Theory of Combinatorial Algorithms

Prof. Emo Welzl and Prof. Bernd Gärtner

Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

 Mittagsseminar Talk Information

Date and Time: Thursday, September 05, 2019, 12:15 pm

Duration: 30 minutes

Location: CAB G 11

Speaker: Evanthia Papadopoulou (Università della Svizzera italiana)

Abstract Voronoi-like diagrams for trees and forests

The Voronoi diagram is a well-known, versatile geometric partitioning structure with numerous applications in science and engineering. Abstract Voronoi diagrams offer a unifying framework to various concrete Voronoi instances in two dimensions. Instead of sites and distance measures, abstract Voronoi diagrams are defined in terms of bisecting curves that satisfy some simple combinatorial properties. They can be constructed in optimal O(n\log n) time for n given sites. In addition, for certain tree-like Voronoi diagrams, linear-time construction algorithms have been well-known to exist. Examples include the Voronoi diagram of points in convex position, updating a Voronoi diagram of point-sites after deletion of one site, constructing the farthest-point Voronoi diagram after the convex hull, and others. Surprisingly, no linear-time constructions have been available for any non-point sites (except the medial axis of a simple polygon) nor for abstract Voronoi diagrams.

In this talk I will present a relaxed Voronoi structure for trees (and forests) within a simply connected domain D, termed a Voronoi-like diagram. The involved sites are represented by a boundary curve that is derived from the boundary of D. The Voronoi-like region of one site (along the boundary curve) is defined in terms of only a simple monotone path in the arrangement of bisectors involving that site. In contrast, the boundary of a real Voronoi region is always an envelope in the same arrangement. This structure turns out to have nice properties and is always well defined. Using Voronoi like-diagrams we can derive simple linear-time randomized algorithms to compute tree- and forrest-like abstract Voronoi diagrams in the aforementioned cases. This implies linear-time constructions for all the corresponding concrete diagrams that fall under their umbrella.

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