Ray shooting and parametric search
STOC '92 Proceedings of the twenty-fourth annual ACM symposium on Theory of computing
SCG '92 Proceedings of the eighth annual symposium on Computational geometry
Ray shooting in convex polytopes
SCG '92 Proceedings of the eighth annual symposium on Computational geometry
Minkowski-type theorems and least-squares partitioning
SCG '92 Proceedings of the eighth annual symposium on Computational geometry
On geometric optimization with few violated constraints
SCG '94 Proceedings of the tenth annual symposium on Computational geometry
ACM Computing Surveys (CSUR)
Iterated snap rounding with bounded drift
Proceedings of the twenty-second annual symposium on Computational geometry
Iterated snap rounding with bounded drift
Computational Geometry: Theory and Applications
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\indent We consider the half-space range reporting problem: Given an $n$ point set $S$ in ${\bf R} ^d$, preprocess it into a data structure, so that, given a query half-space $\gamma$, the points of $S \cap \gamma$ can be reported efficiently. We extend previously known static solutions to dynamic ones, supporting insertions and deletions of points of $S$. For given $m, n \leq m \leq n ^ {\lfloor d/2 \rfloor}$ and an arbitrarily small positive constant $\varepsilon$, we achieve $O(m^{1+ \varepsilon})$ space and preprocessing time, $O \frac {n}{m^{1/ \lfloor d/2 \rfloor}} \, \mbox {log} \, n)$ query time and $O(m^{1 + \varepsilon} / n)$ amortized update time $(d \geq 3)$. We present, among others, the following applications: an $O(n ^ {1 + \varepsilon})$ time algorithm for computing convex layers in ${\bf R}^3$, and an output sensitive algorithm for computing a level in an arrangements of planes in ${\bf R} ^3$, whose time complexity is $O((b+n) \, n^ \varepsilon)$, where $b$ is the size of the level.