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We give a simple lazy randomized incremental algorithm to compute ≤<italic>k</italic>-levels in arrangements of <italic>x</italic>-monotone Jordan curves in the plane, and in arrangements of planes in three-dimensional space. If each pair of curves intersects in at most <italic>s</italic> points, the expected running time of the algorithm is… (More)

In TSP with neighborhoods (TSPN) we are given a collection S of regions in the plane, called neighborhoods, and we seek the shortest tour that visits all neighborhoods. Until now constant-factor approximation algorithms have been known only for cases where the neighborhoods are of approximately the same size. In this paper we present the first polynomial… (More)

We describe a new and simple method for constructing binary space partitions in arbitrary dimensions. We also introduce the concept of uncluttered scenes, which are scenes with a certain property that we suspect many realistic scenes exhibit, and we show that our method constructs a BSP of size O(n) for an uncluttered scene consisting of n objects. The… (More)

We present the Priority R-tree, or PR-tree, which is the first R-tree variant that always answers a window query using O((N/B)<sup>1</sup> <sup>1/d</sup> + T/B) I/Os, where N is the number of d-dimensional (hyper-) rectangles stored in the R-tree, B is the disk block size, and T is the output size. This is provably asymptotically optimal and significantly… (More)

Many algorithms developed in computational geometry are needlessly complicated and slow because they have to be prepared for very complicated, hypothetical inputs. To avoid this, realistic models are needed that describe the properties that realistic inputs have, so that algorithms can de designed that take advantage of these properties. This can lead to… (More)

A box-tree is a \ifasci so-called \emph{bounding-volume hierarchy} \else bounding-volume hierarchy \fi that uses axis-aligned boxes as bounding volumes. The query complexity of a box-tree with respect to a given type of query is the maximum number of nodes visited when answering such a query. We describe several new algorithms for constructing box-trees… (More)

We study the following variant of the well-known line-simpli-ficationproblem: we are getting a possibly infinite sequence of points p<sub>0</sub>,p<sub>1</sub>,p<sub>2</sub>,... in the plane defining a polygonal path, and as wereceive the points we wish to maintain a simplification of the pathseen so far. We study this problem in a streaming setting, where… (More)

In this paper we study several instances of the problem of determining the maximum number of topologically distinct two-dimensional images that three-dimensional scenes can induce. To bound this number, we investigate arrangements of curves and of surfaces that have a certain sparseness property. Given a collection of n algebraic surface patches of constant… (More)

Snap rounding is a method for converting arbitrary-precision arrangements of segments into fixed-precision representation. We present an algorithm for snap rounding with running time O((n + I) log n), where I is the number of intersections between the input segments. In the worst case, our algorithm is an order of magnitude more efficient than the best… (More)