Scientific Research (NWO) under project no. 639.023.301. We present new results for three problems dealing with a set P of n constant-complexity fat objects in 3-space. (i) We describe a data structure for vertical ray shooting in P that has O(log 2 n) query time and uses O(n log 2 n) storage. (ii) We give an algorithm to compute in O(n log 3 n) time a depth order on P, if it exists. (iii) We give an algorithm to verify in O(n log 4 n) time whether a given order on P is a valid depth order. All three results improve on previous results. 1

Abstract. Suppose we want to compute the Delaunay triangulation of a set P whose points are restricted to a collection R of input regions known in advance. Building on recent work by Löffler and Snoeyink [21], we show how to leverage our knowledge of R for faster Delaunay computation. Our approach needs no fancy machinery and optimally handles a wide variety of inputs, eg, overlapping disks of different sizes and fat regions. 1

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

We consider the problem of constructing binary space partitions for the set P of d-dimensional objects in d-dimensional space. There are several classes of objects defined for such settings, which support design of effective algorithms. We extend the existing the de Berg hierarchy of classes [8] by the definition of new classes derived from that one and we show desirability of such an extension. Moreover we propose a new algorithm, which works on generalized λ-low density scenes [20] (defined in this paper) and provides BSP tree of linear size. The tree can be constructed in O(n log 2 n) time and space, where n is the number of objects. Moreover, we can trade-off between size and balance of the BSP tree fairly simply.

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.

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The worst-case model for algorithm design does not always reflect the real world: inputs may have additional structure to be exploited, and sometimes data can be imprecise or become available only gradually. To better understand these situations, we examine several scenarios where additional information can affect the design and analysis of geometric algorithms. First, we consider hereditary convex hulls: given a three-dimensional convex polytope and a two-coloring of its vertices, we can find the individual monochromatic polytopes in linear expected time. This can be generalized in many ways, eg, to more than two colors, and to the offline-problem where we wish to preprocess a polytope so that any large enough subpolytope can be found quickly. Our techniques can also be used to give a simple analysis of the self-improving algorithm for planar Delaunay triangulations by Clarkson and Seshadhri [58]. Next, we assume that the point coordinates have a bounded number of bits, and that we can do standard bit manipulations in constant time. Then Delaunay triangulations can be found in expected time O(n √ log log n). Our result is based on a new connection between quadtrees and Delaunay triangulations, which also lets us generalize a recent result by Löffler and Snoeyink about Delaunay triangulations for imprecise points [110]. Finally, we consider randomized incremental constructions when the input permutation is generated by a bounded-degree Markov chain, and show that the resulting running time is almost optimal for chains with a constant eigenvalue gap.