On Planar Visibility Counting Problem
英文摘要：
For a set $S$ of $n$ disjoint line segments in $\mathbb{R}^{2}$, the visibility counting problem is to preprocess $S$ such that the number of visible segments in $S$ from any query point $p$ can be computed quickly. There have been approximation algorithms for this problem with trade off between space and query time. We propose a new randomized algorithm to compute the exact answer of the problem. For any $0<α<1$, the space, preprocessing time and query time are $O_ε(n^{4-4α})$, $O_ε(n^{4-2α})$ and $O_ε(n^{2α})$, respectively. Where $O_ε(f(n)) = O(f(n)n^ε)$ and $ε>0$ is an arbitrary constant number.
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