Approximate Range Searching Using Binary Space Partitions

Abstract

We show how any BSP tree ${\mathcal T}_P$ for the endpoints of a set of n disjoint segments in the plane can be used to obtain a BSP tree of size $O(n.depth({\mathcal T}_P))$ for the segments themselves, such that the range-searching efficiency remains almost the same. We apply this technique to obtain a BSP tree of size O(n log n) such that ε-approximate range searching queries with any constant-complexity convex query range can be answered in O(min _ε> 0{1/ε + k _ε }log n) time, where k _ε is the number of segments intersecting the ε-extended range. The same result can be obtained for disjoint constant-complexity curves, if we allow the BSP to use splitting curves along the given curves.

We also describe how to construct a linear-size BSP tree for low-density scenes consisting of n objects in ${\mathbb R}^{d}$ such that ε-approximate range searching with any constant-complexity convex query range can be done in $O(log n + {\rm min}_{\epsilon > 0}{\{1/\epsilon^{(d-1)}+k_{\epsilon}\}})$ time.