Time-space Trade-offs for Voronoi Diagrams

Let S be a planar n-point set. Classically, one can find the Voronoi diagram VD(S) for S in O(n log n) time and O(n) space. We study the situation when the available workspace is limited: for s ∈ {1, . . . , 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 Θ(log n) bits for reading and writing intermediate data. We describe a randomized s-workspace algorithm for finding VD(S) in expected time O((n/s) log s + n log s log∗ s). This almost matches the optimal running times for both constant and linear workspace and provides a continuous trade-off between them.