Linear Size Binary Space Partitions for Uncluttered Scenes

Abstract

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.

Approximate Range Searching Using Binary Space Partitions

“…In this section we show how to construct a BSP tree for S that has linear size and very good performance for approximate range searching if the density of S is constant. Our method combines ideas from de Berg [11] with the BAR-tree of Duncan et al [18]. We will call this BSP an object BAR-tree, or oBAR-tree for short.…”

“…Our overall strategy, also used by de Berg [11], is to compute a suitable set of points that will guide the construction of the BSP tree. Unlike in [11], however, we cannot use the bounding-box vertices of the objects in S for this, because that does not work in combination with a BAR-tree.…”

“…Hence, several researchers have proposed algorithms to construct small BSP trees in various different settings [4,5,11,12,25,26,30]. For instance, Paterson and Yao [25] proved that any set of n disjoint line segments in the plane admits a BSP tree of size O(n log n).…”

“…Paterson and Yao also proved that there are sets of n disjoint triangles in R 3 for which any BSP tree must have quadratic size, and they gave a construction algorithm that achieves this bound. Since a quadratic-size BSP tree is useless in most practical applications, de Berg [11] studied BSP trees for so-called uncluttered scenes. He proved that uncluttered scenes admit a BSP tree of linear size, in any fixed dimension.…”

“…Range searching in low-density scenes can be done as well in O(log n) time-again using some additional structures-but only if the diameter of the query range is about the same as the diameter of the smallest object in the scene. Surprisingly, we do not know of any papers that investigate the efficiency of BSP trees, without any additional data structures as in [11], for range searching on non-point objects.…”