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 each cell has the same nearest neighbors 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 a 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 $O((n^2/s) \log s + n \log s \log ^*s)$.

W. Mulzer, P. Seiferth, and Y. Stein—WS and PS were supported in part by DFG Grants MU 3501/1 and MU 3501/2. YS was supported by the DFG within the research training group MDS (GRK 1408).