On overlays and minimization diagrams
Proceedings of the twenty-second annual symposium on Computational geometry
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For a collection $\F$ of $d$-variate piecewise linear functions of overall combinatorial complexity $n$, the lower envelope $\E(\F)$ of $\F$ is the pointwise minimum of these functions. The minimization diagram $\M(\F)$ is the subdivision of $\reals^d$ obtained by vertically (i.e., in direction $x_{d+1}$) projecting $\E(\F)$. The overlay $\O(\F,\G)$ of two such subdivisions $\M(\F)$ and $\M(\G)$ is their superposition. We extend and improve the analysis of de Berg et al. \cite{bgh-vdt3s-96} by showing that the combinatorial complexity of $\O(\F,\G)$ is $\Omega(n^d \alpha^{2}(n))$ and $O(n^{d+\eps})$ for any $\eps0$ when $d \ge 2$, and $O(n^2 \alpha(n) \log n)$ when $d=2$. We also describe an algorithm that constructs $\O(\F,\G)$ in this time.We apply these results to obtain efficient general solutions to the problem of matching two polyhedral terrains in higher dimensions under translation. That is, given two piecewise linear terrains of combinatorial complexity $n$ in $\reals^{d+1}$, we wish to find a translation of the first terrain that minimizes its distance to the second, according to some distance measure. For the perpendicular distance measure, which we adopt from functional analysis since it is natural for measuring the similarity of terrains, we present a matching algorithm that runs in time $O(n^{2d+\eps})$ for any $\eps0$. Sharper running time bounds are shown for $d \le 2$. For the directed and undirected \Hd\ distance measures, we present a matching algorithm that runs in time $O(n^{d^2+d+\eps})$ for any $\eps0$.