Voronoi Diagrams 1.Given a set of points S in the plane, which are the Voronoi sites. 2.Each site s has a Voronoi cell, also called a Dirichlet cell, V(s)

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Presentation transcript:

Voronoi Diagrams 1.Given a set of points S in the plane, which are the Voronoi sites. 2.Each site s has a Voronoi cell, also called a Dirichlet cell, V(s) consisting of all points closer to s than to any other site. 3.The segments of the Voronoi diagram are all the points in the plane that are equidistant to the two nearest sites. 4.The Voronoi nodes are the points equidistant to three (or more) sites.

Where should the shops be built? Ten shops in a flat city and their Voronoi cells Based on this model, what is the assumption regarding how customers are choosing which shop to go to? 1.Distance is only consideration 2.Euclidean Distance is used. 3.Customers travel by most efficient path Point A to Point B

Where should the shops be built? Ten shops in a flat city and their Voronoi cells Based on this model, what is the assumption regarding how customers are choosing which shop to go to and the path they take? 1.Distance is only consideration 2.Manhattan Distance is used. 3.Customers only go to the shops by a vehicle and the traffic paths are parallel to the and axes Note: Voronoi cells depend significantly on the metric used.

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You are the owner…. 1.…of five pizzerias in the town of Squaresville. 2.To ensure minimal delivery times, you devise a system in which customers call a central phone number and are then transferred to the pizzeria that is closest to them. 3.The map of Squaresville shows the position of the five pizzerias. 4.You need to divide the town into five regions so that customers order their pizza from the closest pizzeria.

Workers Needed to Service a Given Area