Improved Time-Space Trade-offs for Computing Voronoi Diagrams

Abstract: Let $P$ be a planar set of $n$ sites in general position. For $k\in{1,\dots,n-1}$, the Voronoi diagram of order $k$ for $P$ is obtained by subdividing the plane into cells such that points in the same cell have the same set of nearest $k$ neighbors in $P$. The (nearest site) Voronoi diagram (NVD) and the farthest site Voronoi diagram (FVD) are the particular cases of $k=1$ and $k=n-1$, respectively. For any given $K\in{1,\dots,n-1}$, the family of all higher-order Voronoi diagrams of order $k=1,\dots,K$ for $P$ can be computed in total time $O(nK^2+ n\log n)$ using $O(K^2(n-K))$ space [Aggarwal et al., DCG'89; Lee, TC'82]. Moreover, NVD and FVD for $P$ can be computed in $O(n\log n)$ time using $O(n)$ space [Preparata, Shamos, Springer'85]. For $s\in{1,\dots,n}$, an $s$-workspace algorithm has random access to a read-only array with the sites of $P$ in arbitrary order. Additionally, the algorithm may use $O(s)$ words, of $\Theta(\log n)$ bits each, for reading and writing intermediate data. The output can be written only once and cannot be accessed or modified afterwards. We describe a deterministic $s$-workspace algorithm for computing NVD and FVD for $P$ that runs in $O((n^2/s)\log s)$ time. Moreover, we generalize our $s$-workspace algorithm so that for any given $K\in O(\sqrt{s})$, we compute the family of all higher-order Voronoi diagrams of order $k=1,\dots,K$ for $P$ in total expected time $O (\frac{n^2 K^5}{s}(\log s+K2^{O(\log^* K)}))$ or in total deterministic time $O(\frac{n^2 K^5}{s}(\log s+K\log K))$. Previously, for Voronoi diagrams, the only known $s$-workspace algorithm runs in expected time $O\bigl((n^2/s)\log s+n\log s\log^* s)$ [Korman et al., WADS'15] and only works for NVD (i.e., $k=1$). Unlike the previous algorithm, our new method is very simple and does not rely on advanced data structures or random sampling techniques.