Box-Trees and R-Trees with Near-Optimal Query Time

Abstract

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.

Optimal External Memory Planar Point Enclosure

“…These bounds are optimal in comparison-based models of computation, and they are the bounds we would like to obtain for more complicated external memory data structure problems. In the study of such problems, a number of different lower bound models have been developed in recent years: the non-replicating index model [21], the external memory pointer machine model [25], the bounding-volume hierarchy model [1], and the indexability model [20,19]. The most general of these models is the indexability model of Hellerstein, Koutsoupias, and Papadimitriou [20,19].…”

“…[17] for a survey); very recently Arge et al [5] designed a variant that answers a query in worst-case O( N/B + K/B) I/Os. This is optimal in the very restrictive bounding-volume hierarchy model [1]. However, in the internal memory pointer machine model a linear size data structure that can answer a query in the optimal O(log N + K) time has been developed [10].…”

“…While a lot of work has been done on developing worst-case efficient external memory data structures for range searching problems (see e.g. [3,17] for surveys), the point enclosure problem has only been considered in some very restricted models of computation [1,14]. In this paper we prove a general lower bound on the tradeoff between the size of an external memory point enclosure structure and its query cost, and show that this tradeoff is asymptotically tight by developing a family of structures with matching space and query bounds.…”

Box-trees for collision checking in industrial installations

“…Agarwal et al [1] recently showed that a box-tree exists that has O(n 2/3 + k) query time for ranges that are axis-parallel boxes, where n is the total number of boxes in S and k is the number of boxes intersecting the query range. This bound is rather disappointing: if the query time would really be that bad, box-trees would not be used so much in practice.…”

“…We describe a new, simple algorithm to construct a box-tree on a set of boxes in 3D. This algorithm generalizes the 2D kd-interval tree described by Agarwal et al [1] to 3D, with one additional crucial twist: We partition the input boxes into three subsets, according to the orientation of their longest edge, and construct separate box-trees for these subsets; these subtrees are then combined to form the final tree. Our main contribution is a rather involved analysis of the worst-case query time of this box-tree in the setting described above, showing it is polylogarithmic.…”