Matthew T. Dickerson
Michael T. Goodrich
ABSTRACT
We provide an efficient algorithm for two-site Voronoi diagrams in geographic networks. A two-site Voronoi diagram labels each vertex in a geographic network with their two nearest neighbors, which is useful in many contexts.
- M. Abellanas, F. Hurtado, V. Sacristán, C. Icking, L. Ma, R. Klein, E. Langetepe, and B. Palop. Voronoi diagram for services neighboring a highway. Inf. Process. Lett., 86(5):283--288, 2003. Google Scholar
Digital Library
- O. Aichholzer, F. Aurenhammer, and B. Palop. Quickest paths, straight skeletons, and the city Voronoi diagram. In SCG '02: Proceedings of the eighteenth annual symposium on Computational geometry, pages 151--159, New York, NY, USA, 2002. ACM. Google ScholarDigital Library
- F. Aurenhammer. Voronoi diagrams: A survey of a fundamental geometric data structure. ACM Comput. Surv., 23(3):345--405, Sept. 1991. Google ScholarDigital Library
- F. Aurenhammer and R. Klein. Voronoi diagrams. In J.-R. Sack and J. Urrutia, editors, Handbook of Computational Geometry, pages 201--290. Elsevier Science Publishers B.V. North-Holland, Amsterdam, 2000.Google Scholar
- S. W. Bae and K.-Y. Chwa. Voronoi diagrams with a transportation network on the euclidean plane. In Proc. Int. Symp. on Algorithms and Computation (ISAAC), volume 3341 of LNCS, pages 101--112. Springer, 2004. Google ScholarDigital Library
- S. W. Bae and K.-Y. Chwa. Shortest paths and Voronoi diagrams with transportation networks under general distances. In Proc. Int. Symp. on Algorithms and Computation (ISAAC), volume 3827 of LNCS, pages 1007--1018. Springer, 2005. Google ScholarDigital Library
- G. Barequet, M. T. Dickerson, and R. L. S. Drysdale. 2-point site Voronoi diagrams. Discrete Appl. Math., 122(1--3):37--54, 2002. Google ScholarDigital Library
- G. Barequet, R. L. Scot, M. T. Dickerson, and D. S. Guertin. 2-point site Voronoi diagrams. In SCG '01: Proceedings of the seventeenth annual symposium on Computational geometry, pages 323--324, New York, NY, USA, 2001. ACM. Google ScholarDigital Library
- B. Chazelle and H. Edelsbrunner. An improved algorithm for constructing kth-order Voronoi diagrams. IEEE Trans. Comput., C-36:1349--1354, 1987. Google ScholarDigital Library
- T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein. Introduction to Algorithms. MIT Press, Cambridge, MA, 2nd edition, 2001. Google ScholarDigital Library
- V. T. de Almeida and R. H. Güting. Using Dijkstra's algorithm to incrementally find the k-nearest neighbors in spatial network databases. In SAC '06: Proceedings of the 2006 ACM symposium on Applied computing, pages 58--62, New York, NY, USA, 2006. ACM. Google ScholarDigital Library
- E. W. Dijkstra. A note on two problems in connexion with graphs. Numerische Mathematik, 1:269--271, 1959.Google ScholarDigital Library
- G. L. Dirichlet. Über die Reduktion der positiven quadratischen Formen mit drei unbestimmten ganzen Zahlen. J. Reine Angew. Math., 40:209--227, 1850.Google ScholarCross Ref
- M. Erwig. The graph Voronoi diagram with applications. Networks, 36(3):156--163, 2000.Google ScholarCross Ref
- S. Fortune. Voronoi diagrams and Delaunay triangulations. In D.-Z. Du and F. K. Hwang, editors, Computing in Euclidean Geometry, volume 1 of Lecture Notes Series on Computing, pages 193--233. World Scientific, Singapore, 1st edition, 1992.Google Scholar
- M. T. Goodrich and R. Tamassia. Algorithm Design: Foundations, Analysis, and Internet Examples. John Wiley & Sons, New York, NY, 2002. Google ScholarDigital Library
- F. Hurtado, R. Klein, E. Langetepe, and V. Sacristán. The weighted farthest color Voronoi diagram on trees and graphs. Comput. Geom. Theory Appl., 27(1):13--26, 2004. Google ScholarDigital Library
- D. E. Knuth. Sorting and Searching, volume 3 of The Art of Computer Programming. Addison-Wesley, Reading, MA, 1973.Google Scholar
- M. Kolahdouzan and C. Shahabi. Voronoi-based k nearest neighbor search for spatial network databases. In VLDB '04: Proceedings of the Thirtieth international conference on Very large data bases, pages 840--851. VLDB Endowment, 2004. Google ScholarDigital Library
- D. T. Lee. On k-nearest neighbor Voronoi diagrams in the plane. IEEE Trans. Comput., C-31:478--487, 1982. Google ScholarDigital Library
- K. Mehlhorn. A faster approximation algorithm for the Steiner problem in graphs. Information Processing Letters, 27:125--128, 1988. Google ScholarDigital Library
- K. Patroumpas, T. Minogiannis, and T. Sellis. Approximate order-k Voronoi cells over positional streams. In GIS '07: Proceedings of the 15th annual ACM international symposium on Advances in geographic information systems, pages 1--8, New York, NY, USA, 2007. ACM. Google ScholarDigital Library
- M. Safar. K nearest neighbor search in navigation systems. Mob. Inf. Syst., 1(3):207--224, 2005. Google ScholarDigital Library
- K. Sugihara. Algorithms for computing Voronoi diagrams. In A. Okabe, B. Boots, and K. Sugihara, editors, Spatial Tesselations: Concepts and Applications of Voronoi Diagrams. John Wiley & Sons, Chichester, UK, 1992.Google Scholar
- G. M. Voronoi. Nouvelles applications des paramètres continus à la théorie des formes quadratiques. premier Mémoire: Sur quelques propriétés des formes quadratiques positives parfaites. J. Reine Angew. Math., 133:97--178, 1907.Google Scholar
Index Terms
- Two-site Voronoi diagrams in geographic networks
Recommendations
Abstract Voronoi diagrams revisited
Abstract Voronoi diagrams [R. Klein, Concrete and Abstract Voronoi Diagrams, Lecture Notes in Computer Science, vol. 400, Springer-Verlag, 1987] were designed as a unifying concept that should include as many concrete types of diagrams as possible. To ...
Voronoi diagrams with overlapping regions
Voronoi diagrams are a tessellation of the plane based on a given set of points (called Voronoi points) into polygons so that all the points inside a polygon are closest to the Voronoi point in that polygon. All the regions of existing Voronoi diagrams ...
Voronoi Diagrams on Periodic Graphs
ISVD '10: Proceedings of the 2010 International Symposium on Voronoi Diagrams in Science and EngineeringA periodic graph models various natural and artficial periodic patterns with repetitions of a given static graph, and have vast applications in crystallography, scheduling, VLSI circuits and systems of uniform recurrence equations. This paper considers ...
Comments