| Presentation | 2006-09-26 Zone Diagram : Existence, Uniquness, and Algorithmic Challenges Tetsuo ASANO, JIRi MATOUSEK, Takeshi TOKUYAMA, |
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| Abstract(in Japanese) | (See Japanese page) |
| Abstract(in English) | A zone diagram is a new variation of the classical notion of Voronoi diagram. Given points (sites) P_1…, P_n in the plane, each P_i is assigned a region R_i, but in contrast to the ordinary Voronoi diagrams, the union of the R_i has a nonempty complement, the neutral zone. The defining property is that each R_i consists of all x ∈ R^2 that lie closer (non-strictly) to P_i than to the union of all the other R_j, 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. |
| Keyword(in Japanese) | (See Japanese page) |
| Keyword(in English) | Computational Geometry / Computation / Fixed Point Theorems / Voronoi Diagram / Trisector |
| Paper # | COMP2006-30 |
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| Conference Information | |
| Committee | COMP |
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| Conference Date | 2006/9/19(1days) |
| Place (in Japanese) | (See Japanese page) |
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| Topics (in Japanese) | (See Japanese page) |
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| Paper Information | |
| Registration To | Theoretical Foundations of Computing (COMP) |
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| Language | JPN |
| Title (in Japanese) | (See Japanese page) |
| Sub Title (in Japanese) | (See Japanese page) |
| Title (in English) | Zone Diagram : Existence, Uniquness, and Algorithmic Challenges |
| Sub Title (in English) | |
| Keyword(1) | Computational Geometry |
| Keyword(2) | Computation |
| Keyword(3) | Fixed Point Theorems |
| Keyword(4) | Voronoi Diagram |
| Keyword(5) | Trisector |
| 1st Author's Name | Tetsuo ASANO |
| 1st Author's Affiliation | School of Information Science, Japan Advanced Institute of Science and Technology() |
| 2nd Author's Name | JIRi MATOUSEK |
| 2nd Author's Affiliation | Department of Applied Mathematics and Institute of Theoretical Computer Science Chales University |
| 3rd Author's Name | Takeshi TOKUYAMA |
| 3rd Author's Affiliation | Graduate School of Information Sciences, Tohoku University |
| Date | 2006-09-26 |
| Paper # | COMP2006-30 |
| Volume (vol) | vol.106 |
| Number (no) | 258 |
| Page | pp.pp.- |
| #Pages | 8 |
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