TY - JOUR

T1 - Time–space trade-offs for triangulations and Voronoi diagrams

AU - Korman, Matias

AU - Mulzer, Wolfgang

AU - van Renssen, André

AU - Roeloffzen, Marcel

AU - Seiferth, Paul

AU - Stein, Yannik

N1 - Funding Information:
MK was supported in part by the ELC project (MEXT KAKENHI Nos. 12H00855 and 15H02665). 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 ?Methods for Discrete Structures? (GRK 1408).
Publisher Copyright:
© 2017 Elsevier B.V.

PY - 2018/8

Y1 - 2018/8

N2 - 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(nlogn) time and O(n) space. We study the situation when the available workspace is limited: given a parameter 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(logn) 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(n2/s+nlognlogs) and a randomized s-workspace algorithm for finding the Voronoi diagram of S in expected time O((n2/s)logs+nlogslog⁎s).

AB - 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(nlogn) time and O(n) space. We study the situation when the available workspace is limited: given a parameter 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(logn) 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(n2/s+nlognlogs) and a randomized s-workspace algorithm for finding the Voronoi diagram of S in expected time O((n2/s)logs+nlogslog⁎s).

KW - Randomized algorithm

KW - Time–space trade-off

KW - Triangulation

KW - Voronoi diagram

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U2 - 10.1016/j.comgeo.2017.01.001

DO - 10.1016/j.comgeo.2017.01.001

M3 - Article

AN - SCOPUS:85045399673

VL - 73

SP - 35

EP - 45

JO - Computational Geometry: Theory and Applications

JF - Computational Geometry: Theory and Applications

SN - 0925-7721

ER -