Approximate Range Searching Using Binary Space Partitions

We show how any BSP tree TP 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(TP )) 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(minε>0{(1/ε) + kε} log n) time, where kε 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 R such that ε-approximate range searching with any constant-complexity convex query range can be done in O(log n + minε>0{(1/εd−1) + kε}) time. Finally we show how to adapt our structures so that they become I/O-efficient.