Zone diagrams: Existence, uniqueness and algorithmic challenge

Tetsuo Asano, Jiří Matoušek, Takeshi Tokuyama

Research output: Chapter in Book/Report/Conference proceedingConference contribution

3 Citations (Scopus)


A zone diagram is a new variation of the classical notion of Voronoi diagram. Given points (sites) p1, ⋯;, pn in the plane, each pi, is assigned a region Ri, but in contrast to the ordinary Voronoi diagrams, the union of the Ri has a nonempty complement, the neutral zone. The defining property is that each Ri consists of all x ∈ ℝ2 that lie closer (non-strictly) to pi than to the union of all the other Rj, j ≠ i. Thus, the zone diagram is defined implicitly, by a "fixed-point property," and neither its existence nor its uniqueness seem obvious. We establish existence using a general fixed-point result (a consequence of Schauder's theorem or Kakutani's theorem); this proof should generalize easily to related settings, say higher dimensions. Then we prove uniqueness of the zone diagram, as well as convergence of a natural iterative algorithm for computing it, by a geometric argument, which also relies on a result for the case of two sites in an earlier paper. Many challenging questions remain open.

Original languageEnglish
Title of host publicationProceedings of the 18th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2007
PublisherAssociation for Computing Machinery
Number of pages10
ISBN (Electronic)9780898716245
Publication statusPublished - 2007 Jan 1
Event18th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2007 - New Orleans, United States
Duration: 2007 Jan 72007 Jan 9


Other18th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2007
Country/TerritoryUnited States
CityNew Orleans

ASJC Scopus subject areas

  • Software
  • Mathematics(all)


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