Maximin location of convex objects in a polygon and related dynamic Voronoi diagrams

Authors:
Hiromi Aonuma;Hiroshi Imai;Keiko Imai;Takeshi Tokuyama
Affiliations:
Department of Computer Science and Communication Engineering, Kyushu University, Fukuoka 812, Japan;Information Science Center, Kyushu Institute of Technology, Iizuka, Fukuoka 820, Japan;-;IBM Research, Tokyo Research Laboratory, 5-11 Sanban-cho, Tokyo 102, Japan
Venue:
SCG '90 Proceedings of the sixth annual symposium on Computational geometry
Year:
1990

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Abstract

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 &Ogr;(m4n &lgr;16(mn) log mn) time, where m and n are the numbers of edges of P and Q, respectively, and &lgr;16(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 &Ogr;(mn log mn) time. The problem of placing multiple translates of P inside Q in a maximum manner is also considered, and in connection with this problem the dynamic Voronoi diagram of &kgr; rigidly moving sets of n points is investigated. The combinatorial complexity of this canonical dynamic diagram for &kgr;n points is shown to be &Ogr;(n2) and &Ogr;(n3&kgr;4 log* &kgr;) for &kgr; = 2, 3 and &kgr; ≥ 4, respectively. Several related problems are also treated in a unified way.