## Theory of Combinatorial Algorithms

Prof. Emo Welzl and Prof. Bernd Gärtner

# Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

 Mittagsseminar Talk Information

Date and Time: Thursday, October 06, 2016, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Matias Korman (Tohoku University)

## Time-Space Trade-offs for Computing (High Order) Voronoi Diagrams

Let P be a planar n-point set in general position. For any k (up to n-1), the Voronoi diagram of order k is obtained by subdividing the plane into regions such that points in the same cell have the same set of nearest k neighbors in P. The nearest point Voronoi diagram (NVD) and farthest point Voronoi diagram (FVD) are the particular cases of k=1 and k=n-1. It is known that the Voronoi diagram of orders 1 to k for P can be computed in total time O(nk^2+ n log n) using O(n) space (or O(n log n) time for FVD).

For s up to 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 afterwards.

We describe a deterministic s-workspace algorithm for computing an NVD and also an FVD for P that runs in O((n^2/s) log s) time. Moreover, we generalize our algorithm for computing the family of all higher-order Voronoi diagrams of P up to order K=O(sqrt(s)) in total time O(n^2 K^6 polylog(K,s)).

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