and finally, a number of recent, representative and practically relevant algorithms from computer graphics will be described in detail, showing the utilization of the data structure in a creative and enlightening way. We have arranged the topics in roughly increasing degree of difficulty. The hierarchical data structures are ordered by increasing flexibility, while the non-hierarchical topics build on each other. Finally, the last topic presents a generic technique for

We present a novel framework for hierarchical collision detection that can be applied to virtually all bounding volume (BV) hierarchies. It allows an application to trade quality for speed. Our algorithm yields an estimation of the quality, so that applications can specify the desired quality. In a timecritical system, applications can specify the maximum time budget instead, and quantitatively assess the quality of the results returned by the collision detection afterwards. Our framework

We present a novel, generic framework and algorithm for hierarchical collision detection, which allows an application to balance speed and quality of the collision detection. We pursue

Abstract: We present a novel sorting algorithm, Vis-Sort, to sort 1D and 3D geometric elements. Given a set of acyclic and non-intersecting 3D geometric primitives, Vis-Sort computes the visibility ordering from a viewpoint. The running time of our algorithm is dependent upon the degree of sortedness in the 3D sequence and is bounded by O(�Y �n), where n is the number of primitives and �Y � is the Knuth’s measure of disorder. The Knuth’s measure of disorder computes the minimum number of elements that need to be removed from the sequence for the remaining sequence to be sorted [35]. Vis-Sort exploits the spatial and temporal coherence between successive instances in a dynamic environment and performs incremental computations. Our algorithm requires no preprocessing and is applicable to all kind of models, including polygon soups and deformable models. We have used our algorithm for order-independent transparency computations in high-depth complexity environments and performing N-body collision culling in dynamic environments. We have implemented our algorithm and tested the system on a PC with a 3.4 GHz Pentium IV CPU with an NVIDIA GeForce FX 6800 Ultra GPU and applied it to complex environments with tens or hundreds of thousands of polygons. Our algorithm can compute a visibility ordering among the objects and triangles at interactive frame rates.

A binary space partition (BSP) for a set of disjoint objects in Euclidean space is a recursive decomposition, where each step partitions the space (and some of the objects) along a hyperplane and recurses on the objects clipped in each of the two open halfspaces. The size of a BSP is defined as the number of resulting fragments of the input objects. It is shown that every set of n disjoint line segments in the plane admits a BSP of size O(n log n / log log n). This bound is best possible apart from the constant factor. 1

A binary space partition tree is a data structure for the representation of a set of objectsin space. It found an increasing number of applications over the last decades. In recent years, intensifying research focused on its combinatorial properties, which affect directly the efficiency of applications. Important advances were made on binary space partitions for disjoint line segments in the plane and for axis-aligned objects in higher dimensions. New research directions were also initiated on some realistic polygonal scenes and on kinetic binary space partitions. This paper attempts to give an overview of these results and reiterates some of the most pressing open problems.

We present a novel framework for hierarchical collision detection that can be applied to virtually all bounding volume (BV) hierarchies. It allows an application to trade quality for speed. Our algorithm yields an estimation of the quality, so that applications can specify the desired quality. In a timecritical system, applications can specify the maximum time budget instead, and quantitatively assess the quality of the results returned by the collision detection afterwards.

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