Solving the Post O ce Problem Voronoi Diagrams · Institute of Software Technology Birgit Vogtenhuber January 2016 Voronoi Diagrams The Post O ce Problem Goal: A data structure that

Institute of Software Technology

Birgit Vogtenhuber January 2016 Voronoi Diagrams

Voronoi Diagrams:Solving the Post Office Problem

Discrete and Computational GeometryWS 2015/16



The Post Office Problem

Goal: A data structure that solves this problem efficiently.

What is the closest post office for a given position?

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First: Partition into regions with the same answer.


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Simplest Case: Just two post offices p1 and p2

p1

p2

b(p1, p2)

• Regions bounded by the bisector b(p1, p2) of p1 and p2.

• Region of p1: closed halfspace H(p1, p2) containing p1.

H(p1, p2)

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General: Set P ={p1, . . . , pn} of n post office points in lR2

• Voronoi Cell VP (pi): all points of lR2 that are at leastas close to pi as to any other point of P .

pi

p1

p2

p3

p4

p5

VP (pi)

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• Voronoi Cell VP (pi): all points of lR2 that are at leastas close to pi as to any other point of P .

Observation:◦ VP (pi) is the intersection of all halfspaces H(pj , pi)

with i ∈ {1, . . . , n}\{j}◦ VP (pi) is convex and possibly unbounded

• Voronoi Diagram VD(P ): subdivision of lR2 induced bythe voronoi cells VP (pi), for i = 1, . . . , n.

• pi, pj neighbors in VD(P ): VP (pi) and VP (pj) sharean edge in VD(P )

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Note: inside a cell of VD(P ): one unique closest point.2





Question: on an edge of VD(P )? ⇒ two closest points2





Question: on a vertex of VD(P )? ⇒ ≥ 3 closest points2



These Circles look familiar...

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Voronoi Diagram / Delaunay Graph

• The Voronoi diagram VD(P ) of a point set P is unique.• VD(P ) is the dual of the Delaunay graph DG(P ).• If P has no 4 co-circular points then VD(P ) is the dual

of the unique Delaunay triangulation DT(P ).• The unbounded cells of VD(P ) contain the extreme

vertices of P .• The duals of the unbounded edges of VD(P ) form the

convex hull of the point set.• An edge of DT(P ) might not intersect its dual edge in

VD(P ) and a triangle in DT(P ) might not contain itsdual vertex in VD(P ), independent of whether or notDT(P ) is unique.

• VD(P ) has O(n) vertices, edges, and regions.

• An edge of DT(P ) might not intersect its dual edge inVD(P ) and a triangle in DT(P ) might not contain itsdual vertex in VD(P ), independent of whether or notDT(P ) is unique.

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Constructing Voronoi Diagrams

One approach: Construct from Delaunay Triangulation

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Vertices: Circumcenters of Delaunay triangles

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Vertices: Circumcenters of Delaunay triangles

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Edges: (parts of) bisectors of Delaunay edges.

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Edges: (parts of) bisectors of Delaunay edges.

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One approach: Construct from Delaunay Triangulation⇒ Time: O(n log n) expected

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`

Different idea: Plane sweep: Maintain VD left of `.

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`

Problem: How to discover new Voronoi regions in time?

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Solution: See sweep-line as additional line-shaped site.

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⇒ Left of `: Voronoi diagram of swept points and `.

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`

⇒ Left of `: Voronoi edges and a Wavefront of parabolic arcs.

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`

Point scanned: spike event

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`

Parabula vanishes: site event

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`5





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⇒ Time: O(n log n) deterministicDifferent idea: Plane sweep: Maintain VD left of `.

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⇒ Time: O(n log n) deterministicDifferent idea: Plane sweep: Maintain VD left of `.

⇒ DT in O(n log n) deterministic!

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Goal: A data structure that solves this problem efficiently.


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Theorem: Given a triangulation T for a set P of n pointsin lR2, one can build in O(n) time an data structure of sizeO(n) that allows for any query point q ∈ conv(P ) to tofind in O(log n) time a triangle from T containing q.

For example: Kirkpatrick’s Hierarchy

Corollary: The postoffice problem can be solved withO(n log n) preprocessing time and O(log n) query timewith a data structure that uses O(n) space.

• add Bounding box to VD(P )• triangulate regions• apply the theorem

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More About Voronoi Diagrams

• . . . exist also in higher dimensions• . . . are used also for other objects than points• . . . can be defined also for other metrics than the. . . euclidean distance

Voronoi Diargrams . . .

Thank you for your attention!

And there’s much more to know about them . . .

. . . but: not anymore in this lecture. See for example thebook Voronoi Diagrams and Delaunay Triangulations byAurenhammer, Klein, Lee.

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Documents

Solving the Post O ce Problem Voronoi Diagrams · Institute of Software Technology Birgit Vogtenhuber January 2016 Voronoi Diagrams The Post O ce Problem Goal: A data structure that