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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.

Keywords

Query Range Query Time Recursive Call Splitting Line Input Object 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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