We prove a general lemma on the existence of (1/r)-cuttings of geometric objects in Ed that satisfy certain properties. We use this lemma to construct (1/r)-cuttings of small size for arrangements of line segments in the plane and arrangements of triangles in 3-space; for line segments in the plane we obtain a cutting of size O(r+Ar2/n2), and for triangles in 3-space our cutting has size O(r2+c+Ar3/n3). Here A is the combinatorial complexity of the arrangement. Finally, we use these results to obtain new results for several problems concerning line segments in the plane and triangles in 3-space.

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 On for an uncluttered scene consisting of n objects. The construction time is On log n. Because any set of disjoint fat objects is uncluttered, our result implies an e cient method to construct a linear size BSP for fat objects.We use our BSP to develop a data structure for point location in uncluttered scenes. The query time of our structure is Olog n, and the amount of storage is On. This result can in turn be used to perform range queries with not-too-small ranges in scenes consisting of disjoint fat objects or, more generally, in so-called low-density scenes.

Abstract. We present a simple and efficient paradigm for computing the exact solution of the motion planning problem in environments with a low obstacle density. Such environments frequently occur in practical instances of the motion planning problem. The complexity of the free space for such environments is known to be linear in the number of obstacles. Our paradigm is a new cell decomposition approach to motion planning and exploits properties that follow from the low density of the obstacles in the robot's workspace. These properties allow us to decompose the workspace, subject to some constraints, rather than to decompose the higher-dimensional free configuration space directly. A sequence of uniform steps transforms the workspace decomposition into a free space decomposition of asymptotically the same size. The approach leads to nearly optimal O(n log n) motion planning algorithms for free-flying robots with any fixed number of degrees of freedom in workspaces with low obstacle density.

A box-tree is a bounding-volume hierarchy that uses axisaligned 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 with small worst-case query complexity with respect to queries with axis-parallel boxes and with points. We also prove lower bounds on the worst-case query complexity for box-trees, which show that our results are optimal or close to optimal. Finally, we present algorithn^ to convert box-trees to R-trees, resulting in R-trees with (almost) optimal query complexity.

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