Farthest neighbors and center points in the presence of rectngular obstacles

• Authors: Boaz Ben-Moshe;Matthew J. Katz;Joseph S. B. Mitchell
• Venue: SCG '01 Proceedings of the seventeenth annual symposium on Computational geometry
• Year: 2001

We study several natural proximity and facility location problems that arise for a set ${\cal P}$ of $n$ points and a set $\R$ of $m$ disjoint rectangular obstacles in the plane, where distances are measured according to the $L_1$ shortest path (geodesic) metric. In particular, we compute, in time $O(mn\log(m+n))$, a data structure of size $O(mn)$ that supports $O(\log(m+n))$-time farthest point queries; we avoid computing the more complicated farthest neighbor Voronoi diagram, whose combinatorial complexity we show to be $\Theta(mn)$. We study the center point problem, finding in $O(mn\log(m+n))$ time a center point (and the set of center points) that minimize the maximum distance to sites of ${\cal P}$; this result improves the best previous bound by a factor of roughly $m$. In addition, we give algorithms for approximating the diameter, $D$, and radius, $r$, of ${\cal P}$, including methods to (i) compute a pair of points $a,b \in {\cal P}$, such that $d(a,b) \ge (1-\eps)D$, in $O(n\log n + \frac{1}{\eps}(n+m) \log m)$ time; and (ii) compute a point $c'$, such that $\max \{d(p, c') \ | \ p \in {\cal P}\} \le (1+\eps)r$, in $O(n\log(m+n) + (m/\eps)\log(m+1/\eps))$ time. Finally, we show that for all the problems above it is enough to consider only a subset of ${\cal P}$. This subset is likely to be much smaller than ${\cal P}$, it is computable in $O(n \log n)$ time, and using it results in significantly decreased runtime in practice.