Output-Sensitive Tools for Range Searching in Higher Dimensions

Abstract: Let $P$ be a set of $n$ points in ${\mathbb R}<sup>{d}$. A point $p \in P$ is $k$\emph{-shallow} if it lies in a halfspace which contains at most $k$ points of $P$ (including $p$). We show that if all points of $P$ are $k$-shallow, then $P$ can be partitioned into $\Theta(n/k)$ subsets, so that any hyperplane crosses at most $$O((n/k)^{1-1/(d-1)} \log<sup>{2/(d-1)}(n/k))$$ subsets. Given such a partition, we can apply the standard construction of a spanning tree with small crossing number within each subset, to obtain a spanning tree for the point set $P$, with crossing number $$O(n^{{1-1/(d-1)}k^{1/d(d-1)}} \log<sup>{2/(d-1)}(n/k))$$. This allows us to extend the construction of Har-Peled and Sharir \cite{hs11} to three and higher dimensions, to obtain, for any set of $n$ points in ${\mathbb R}<sup>{d}$ (without the shallowness assumption), a spanning tree $T$ with {\em small relative crossing number}. That is, any hyperplane which contains $w \leq n/2$ points of $P$ on one side, crosses $$O(n^{{1-1/(d-1)}w^{1/d(d-1)}} \log<sup>{2/(d-1)}(n/w))$$ edges of $T$. Using a similar mechanism, we also obtain a data structure for halfspace range counting, which uses $O(n \log \log n)$ space (and somewhat higher preprocessing cost), and answers a query in time $$O(n^{{1-1/(d-1)}k^{1/d(d-1)}} (\log (n/k))<sup>{O(1)})$$, where $k$ is the output size.