arXiv 0807.12212008-07-08cs.CG

$\reals^3$中凸多面体的线横截

Line Transversals of Convex Polyhedra in $\reals^3$

Haim Kaplan, Natan Rubin, Micha Sharir

备注：: 10 pages+ 15 page appendix

摘要：

我们建立了一个界的$O(n^2k^{1+\eps})$，对于任何$\eps>0$，对集合$\T$的线横截的集合$¶$的$k$凸多面体在$\reals^3$与$n$方面的总和$\reals^3$的组合复杂性，并提出了一个随机算法计算的边界$\T$在可比的预期时间。因此，当使用$k\ll n$时，$\T$的复杂性（和构造成本）的新界限改进了先前最佳已知界限，其在$n$中几乎是立方的。为了得到上述结果，我们研究了由一条固定的线$\ell_0$生成的线横截集$\TL$，建立了$O(nk^{1+\eps})$关于$\TL$复杂度的几乎紧界，并给出了一个在相当的期望时间内计算$\TL$的随机算法.略微改进的组合边界的复杂性$\TL$，和可比的改进，在成本上构建这一套，建立两个特殊的情况下，都假设多面体的$¶$是成对不相交的情况下，其中$\ell_0$是不相交的多面体的$¶$，和的情况下，多面体的$¶$是无界的方向平行于$\ell_0$。

英文摘要：

We establish a bound of $O(n^2k^{1+\eps})$, for any $\eps>0$, on the combinatorial complexity of the set $\T$ of line transversals of a collection $¶$ of $k$ convex polyhedra in $\reals^3$ with a total of $n$ facets, and present a randomized algorithm which computes the boundary of $\T$ in comparable expected time. Thus, when $k\ll n$, the new bounds on the complexity (and construction cost) of $\T$ improve upon the previously best known bounds, which are nearly cubic in $n$. To obtain the above result, we study the set $\TL$ of line transversals which emanate from a fixed line $\ell_0$, establish an almost tight bound of $O(nk^{1+\eps})$ on the complexity of $\TL$, and provide a randomized algorithm which computes $\TL$ in comparable expected time. Slightly improved combinatorial bounds for the complexity of $\TL$, and comparable improvements in the cost of constructing this set, are established for two special cases, both assuming that the polyhedra of $¶$ are pairwise disjoint: the case where $\ell_0$ is disjoint from the polyhedra of $¶$, and the case where the polyhedra of $¶$ are unbounded in a direction parallel to $\ell_0$.

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摘要：

英文摘要：

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