Time-Space Trade-offs for Triangulations and Voronoi Diagrams

TL;DR

This paper explores algorithms for computing triangulations and Voronoi diagrams of planar point sets under limited workspace constraints, providing new time bounds for both deterministic and randomized approaches.

Contribution

It introduces novel time-efficient algorithms for triangulations and Voronoi diagrams that operate within restricted memory environments, extending classical methods.

Findings

Deterministic algorithm for triangulation in O(n^2/s + n log n log s) time.

Randomized algorithm for Voronoi diagram in expected O((n^2/s) log s + n log s log* s) time.

Effective trade-offs between workspace size and computational time.

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 $O((n^2/s) \log s + n \log s \log^*s)$.