Time-Space Trade-offs for Triangulations and Voronoi Diagrams

Abstract: Let $S$ be a planar $n$-point set. A triangulation for $S$ is a maximal plane straight-line graph with vertex set $S$. The Voronoi diagram for $S$ is the subdivision of the plane into cells such that all points in a cell have the same nearest neighbor in $S$. Classically, both structures can be computed in $O(n \log n)$ time and $O(n)$ space. We study the situation when the available workspace is limited: given a parameter $s \in {1, \dots, n}$, an $s$-workspace algorithm has read-only access to an input array with the points from $S$ in arbitrary order, and it may use only $O(s)$ additional words of $\Theta(\log n)$ bits for reading and writing intermediate data. The output should then be written to a write-only structure. We describe a deterministic $s$-workspace algorithm for computing an arbitrary triangulation of $S$ in time $O(n^2/s + n \log n \log s )$ and a randomized $s$-workspace algorithm for finding the Voronoi diagram of $S$ in expected time $O((n^2/s) \log s + n \log s \log^*s)$.