de Mt Mark Berg

Trekking in the Alps without freezing or getting tired

Abstract. Let F be a polyhedral terrain with n vertices. We show how to preprocess F such that for any two query points on F it can be decided whether there exists a path on F between the two points whose height decreases monotonically. More generally, the minimum total ascent or descent along any path between the two points can be computed. It is also possible to decide, given two query points and a height, whether there is a path that stays below this height. All these queries can be answered with one data structure which stores the so-called height-level map of the terrain. Although the height-level map has quadratic worst-case complexity, it is stored implicitly using only linear storage. The query time for all the above queries is

Linear size binary space partitions for uncluttered scenes

O(\log n)

Motion planning for multiple robots

and the structure can be built in

Vertical decompositions for triangles in 3-space

O(n\log n)

Computing the Maximum Overlap of Two Convex Polygons Under Translations.

time. A path with the desired property can be reported in additional time that is linear in the description size of the path.Let F be a polyhedral terrain with n vertices. We show how to preprocess F such that for any two query points on F it can be decided whether there exists a path on F between the two points whose height decreases monotonically. More generally, the minimum total ascent or descent along any path between the two points can be computed. It is also possible to decide, given two query points and a height, whether there is a path that stays below this height. All these queries can be answered with one data structure which stores the so-called height-level map of the terrain. Although the height-level map has quadratic worst-case complexity, it is stored implicitly using only linear storage. The query time for all the above queries is

Motion Planning in Environments with Low Obstacle Density

O(\log n)

Efficient ray shooting and hidden surface removal

and the structure can be built in

Finding shortest paths in the presence of orthogonal obstacles using a combined L 1 and link metric

O(n\log n)

Streaming Algorithms for Line Simplification

time. A path with the desired property can be reported in additional time that is linear in the description size of the path.

Hidden surface removal for axis-parallel polyhedra

We describe a new and simple method for constructing binary space partitions (BSPs) 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 construction time is O(n log n) . Because any set of disjoint fat objects is uncluttered, our result implies an efficient 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 O( log n) , and the amount of storage is O(n) . 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 describe a new and simple method for constructing binary space partitions (BSPs) 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 construction time is O(n log n) . Because any set of disjoint fat objects is uncluttered, our result implies an efficient 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 O( log n) , and the amount of storage is O(n) . 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.