Zone diagrams: Existence, uniqueness, and algorithmic challenge

Tetsuo Asano, Jir̃í Matous̃ek, Takeshi Tokuyama

Research output: Contribution to journalArticlepeer-review

21 Citations (Scopus)


A zone diagram is a new variation of the classical notion of the 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 ε ℝ2 that lie closer (nonstrictly) 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
Pages (from-to)1182-1198
Number of pages17
JournalSIAM Journal on Computing
Issue number4
Publication statusPublished - 2007 Dec 1


  • Computational geometry
  • Distance trisector curve
  • Voronoi diagram
  • Zone diagram

ASJC Scopus subject areas

  • Computer Science(all)
  • Mathematics(all)


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