Computational Geometry – Voronoi Diagram COMPUTATIONAL GEOMETRY -- VORONOI DIAGRAM Sophie Che.

Computational Geometry – Voronoi Diagram

COMPUTATIONAL GEOMETRY

-- VORONOI DIAGRAM

Sophie Che


THE LARGEST EMPTY CIRCLE PROBLEM

An application of Voronoi Diagram2


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LARGEST EMPTY CIRCLE

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e : Voronoi edge

vv : Voronoi vertex

S : site points

e

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DEFINITIONS &BASIC PROPERTIES

Of Voronoi Diagram5


DEFINITION

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pi

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FORMAL DEFINITION

Let P be a set of n distinct points (sites) in the plane.

The Voronoi diagram of P is the subdivision of the plane into n cells, one for each site.

A point q lies in the cell corresponding to a site pi P

iff Euclidean_Distance(q, pi) < Euclidean_distance(q, pj)

for each pi P, j i.

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STRUCTURE

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2 sites Perpendicular

Bisector

3 sites Non-collinear sites Half lines meeting at

a Voronoi vertex

4 sites A bounded cell


STRUCTURE (CONT’D)

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More sites…

V(pi) = 1 j n, j i h(pi,pj)

pi

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COMPLEXITY (SIZE)

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Intuitively: n2

n sites At most n-1 vertices

and edges per cell

Actually: O(n) For n 3,

|v| 2n 5,|e| 3n 6

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LINEAR COMPLEXITY – PROOF

Euler’s Formula: For planar graphs, |V| – |E| + F = 2 Connecting all half-infinite edges to an extra

vertex

(|v| + 1) – |e| + n = 2 Plus,

|v| 2n 5|e| 3n 6

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)1|(|33)deg(||2)()(

vvePVorvPVorv

epi

p

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DELAUNAY TRIANGULATIONDual graph of Voronoi Diagram12


DELAUNAY TRIANGULATION

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Dual graph Nodes: site points Edges:

“neighboring” sites in VD

Properties Planar graph Linear complexity

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DELAUNAY PROPERTIES

1. DT is the straight-line dual of VD.2. DT is a triangulation if no four sites are

cocircular.3. Each face of DT corresponds to a vertex of

VD.4. Each edge of DT corresponds to an edge of

VD.5. Each node of DT corresponds to a cell of VD.6. The boundary of DT is the convex hull of the

sites.7. The interior of each face of DT contains no

sites.14

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VORONOI PROPERTIES

1. Each Voronoi cell is convex.

2. The cell of pi is unbounded iff pi is on the convex hull of the sites. [c.f. D6]

3. If v is a Voronoi vertex at the junction of p1, p2 and p3, then v is the center of the circle C(v) determined by p1, p2 and p3.

4. C(v) is the circumcircle for the Delaunay triangle corresponding to v.

5. The interior of C(v) contains no sites. [c.f. D7]

6. If pi is a nearest neighbor to pj, then pipj is an edge of DT.

7. If there is some circle through pi and pj that contains no other sites, then pipj is an edge of DT.

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VORONOI VERTEX

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A Voronoi vertex is an intersection of 3 (or more) segments, each equidistant from a pair of sites

A point q is a Voronoi vertex iff its largest empty circle centered at q touches at least 3 (or more) sites

pk

qpi

pj

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VORONOI EDGE

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A Voronoi edge is a subset of locus of points equidistant from pi and pj

A point q lies on a Voronoi edge between sites pi and pj iff its largest empty circle centered at q touches only pi and pj

q

pi

pj

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ALGORITHMSTo compute a Voronoi Diagram18


COMPUTING VORONOI DIAGRAM

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HALF-PLANE INTERSECTION

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For each site pi,

Compute the common intersection of the half-planes h(pi, pj), with j≠i

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For each site pi,


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For each site pi,


O(nlogn) time per Voronoi cell

O( n2 log n ) in total

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INCREMENTAL CONSTRUCTION

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Incremental insertion Obtain VDn from VDn-

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Steps: Locate pi in some

Voronoi cell of pj; Draw bisector

between pi and pj; Expand boundary of

Voronoi cell of pi;

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pi

pj



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1




Voronoi cell of pi;

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pi

pj



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1




Voronoi cell of pi;

pi

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pj



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1

O(n) time per point insertion

O(n2) in total

Demo

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GENERAL SWEEP LINE FRAMEWORK

Sweep a horizontal line (sweep line) from top to bottom over the plane

Incremental constructionMaintain unchanged

informationHandle changes at

certain special points(event points)

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l

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FORTUNE’S ALGORITHM

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Problems:What to maintain?

Intersections?

Beach lineunchanged

information – Distance

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BEACH LINE

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Sequence of Parabolic arcsSet of points

equidistantfrom nearest site

above sweep line& sweep line itself

breakpoints lie on Voronoi edges

How to store it?? Sites consist of the

line

Equidistancebreakpoint

arc

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FORTUNE’S ALGORITHM

Demo

A visual implementation of Fortune's Voronoi algorithm

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EVENTS NEEDED TO BE HANDLED – SITE EVENT

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When the sweep line reaches a new site…

l

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l

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A new arc appears

l

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HANDLING SITE EVENTS

Find where it projects on beach line

Insert its arc in proper position in the list of beach line

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EVENTS NEEDED TO BE HANDLED – CIRCLE EVENT

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When the sweep line encounters the bottom of an empty circle touching 3 or more sites…

l

pi pj pk

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EVENTS NEEDED TO BE HANDLED – CIRCLE EVENT

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When the sweep line encounters the bottom of an empty circle touching 3 or more sites…

An arc disappears

l

y

pi pj pk

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HANDLING CIRCLE EVENTS

Remove pj from beach line

Create a Voronoi vertex yCenter of empty circle is a Voronoi vertex

Tie two Voronoi edges into y

Create a Voronoi edge between pi and pk 37

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RUNNING TIME ANALYSIS

How many arcs in beach line?Each site split at most 1 existing arc into

2 At most 2n-1 arcs

How many events?Site events

O(n) – number of sitesCircle events

O(n) – number of arcs 38

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RUNNING TIME ANALYSIS – DATA STRUCTURE

Beach line – sites that consist of the line A binary search tree

Event queue A priority queue

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RUNNING TIME ANALYSIS – SITE EVENT

Find where it projects on beach lineO(log n)

Insert its “arc” in proper position in the list of beach lineO(1)

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RUNNING TIME ANALYSIS – CIRCLE EVENT

Remove pj from beach lineO(log n)

Create a Voronoi vertex yO(1)

Tie two Voronoi edges into yO(1)

Create a Voronoi edge between pi and pk

O(1)41

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TOTAL RUNNING TIME

Each site/circle event handler O(log n)

At most O(n) site and circle events

O(n log n) total running time

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IS FORTUNE’S ALGORITHM OPTIMAL?

Sorting n real numbers is reducible to computing Voronoi diagrams

Lower bound O(n log n)

Fortune’s algorithm is optimal!

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SUMMARY

VD is a useful planar subdivision of a discrete point set

VD has linear complexity/sizeO(n) edges and O(n) vertices

Can be constructed in O(n log n) timeFortune’s algorithm (optimal)

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THANK YOU

Questions ??

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SPECIAL CASES

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Collinear sites Co-circle sites

v


LINEAR COMPLEXITY – PROOF

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Case 1: collinear|v| = 0,|e| = n – 1

Computational Geometry – Voronoi Diagram COMPUTATIONAL GEOMETRY -- VORONOI DIAGRAM Sophie Che.

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