TY - GEN

T1 - Maximin location of convex objects in a polygon and related dynamic voronoi diagrams

AU - Aonuma, Hiromi

AU - Imai, Hiroshi

AU - Imai, Keiko

AU - Tokuyama, Takeshi

N1 - Funding Information:
The authors would like to thank a referee for useful comments. A preliminary version of this paper was presented at the 6th ACM Symposium on Computational Geometry [3]. The research by the first author has been conducted under the research project “Integrated Geographic Information Systems” at the Institute of Science and Engineering of Chuo University (one of the Research Centers for High Technology of Private Universities, financially supported in part by the Ministry of Education, Science, Sports and Culture of Japan). The work by the second author was partially supported by the Grant-in-Aid for Scientific Research of the Ministry of Education, Science, Sports and Culture of Japan.

PY - 1990

Y1 - 1990

N2 - This paper considers the maximin placement of a convex polygon P inside a polygon Q, and introduce several new static and dynamic Voronoi diagrams to solve the problem. It is shown that P can be placed inside Q, using translation and rotation, so that the minimum Euclidean distance between any point on P and any point on Q is maximized in O(m4nλ16(mn)log mn) time, where m and n are the numbers of edges of P and Q, respectively, and λ16(N) is the maximum length of Davenport-Schinzel sequences on N alphabets of order 16. If only translation is allowed, the problem can be solved in O(mn log mn) time. The problem of placing multiple translates of P inside Q in a maximin manner is also considered, and in connection with this problem the dynamic Voronoi diagram of k rigidly moving sets of n points is investigated. The combinatorial complexity of this canonical dynamic diagram for kn points is shown to be O(n2) and O(n3k4 log* k) for k = 2,3 and k ≥ 4, respectively. Several related problems are also treated in a unified way.

AB - This paper considers the maximin placement of a convex polygon P inside a polygon Q, and introduce several new static and dynamic Voronoi diagrams to solve the problem. It is shown that P can be placed inside Q, using translation and rotation, so that the minimum Euclidean distance between any point on P and any point on Q is maximized in O(m4nλ16(mn)log mn) time, where m and n are the numbers of edges of P and Q, respectively, and λ16(N) is the maximum length of Davenport-Schinzel sequences on N alphabets of order 16. If only translation is allowed, the problem can be solved in O(mn log mn) time. The problem of placing multiple translates of P inside Q in a maximin manner is also considered, and in connection with this problem the dynamic Voronoi diagram of k rigidly moving sets of n points is investigated. The combinatorial complexity of this canonical dynamic diagram for kn points is shown to be O(n2) and O(n3k4 log* k) for k = 2,3 and k ≥ 4, respectively. Several related problems are also treated in a unified way.

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U2 - 10.1145/98524.98575

DO - 10.1145/98524.98575

M3 - Conference contribution

AN - SCOPUS:0025022835

SN - 0897913620

SN - 9780897913621

T3 - Proc Sixth Annu Symp Comput Geom

SP - 225

EP - 234

BT - Proc Sixth Annu Symp Comput Geom

PB - Publ by ACM

T2 - Proceedings of the Sixth Annual Symposium on Computational Geometry

Y2 - 6 June 1990 through 8 June 1990

ER -