Voronoi Diagram Sasanka Roy Indian Institute of Science Education - - PowerPoint PPT Presentation

Voronoi Diagram

Sasanka Roy Indian Institute of Science Education and Research

Organization of the Talk

Organization of the Talk

• 1. Preliminaries
• 2. Generic Definition
• 3. Some Technical Details
• 4. Conclusion

Organization of the Talk

• 1. Preliminaries
• 2. Generic Definition
• 3. Some Technical Details
• 4. Conclusion

What are we going to talk about?

What are we going to talk about?

We have some data

What are we going to talk about?

We have some data Geometric Data

What are we going to talk about?

We have some data Geometric Data Geometric Data ????

What are we going to talk about?

We have some data Geometric Data Geometric Data ???? What do I mean ????

What are we going to talk about?

We have some data Geometric Data Geometric Data ???? What do I mean ???? I mean: we have

What are we going to talk about?

We have some data Geometric Data Geometric Data ???? What do I mean ???? I mean: we have points,

What are we going to talk about?

We have some data Geometric Data Geometric Data ???? What do I mean ???? I mean: we have points, line segments,

What are we going to talk about?

We have some data Geometric Data Geometric Data ???? What do I mean ???? I mean: we have points, line segments, polygons etc.

What are we going to talk about?

We have some data Geometric Data Geometric Data ???? What do I mean ???? I mean: we have points, line segments, polygons etc.

Then what?????

What are we going to talk about?

We have some data Geometric Data Geometric Data ???? What do I mean ???? I mean: we have points, line segments, polygons etc.

Then what????? We want to get answers to the specific questions

What are we going to talk about?

We have some data Geometric Data Geometric Data ???? What do I mean ???? I mean: we have points, line segments, polygons etc.

Then what????? We want to get answers to the specific questions Closest points to the line segments

What are we going to talk about?

We have some data Geometric Data Geometric Data ???? What do I mean ???? I mean: we have points, line segments, polygons etc.

Then what????? We want to get answers to the specific questions Closest points to the line segments

What are we going to talk about?

We have some data Geometric Data Geometric Data ???? What do I mean ???? I mean: we have points, line segments, polygons etc.

Then what????? We want to get answers to the specific questions Closest points to the line segments Point inside the simple polygon

What are we going to talk about?

We have some data Geometric Data Geometric Data ???? What do I mean ???? I mean: we have points, line segments, polygons etc.

Then what????? We want to get answers to the specific questions Closest points to the line segments Point inside the simple polygon

What are we going to talk about?

We have some data Geometric Data Geometric Data ???? What do I mean ???? I mean: we have points, line segments, polygons etc.

Then what????? We want to get answers to the specific questions Closest points to the line segments Point inside the simple polygon

what are all these?

Can you be a bit Practical??

Planar Point Location

Planar Point Location

Which state has the site/point with Latitude= 26° 11' 0'' N Longitude= 91° 44' 0''E

Planar Point Location

Guwahati in Assam

Which state has the site/point with Latitude= 26° 11' 0'' N Longitude= 91° 44' 0''E

Planar Point Location

Can we view States as simple polygon?

Guwahati in Assam

Which state has the site/point with Latitude= 26° 11' 0'' N Longitude= 91° 44' 0''E

Planar Point Location

Can we view States as simple polygon? simple polygon: Closed region whose boundary is formed by non-intersecting line segments

Guwahati in Assam

Which state has the site/point with Latitude= 26° 11' 0'' N Longitude= 91° 44' 0''E

Planar Point Location

Can we view States as simple polygon? Yes simple polygon: Closed region whose boundary is formed by non-intersecting line segments

Guwahati in Assam

Which state has the site/point with Latitude= 26° 11' 0'' N Longitude= 91° 44' 0''E

28

Given a planar subdivision S of O(n) vertices/faces/edges

Formally Planar Point Location

29

Given a planar subdivision S of O(n) vertices/faces/edges

S

Formally Planar Point Location

30

Given a planar subdivision S of O(n) vertices/faces/edges

S

vertex

Formally Planar Point Location

31

Given a planar subdivision S of O(n) vertices/faces/edges

S

face vertex

Formally Planar Point Location

32

Given a planar subdivision S of O(n) vertices/faces/edges

S

face edge vertex

Formally Planar Point Location

33

Given a planar subdivision S Preprocess S such that:

S

Formally Planar Point Location

34

Given a planar subdivision S Preprocess S such that: For any query point q,

q S

Formally Planar Point Location

35

Given a planar subdivision S Preprocess S such that: For any query point q

R q

The region/face R containing q can be reported efficiently.

S

Formally Planar Point Location

36

Given a planar subdivision S Preprocess S such that: For any query point q

R q

The region/face R containing q can be reported efficiently.

S

Formally Planar Point Location

37

R q S

Formally Planar Point Location

38

Preprocessing Time:

R q S

Formally Planar Point Location

39

Preprocessing Time:

R q S

Preprocessing space requirement:

Questions?

40

Preprocessing Time:

R q S

Preprocessing space requirement: Query Time:

Questions?

41

Preprocessing Time:

R q S

Preprocessing space requirement: Query Time:

O(n)

Questions?

42

Preprocessing Time:

R q S

Preprocessing space requirement: Query Time:

O(n) O(n)

Questions?

43

Preprocessing Time:

R q S

Preprocessing space requirement: Query Time:

O(n) O(n) O(log n)

Questions?

Back to Voronoi Diagram

Subhas C. Nandy Advanced Computing and Microelectronics Unit Indian Statistical Institute Kolkata 700108

Organization of the Talk

• 1. Preliminaries
• 2. Generic Definition
• 3. Some Technical Details
• 4. Conclusion

Thank you Google

Viewpoint 1: Locate the nearest dentistry. Viewpoint 2: Find the ‘service area’ of potential customers for each dentist.

Thank you Google

Viewpoint 1: Locate the nearest dentistry. Viewpoint 2: Find the ‘service area’ of potential customers for each dentist.

Thank you Google

I am here.

Formal Definition

Formal Definition

P → A set of n distinct points (Geometric Objects) in the plane.

Formal Definition

P → A set of n distinct points (Geometric Objects) in the plane.

Formal Definition

P → A set of n distinct points (Geometric Objects) in the plane. Preprocess P such that closest point x ∈ P of any query point q can be found efficiently

Formal Definition

P → A set of n distinct points (Geometric Objects) in the plane. Preprocess P such that closest point x ∈ P of any query point q can be found efficiently q

Formal Definition

P → A set of n distinct points (Geometric Objects) in the plane. Preprocess P such that closest point x ∈ P of any query point q can be found efficiently q How to solve this efficiently?

Formal Definition

P → A set of n distinct points (Geometric Objects) in the plane. Preprocess P such that closest point x ∈ P of any query point q can be found efficiently Subdivision of the plane into n cells such that q How to solve this efficiently?

Formal Definition

P → A set of n distinct points (Geometric Objects) in the plane.

• each cell contains exactly one site,
• if a point q lies in a cell containing pi then

d(q, pi ) < d(q, pj ) for all pi ∈ P, j ≠ i. Preprocess P such that closest point x ∈ P of any query point q can be found efficiently Subdivision of the plane into n cells such that q How to solve this efficiently?

Formal Definition

P → A set of n distinct points (Geometric Objects) in the plane. Preprocess P such that closest point x ∈ P of any query point q can be found efficiently Subdivision of the plane into n cells such that q How to solve this efficiently?

• each cell contains exactly one site,
• if a point q lies in a cell containing pi then

d(q, pi ) < d(q, pj ) for all pi ∈ P, j ≠ i.

Formal Definition

P → A set of n distinct points (Geometric Objects) in the plane. Preprocess P such that closest point x ∈ P of any query point q can be found efficiently Subdivision of the plane into n cells such that Voronoi diagram of P: V(P): q How to solve this efficiently?

• each cell contains exactly one site,
• if a point q lies in a cell containing pi then

d(q, pi ) < d(q, pj ) for all pi ∈ P, j ≠ i.

Formal Definition

P → A set of n distinct points (Geometric Objects) in the plane. Preprocess P such that closest point x ∈ P of any query point q can be found efficiently Subdivision of the plane into n cells such that Voronoi diagram of P: V(P): q This is Planar Subdivision so what can we do? How to solve this efficiently?

• each cell contains exactly one site,
• if a point q lies in a cell containing pi then

d(q, pi ) < d(q, pj ) for all pi ∈ P, j ≠ i.

Formal Definition

P → A set of n distinct points (Geometric Objects) in the plane. Preprocess P such that closest point x ∈ P of any query point q can be found efficiently Subdivision of the plane into n cells such that Voronoi diagram of P: V(P): q This is Planar Subdivision so what can we do? Planar point location How to solve this efficiently?

• each cell contains exactly one site,
• if a point q lies in a cell containing pi then

d(q, pi ) < d(q, pj ) for all pi ∈ P, j ≠ i.

Computing the Voronoi Diagram

Input: A set of points on a line (special case)

p1 p2

Computing the Voronoi Diagram

Input: A set of points on a line (special case)

p1 p2

Output: A partitioning of the plane into regions of nearest neighbors

Computing the Voronoi Diagram

Input: A set of points on a line (special case) Output: A partitioning of the plane into regions of nearest neighbors

p1 p2

Computing the Voronoi Diagram

Input: A set of points on a line (special case) Output: A partitioning of the plane into regions of nearest neighbors

p1 p2

half plane containing

p2 = V(p2)

half plane containing

p1 = V(p1)

Computing the Voronoi Diagram

Input: A set of points on a line (special case) Output: A partitioning of the plane into regions of nearest neighbors

p1 p2

half plane containing

p2 = V(p2)

half plane containing

p1 = V(p1)

Closest point of x ∈ V(p2) =

x

Computing the Voronoi Diagram

Input: A set of points on a line (special case) Output: A partitioning of the plane into regions of nearest neighbors

p1 p2

half plane containing

p2 = V(p2)

half plane containing

p1 = V(p1)

Closest point of x ∈ V(p2) = p2

x

p3

Computing the Voronoi Diagram

Input: A set of points on a line (special case) Output: A partitioning of the plane into regions of nearest neighbors

p1 p2

p3

Computing the Voronoi Diagram

Input: A set of points on a line (special case) Output: A partitioning of the plane into regions of nearest neighbors

p1 p2 pn-1 pn

p3

Computing the Voronoi Diagram

Input: A set of points on a line (special case) Output: A partitioning of the plane into regions of nearest neighbors

p1 p2 pn-1 pn

What are these lines?

p3

Computing the Voronoi Diagram

Input: A set of points on a line (special case) Output: A partitioning of the plane into regions of nearest neighbors

p1 p2 pn-1 pn

What are these lines? Perpendicular bisector of line segment [ pi pi+1 ]

Computing the Voronoi Diagram

Computing the Voronoi Diagram

I n p u t : A s e t

• f

p

• i

n t s

P

= (p

1

, p

2

, . . . , p

n

)

• n

a l i n e ( s p e c i a l c a s e )

p3

Computing the Voronoi Diagram

I n p u t : A s e t

• f

p

• i

n t s

P

= (p

1

, p

2

, . . . , p

n

)

• n

a l i n e ( s p e c i a l c a s e ) Output: A partitioning of the plane into regions of nearest neighbors

p1 p2 pn-1 pn

p3

Computing the Voronoi Diagram

I n p u t : A s e t

• f

p

• i

n t s

P

= (p

1

, p

2

, . . . , p

n

)

• n

a l i n e ( s p e c i a l c a s e ) Output: A partitioning of the plane into regions of nearest neighbors

p1 p2 pn-1 pn

Find Perpendicular bisector li of line segment [ pi pi+1 ]

p3

Computing the Voronoi Diagram

I n p u t : A s e t

• f

p

• i

n t s

P

= (p

1

, p

2

, . . . , p

n

)

• n

a l i n e ( s p e c i a l c a s e ) Output: A partitioning of the plane into regions of nearest neighbors

p1 p2 pn-1 pn l1 l2 ln-1

Find Perpendicular bisector li of line segment [ pi pi+1 ]

p3

Computing the Voronoi Diagram

I n p u t : A s e t

• f

p

• i

n t s

P

= (p

1

, p

2

, . . . , p

n

)

• n

a l i n e ( s p e c i a l c a s e ) Output: A partitioning of the plane into regions of nearest neighbors

p1 p2 pn-1 pn l1 l2 ln-1

Let ai be the intersection point of li and [ pi pi+1 ] Find Perpendicular bisector li of line segment [ pi pi+1 ]

p3

Computing the Voronoi Diagram

I n p u t : A s e t

• f

p

• i

n t s

P

= (p

1

, p

2

, . . . , p

n

)

• n

a l i n e ( s p e c i a l c a s e ) Output: A partitioning of the plane into regions of nearest neighbors

p1 p2 pn-1 pn l1 l2 ln-1

Let ai be the intersection point of li and [ pi pi+1 ]

a1 a2 an-1

Find Perpendicular bisector li of line segment [ pi pi+1 ]

p3

Computing the Voronoi Diagram

I n p u t : A s e t

• f

p

• i

n t s

P

= (p

1

, p

2

, . . . , p

n

)

• n

a l i n e ( s p e c i a l c a s e ) Output: A partitioning of the plane into regions of nearest neighbors

p1 p2 pn-1 pn l1 l2 ln-1

Let ai be the intersection point of li and [ pi pi+1 ] Sort ai in increasing x-coordinate

a1 a2 an-1

Find Perpendicular bisector li of line segment [ pi pi+1 ]

p3

Computing the Voronoi Diagram

I n p u t : A s e t

• f

p

• i

n t s

P

= (p

1

, p

2

, . . . , p

n

)

• n

a l i n e ( s p e c i a l c a s e ) Output: A partitioning of the plane into regions of nearest neighbors

p1 p2 pn-1 pn l1 l2 ln-1

Let ai be the intersection point of li and [ pi pi+1 ] Sort ai in increasing x-coordinate

a1 a2 an-1

Find Perpendicular bisector li of line segment [ pi pi+1 ] This gives us Voronoi Diagram V(P)

p3

Computing the Voronoi Diagram

I n p u t : A s e t

• f

p

• i

n t s

P

= (p

1

, p

2

, . . . , p

n

)

• n

a l i n e ( s p e c i a l c a s e ) Output: A partitioning of the plane into regions of nearest neighbors

p1 p2 pn-1 pn l1 l2 ln-1

Let ai be the intersection point of li and [ pi pi+1 ] Sort ai in increasing x-coordinate

a1 a2 an-1

Find Perpendicular bisector li of line segment [ pi pi+1 ] This gives us Voronoi Diagram V(P)

V(p1) V(p2) V(p3) V(pn-1) V(pn)

p3

Query Answering

p1 p2 pn-1 pn l1 l2 ln-1 a1 a2 an-1 V(p1) V(p2) V(p3) V(pn-1) V(pn)

p3

Query Answering

p1 p2 pn-1 pn l1 l2 ln-1

We have ai‘s sorted in increasing x-coordinate

a1 a2 an-1 V(p1) V(p2) V(p3) V(pn-1) V(pn)

p3

Query Answering

p1 p2 pn-1 pn l1 l2 ln-1

We have ai‘s sorted in increasing x-coordinate

a1 a2 an-1 V(p1) V(p2) V(p3) V(pn-1) V(pn)

Given a query point p[x,y]

p3

Query Answering

p1 p2 pn-1 pn l1 l2 ln-1

We have ai‘s sorted in increasing x-coordinate

a1 a2 an-1 V(p1) V(p2) V(p3) V(pn-1) V(pn)

Given a query point p[x,y] p[x,y]

p3

Query Answering

p1 p2 pn-1 pn l1 l2 ln-1

We have ai‘s sorted in increasing x-coordinate

a1 a2 an-1 V(p1) V(p2) V(p3) V(pn-1) V(pn)

Given a query point p[x,y] What we have to do? p[x,y]

p3

Query Answering

p1 p2 pn-1 pn l1 l2 ln-1

We have ai‘s sorted in increasing x-coordinate

a1 a2 an-1 V(p1) V(p2) V(p3) V(pn-1) V(pn)

Given a query point p[x,y] What we have to do? p[x,y] Locate x correctly between ai and ai+1

p3

Query Answering

p1 p2 pn-1 pn l1 l2 ln-1

We have ai‘s sorted in increasing x-coordinate

a1 a2 an-1 V(p1) V(p2) V(p3) V(pn-1) V(pn)

Given a query point p[x,y] What we have to do? p[x,y] Locate x correctly between ai and ai+1 We can forget about y coordinate

Time Complexity analysis

Time Complexity analysis

Preprocessing Time =O(n log n)

Time Complexity analysis

Preprocessing Time =O(n log n) Query Time =O(log n)

Computing the Voronoi Diagram

p3

Computing the Voronoi Diagram

p1 p2 p4

I n p u t : A s e t

• f

p

• i

n t s

P

= (p

1

, p

2

, . . . , p

n

)

• n

2 D

Computing the Voronoi Diagram

I n p u t : A s e t

• f

p

• i

n t s

P

= (p

1

, p

2

, . . . , p

n

)

• n

2 D Output: A partitioning of the plane into regions of nearest neighbors

p3 p1 p2 p4

Computing the Voronoi Diagram

I n p u t : A s e t

• f

p

• i

n t s

P

= (p

1

, p

2

, . . . , p

n

)

• n

2 D Output: A partitioning of the plane into regions of nearest neighbors Find cell for each point one by one?

p3 p1 p2 p4

Computing the Voronoi Diagram

I n p u t : A s e t

• f

p

• i

n t s

P

= (p

1

, p

2

, . . . , p

n

)

• n

2 D Output: A partitioning of the plane into regions of nearest neighbors Find cell for each point one by one? use perpendicular bisector argument

p3 p1 p2 p4

Computing the Voronoi Diagram

I n p u t : A s e t

• f

p

• i

n t s

P

= (p

1

, p

2

, . . . , p

n

)

• n

2 D Output: A partitioning of the plane into regions of nearest neighbors Find cell for each point one by one? use perpendicular bisector argument

p3 p1 p2 p4

Find region for p1

p3

Computing the Voronoi Diagram

p1 p2 p4

I n p u t : A s e t

• f

p

• i

n t s

P

= (p

1

, p

2

, . . . , p

n

)

• n

2 D Output: A partitioning of the plane into regions of nearest neighbors Find cell for each point one by one? use perpendicular bisector argument

V(p1)

Find region for p1 P e r p e n d i c u l a r b i s e c t

• r
• f

[p

1p 2

]

p3

Computing the Voronoi Diagram

p1 p2 p4

I n p u t : A s e t

• f

p

• i

n t s

P

= (p

1

, p

2

, . . . , p

n

)

• n

2 D Output: A partitioning of the plane into regions of nearest neighbors Find cell for each point one by one? use perpendicular bisector argument

V(p1)

Find region for p1 P e r p e n d i c u l a r b i s e c t

• r
• f

[p

1p 2

]

any point x here p1 is closer than p2

Computing the Voronoi Diagram

I n p u t : A s e t

• f

p

• i

n t s

P

= (p

1

, p

2

, . . . , p

n

)

• n

2 D Output: A partitioning of the plane into regions of nearest neighbors Find cell for each point one by one?

p2 p1

use perpendicular bisector argument

p3 p4 V(p1)

Find region for p1 P e r p e n d i c u l a r b i s e c t

• r
• f

[p

1p 2

] P e r p e n d i c u l a r b i s e c t

• r
• f

[p

1p 4

]

Computing the Voronoi Diagram

I n p u t : A s e t

• f

p

• i

n t s

P

= (p

1

, p

2

, . . . , p

n

)

• n

2 D Output: A partitioning of the plane into regions of nearest neighbors Find cell for each point one by one?

p2 p1

use perpendicular bisector argument

p3 p4 V(p1)

Find region for p1 P e r p e n d i c u l a r b i s e c t

• r
• f

[p

1p 2

] P e r p e n d i c u l a r b i s e c t

• r
• f

[p

1p 4

] P e r p e n d i c u l a r b i s e c t

• r
• f

[p

1p 3

]

Computing the Voronoi Diagram

I n p u t : A s e t

• f

p

• i

n t s

P

= (p

1

, p

2

, . . . , p

n

)

• n

2 D Output: A partitioning of the plane into regions of nearest neighbors Find cell for each point one by one?

p2 p1

use perpendicular bisector argument

p3 p4 V(p1)

Find region for p1 P e r p e n d i c u l a r b i s e c t

• r
• f

[p

1p 2

] P e r p e n d i c u l a r b i s e c t

• r
• f

[p

1p 4

] P e r p e n d i c u l a r b i s e c t

• r
• f

[p

1p 3

] F i n d V (p

1

)

Computing the Voronoi Diagram

I n p u t : A s e t

• f

p

• i

n t s

P

= (p

1

, p

2

, . . . , p

n

)

• n

2 D Output: A partitioning of the plane into regions of nearest neighbors Find cell for each point one by one?

p2

use perpendicular bisector argument

p3 p4

Find region for p1 P e r p e n d i c u l a r b i s e c t

• r
• f

[p

1p 2

] P e r p e n d i c u l a r b i s e c t

• r
• f

[p

1p 4

] P e r p e n d i c u l a r b i s e c t

• r
• f

[p

1p 3

] F i n d V (p

1

)

p1 V(p1)

Computing the Voronoi Diagram

p2 p3 p4 p1 V(p1)

H

• w

d

• w

e f i n d V (p

1

) ?

Computing the Voronoi Diagram

H

• w

d

• w

e f i n d V (p

1

) ? G

• b

a c k

p3

Computing the Voronoi Diagram

p1 p2 p4 V(p1)

H

• w

d

• w

e f i n d V (p

1

) ? G

• b

a c k W h a t i s t h i s r e g i

• n

?

p3

Computing the Voronoi Diagram

p1 p2 p4 V(p1)

H

• w

d

• w

e f i n d V (p

1

) ? G

• b

a c k H a l f

• p

l a n e , s a y H

1

, c

• n

t a i n i n g p

1

W h a t i s t h i s r e g i

• n

?

p3

Computing the Voronoi Diagram

p1 p2 p4 V(p1)

H

• w

d

• w

e f i n d V (p

1

) ? G

• b

a c k H a l f

• p

l a n e , s a y H

1

, c

• n

t a i n i n g p

1

W h a t i s t h i s r e g i

• n

? for any point x here, p1 is closer than p2

Computing the Voronoi Diagram

p2 p1 p3 p4 V(p1)

H

• w

d

• w

e f i n d V (p

1

) ? G

• b

a c k W h a t i s t h i s r e g i

• n

?

Computing the Voronoi Diagram

p2 p1 p3 p4 V(p1)

W h a t i s t h i s r e g i

• n

? H a l f

• p

l a n e , s a y H

2

, c

• n

t a i n i n g p

1

H

• w

d

• w

e f i n d V (p

1

) ? G

• b

a c k

Computing the Voronoi Diagram

p2 p1 p3 p4 V(p1)

W h a t i s t h i s r e g i

• n

? H a l f

• p

l a n e , s a y H

2

, c

• n

t a i n i n g p

1

H

• w

d

• w

e f i n d V (p

1

) ? G

• b

a c k for any point x here, p1 is closer than p4

Computing the Voronoi Diagram

p2 p1 p3 p4 V(p1)

W h a t i s t h i s r e g i

• n

? H a l f

• p

l a n e , s a y H

2

, c

• n

t a i n i n g p

1

H

• w

d

• w

e f i n d V (p

1

) ? G

• b

a c k for any point x here, p1 is closer than p4

Computing the Voronoi Diagram

p2 p1 p3 p4 V(p1)

W h a t i s t h i s r e g i

• n

?

Computing the Voronoi Diagram

p2 p1 p3 p4 V(p1)

W h a t i s t h i s r e g i

• n

? H a l f

• p

l a n e , s a y H

3

, c

• n

t a i n i n g p

1

Computing the Voronoi Diagram

p2 p1 p3 p4 V(p1)

W h a t i s t h i s r e g i

• n

? H a l f

• p

l a n e , s a y H

3

, c

• n

t a i n i n g p

1

for any point x here, p1 is closer than p3

Computing the Voronoi Diagram

p2 p1 p3 p4 V(p1)

Computing the Voronoi Diagram

p2 p1 p3 p4 V(p1)

W h a t i s t h i s r e g i

• n

?

Computing the Voronoi Diagram

p2 p1 p3 p4 V(p1)

W h a t i s t h i s r e g i

• n

? for any point x here, p1 is closer than p2 and p4

Computing the Voronoi Diagram

p2 p1 p3 p4 V(p1)

W h a t i s t h i s r e g i

• n

?

H1 ∩ H2

for any point x here, p1 is closer than p2 and p4

Computing the Voronoi Diagram

p2 p3 p4 p1 V(p1)

W h a t i s V (p

1

) ?

Computing the Voronoi Diagram

p2 p3 p4 p1 V(p1)

W h a t i s V (p

1

) ?

H1 ∩ H2 ∩ H3

Computing the Voronoi Diagram

p2 p3 p4 p1 V(p1)

W h a t i s V (p

1

) ?

H1 ∩ H2 ∩ H3

I n g e n e r a l , w h a t w

• u

l d b e V (p

1

) ?

Computing the Voronoi Diagram

p2 p3 p4 p1 V(p1)

W h a t i s V (p

1

) ?

H1 ∩ H2 ∩ H3

I n g e n e r a l , w h a t w

• u

l d b e V (p

1

) ? I n t e r s e c t i

• n
• f

( n

• 1

) h y p e r p l a n e s

Computing the Voronoi Diagram

p2 p3 p4 p1 V(p1)

W h a t i s V (p

1

) ?

H1 ∩ H2 ∩ H3

I n g e n e r a l , w h a t w

• u

l d b e V (p

1

) ? I n t e r s e c t i

• n
• f

( n

• 1

) h y p e r p l a n e s

H1 ∩ H2 ... ∩ ∩ Hn-1

Time complexity of this Brute Force Algorithm

Time complexity of this Brute Force Algorithm

I n t e r s e c t i

• n
• f

( n

• 1

) h y p e r p l a n e s c a n b e f

• u

n d i n O ( n l

• g

n ) t i m e

Time complexity of this Brute Force Algorithm

I n t e r s e c t i

• n
• f

( n

• 1

) h y p e r p l a n e s c a n b e f

• u

n d i n O ( n l

• g

n ) t i m e T

• t

a l t i m e c

• m

p l e x i t y :

Time complexity of this Brute Force Algorithm

I n t e r s e c t i

• n
• f

( n

• 1

) h y p e r p l a n e s c a n b e f

• u

n d i n O ( n l

• g

n ) t i m e T

• t

a l t i m e c

• m

p l e x i t y : O(n2 log n)

Time Complexity of Best Algorithms for Voronoi Diagram

Time Complexity of Best Algorithms for Voronoi Diagram

Voronoi Diagram can be constructed in O(n log n) time

Time Complexity of Best Algorithms for Voronoi Diagram

Voronoi Diagram can be constructed in O(n log n) time There are well-known algorithms like:

• 1. Fortune’s Line Sweep
• 2. Divide and Conquer
• 3. Lifting points in 3D

Size of the Voronoi Diagram

Size of the Voronoi Diagram

Size means: number of vertices, edges and faces

Size of the Voronoi Diagram

Size means: number of vertices, edges and faces Lower bound (Smallest Size possible):

Size of the Voronoi Diagram

Size means: number of vertices, edges and faces Lower bound (Smallest Size possible): n, where n is number of sites

Size of the Voronoi Diagram

Size means: number of vertices, edges and faces Lower bound (Smallest Size possible): Trivial Upper bound (Biggest Size possible): n, where n is number of sites

Size of the Voronoi Diagram

Size means: number of vertices, edges and faces Lower bound (Smallest Size possible): Trivial Upper bound (Biggest Size possible): n, where n is number of sites O(n log n )

Size of the Voronoi Diagram

Size means: number of vertices, edges and faces Lower bound (Smallest Size possible): Trivial Upper bound (Biggest Size possible): n, where n is number of sites Ultimate Upper Bound (Biggest Size possible): O(n log n )

Size of the Voronoi Diagram

Size means: number of vertices, edges and faces Lower bound (Smallest Size possible): Trivial Upper bound (Biggest Size possible): n, where n is number of sites Ultimate Upper Bound (Biggest Size possible): O(n log n ) O(n)

Why to bother about Size?

Why to bother about Size?

Voronoi Diagram is

Why to bother about Size?

Voronoi Diagram is Planar Subdivision

Why to bother about Size?

Voronoi Diagram is Want to do Planar point Location to get closest point Efficiently Planar Subdivision

Why to bother about Size?

Voronoi Diagram is Want to do Planar point Location to get closest point Efficiently Preprocessing Time: Preprocessing space requirement: Query Time:

O(n) O(n) O(log n)

Planar Subdivision For Planar point Location:

Why to bother about Size?

Voronoi Diagram is Want to do Planar point Location to get closest point Efficiently Preprocessing Time: Preprocessing space requirement: Query Time:

O(n) O(n) O(log n)

Planar Subdivision For Planar point Location:

But there is a big if, What is that if?

Why to bother about Size?

Voronoi Diagram is Want to do Planar point Location to get closest point Efficiently Preprocessing Time: Preprocessing space requirement: Query Time:

O(n) O(n) O(log n)

Planar Subdivision For Planar point Location: The size of planar subdivision=

But there is a big if, What is that if?

Why to bother about Size?

Voronoi Diagram is The size of planar subdivision= Want to do Planar point Location to get closest point Efficiently Preprocessing Time: Preprocessing space requirement: Query Time:

O(n) O(n) O(log n) But there is a big if, What is that if? O(n)

Planar Subdivision For Planar point Location:

Summary

P → A set of n distinct points (Geometric Objects) in the plane.

Summary

P → A set of n distinct points (Geometric Objects) in the plane. We can Preprocess P such that closest point x ∈ P of any query point q can be found in O(log n) time Using Planar point location

q x

Summary

P → A set of n distinct points (Geometric Objects) in the plane. We can Preprocess P such that closest point x ∈ P of any query point q can be found in O(log n) time Using Planar point location

q x

Preprocess structure is called Voronoi Diagram V(P)

Summary

P → A set of n distinct points (Geometric Objects) in the plane. We can Preprocess P such that closest point x ∈ P of any query point q can be found in O(log n) time Using Planar point location

q x

Preprocess structure is called Voronoi Diagram V(P) V(P) can be constructed in O( n log n) time and can be stored in O(n) space

Other Kind of Voronoi Diagrams

Furthest Point Voronoi Diagram

Furthest Point Voronoi Diagram

FV(P): the partition of the plane formed by the farthest point Voronoi regions, their edges, and vertices

Furthest Point Voronoi Diagram

FV(P): the partition of the plane formed by the farthest point Voronoi regions, their edges, and vertices V-1(pi): the set of point of the plane farther from pi than from any other site

Voronoi diagram for line segments

Voronoi diagram for line segments

Moving a disk from s to t in the presence of barriers

Organization of the Talk

• 1. Preliminaries
• 2. Generic Definition
• 3. Some Technical Details
• 4. Conclusion

Organization of the Talk

• 1. Preliminaries
• 2. Generic Definition
• 3. Some Technical Details
• 4. Conclusion

Conclusion

Conclusion

Voronoi Diagram is a very Fundamental Interesting Geometric Structure

Conclusion

Voronoi Diagram is a very Fundamental Interesting Geometric Structure This has wide range of application to solve different problem in Geometry, Facility Location, Engineering Sciences, Biological Sciences, Nano Sciences to name a few

Conclusion

Voronoi Diagram is a very Fundamental Interesting Geometric Structure This has wide range of application to solve different problem in Geometry, Facility Location, Engineering Sciences, Biological Sciences, Nano Sciences to name a few The Applications of this structure are so wide that

Conclusion

Voronoi Diagram is a very Fundamental Interesting Geometric Structure This has wide range of application to solve different problem in Geometry, Facility Location, Engineering Sciences, Biological Sciences, Nano Sciences to name a few The Applications of this structure are so wide that There is dedicated Symposiums on Voronoi Diagram:

INTERNATIONAL SYMPOSIUM on VORONOI DIAGRAMS in science and engineering

Thank You