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.


Query Range Query Time Recursive Call Splitting Line Input Object 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Mark de Berg
    • 1
  • Micha Streppel
    • 1
  1. 1.Department of Computer ScienceTU EindhovenEindhovenThe Netherlands

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