Farthest Point         - PowerPoint Presentation

Slide1

Farthest Point

 

 

 

 

 

 

 

 

 

Sites

 

Given a point

,

is its

farthest point

if for all

 

 

The farthest point of

is

 

 

Slide2

Not Every Site Can be the Farthest

 

 

 

 

 

 

 

 

Claim

A point

is the farthest site of some point

in the plane if and only if

is a vertex of the convex hull

CH

of

.

 

 

Slide3

Proof of Necessity

Suppose there exists some

such that

is its farthest site.

 

Assume that

is not a vertex of CH

 

 

 

 

 

 

 

Then

is either in the interior CH

or on one of its edge.

 

Consider the line

through

and

(clearly

).

 

intersects CH

with two of its edges

and

.

 

 

 

 

One of the four endpoints of

and

must be father from

than

.

 

Contradiction.

Slide4

Proof of Sufficiency

Suppose

is a vertex of CH

 

must be extreme in some direction

.

 

 

Let

be the line through

in

.

 

The point

,

for large enough

, is

farther from

than any other site.

 

 

 

 



Start at

.

 



Move on

in the direction

.

 

Slide5

Two Sites

Perpendicular

bisector

 

 

,

 

Half-plane

than to

 

 

Slide6

Voronoi Cell

 

 

 

 

 

 

 



Open convex region



vertices

 



edges

 

 

 

 

 

 

 

Slide7

Unboundedness

 

 

 

 

 

 

 

The cell contains a ray

collinear with

.

 

 

 

farthest point from

.

 

the line through

and

.

 

: half-line starting at

and away

from

.

 

All the points on

have

as the farthest point!

 

 

Slide8

Farthest-Point Voronoi Diagram

Tree-like structure

 

 

 

 

 

 

 

 

 

 



No cycles

A cycle would imply

a bounded cell.



Edges include segments

and half-infinite lines.

 



A vertex has

farthest sites.

 

Slide9

More Properties

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 



Any site that is not a vertex

of the convex hull has

no

Voronoi cell

.



vertices, edges and

cells

 

It contributes no Voronoi edge.



Every Voronoi edge is part

of a bisector of two convex

hull vertices.

Slide10

Center of Smallest Enclosing Disk

 

 

 

Two possibilities:



Vertex

equidistant farthest sites

 



M

idpoint of two sites defining an edge

two equidistant farthest sites

 

 

 

 

Slide11

Storage

Doubly-connected edge list (DCEL) with modifications

Half-infinite edge

If no origin, stores the direction of the edge

(

) instead of coordinates.

 

 

 

 

 Either next(

) or

prev() is undefined.

 

Slide12

 

Preprocessing for Construction

1. Compute the convex hull CH

--

vertices.

 

2. Order vertices of the hull randomly.

(new indices)

 

 

 

 

 

3. Remove

one by one in the order.

 

 

cw

(

 

ccw

(

 

For each

, store its clockwise

neighbor cw(

and counterclockwise

neighbor

ccw

(

at the time of removal.

 

cannot be a neighbor of any point

removed later.

 

Slide13

Construction

1. Initialize with the FPVD of

.

 

 

 

 

 

 

 

Slide14

Construction (cont’d)

 

2. Insert

one by one in the order.

 

 

for

 

most counterclockwise

half-edge in a traversal of

the boundary of

 

Slide15

How to Add

?

 

 

ccw

 

cw

 

ccw

(

 

cw

(

 

The cell

of

will come in between

the adjacent cells

cw

and

ccw

.

 

 

ccw

has a pointer to bisector

(most counterclockwise edge in its cell.

 

Bisector of

ccw

and

will contribute

a half-edge

to

.

 

Traverse the boundary of

ccw

starting at

, in a clockwise way to find

the intersection

of

with a boundary

edge

between

ccw

and, say,

of another site

.

 

 

Move along

to

and cross into the cell of

.

 

 

 

 

Slide16

Moving on …

 

ccw

 

cw

 

ccw

(

 

cw

(

 

 

 

 

At

start a clockwise traversal of the

boundary of the cell

.

 

Trace out the boundary of

by traversing a sequence of cells,

each in a clockwise way.

 

Last bisector will be between

and

cw

.

 

 

Traversal stops at an edge

that

intersects the bisector of

and

.

 

Exit the cell

, and so on.

 

 

 

Slide17

Summary

 

ccw

 

cw

 

ccw

(

 

cw

(

 

 



All new edges are added to DCEL.



Afterward, all the edges lying

inside the cell of

are removed.

 

Theorem

FPVD can be constructed in

expected

time using

storage.

 

time to compute the convex hull.

 

time to construct FPVD (backward analysis).

 

Slide18

Roundness of a Point Set

The

roundness

of a set of points is measured by the

minimum width

of any annulus that contains the points.

Observation

:

There must be one point each

on

and

.

 

 

 

Otherwise, we can always

reduce the size of

, or

increase that of

.

 

But one point on each bounding

circle does not yield the smallest-

width annulus.

Slide19

Only Three Different Cases

points on

 

point on

 

(a)

point on

 

points on

 

(b)

points on

 

points on

 

(c)

Slide20

Smallest-Width Annulus – Case (a)

The problem is equivalent to finding the center point

of the annulus.

 

points on

 

point on

 

(a)

 

must be a vertex of the

farthest-point Voronoi diagram.

 

points on

 

In case (a)

Slide21

Case (b)

point on

 

points on

 

(b)

must be a vertex of the

(nearest-point) Voronoi diagram.

 

 

points on

 

Slide22

Case (c)

points on

 

points on

 

(c)

must be at the intersection of

an VD edge with an FPVD edge.

 

must be on an edge of the FPVD.

 

must be on an edge of the VD.

 

Slide23

Overlay of VD and FPVD

Site

Vertex of VD

Vertex of FPVD

Intersection of VD and FPVD

Vertices of

the overlay

Exactly the candidate centers of the smallest-width annulus.

No need to compute the overlay!

Slide24

Smallest-Width Annulus Algorithm

1. Construct the Voronoi diagram and farthest-point Voronoi diagram.

2. For every vertex

of the FPVD (

)

vertices)

 

 Determine its closest site using VD (

in

) time).

  Its farthest sites

(equidistant) are known (

).  This yields a candidate annulus.

3. For every vertex

of the VD () vertices)

 

 Determine its farthest site using FPVD (

in

) time).

 

 Its closest sites

(equidistant) are known (

).

 

 This yields a candidate annulus.

Slide25

Algorithm (cont’d)

4. For every pair of edges, one from VD and the other form FPVD

(

pairs)

 

 Test if they intersect.

 If so, the two closest sites

and two farthest sites

are

known. Construct the annulus in time.

 5. Choose the smallest-width annulus of all constructed annuli.

Theorem Given a set of points in the plane, the smallest-width annulus can be determined in time using

storage.