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Geometric Problems with Imprecise Input Points Maarten L offler - - PowerPoint PPT Presentation

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Geometric Problems with Imprecise Input Points Maarten L offler Marc van Kreveld Center for Geometry, Imaging and Virtual Environments Utrecht University 1-1 Overview Introduction Geometric problems Imprecise input points

slide-1
SLIDE 1

1-1

Geometric Problems with Imprecise Input Points

Maarten L¨

  • ffler

Marc van Kreveld

Center for Geometry, Imaging and Virtual Environments

Utrecht University

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SLIDE 2

2-1 Geometric Problems with Imprecise Input Points Maarten L¨

  • ffler and Marc van Kreveld, July 4, 2007
  • Introduction

Overview

  • Overview of problems and results
  • Largest diameter of squares
  • Smallest diameter of squares
  • Concluding remarks
  • Geometric problems
  • Imprecise input points
  • Algorithms
slide-3
SLIDE 3

3-1 Geometric Problems with Imprecise Input Points Maarten L¨

  • ffler and Marc van Kreveld, July 4, 2007

Geometric Structures on Point Sets

  • Given a set P of n

points in the plane

P

  • Geometric structures:
slide-4
SLIDE 4

3-2 Geometric Problems with Imprecise Input Points Maarten L¨

  • ffler and Marc van Kreveld, July 4, 2007

Geometric Structures on Point Sets

  • Given a set P of n

points in the plane

  • Bounding box

P

  • Geometric structures:
slide-5
SLIDE 5

3-3 Geometric Problems with Imprecise Input Points Maarten L¨

  • ffler and Marc van Kreveld, July 4, 2007

Geometric Structures on Point Sets

  • Given a set P of n

points in the plane

  • Bounding box
  • Diameter

P

  • Geometric structures:
slide-6
SLIDE 6

3-4 Geometric Problems with Imprecise Input Points Maarten L¨

  • ffler and Marc van Kreveld, July 4, 2007

Geometric Structures on Point Sets

  • Given a set P of n

points in the plane

  • Bounding box
  • Diameter
  • Convex hull

P

  • Geometric structures:
slide-7
SLIDE 7

3-5 Geometric Problems with Imprecise Input Points Maarten L¨

  • ffler and Marc van Kreveld, July 4, 2007

Geometric Structures on Point Sets

  • Given a set P of n

points in the plane

  • Bounding box
  • Diameter
  • Convex hull
  • Minimum spanning tree

P

  • Geometric structures:
slide-8
SLIDE 8

3-6 Geometric Problems with Imprecise Input Points Maarten L¨

  • ffler and Marc van Kreveld, July 4, 2007

Geometric Structures on Point Sets

  • Given a set P of n

points in the plane

  • Bounding box
  • Diameter
  • Convex hull
  • Minimum spanning tree
  • Many others
  • Optimal algorithms

are known

P

  • Geometric structures:
slide-9
SLIDE 9

4-1 Geometric Problems with Imprecise Input Points Maarten L¨

  • ffler and Marc van Kreveld, July 4, 2007

Imprecision

  • Traditional algorithms assume exact input
  • In practice, input data is often not exact
  • Measured from the real world
  • Stored with limited precision
  • Computed by inexact algorithms
  • Output of traditional algorithms is unreliable
  • Given exact description of input imprecision,

we can exactly predict output imprecision

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SLIDE 10

5-1 Geometric Problems with Imprecise Input Points Maarten L¨

  • ffler and Marc van Kreveld, July 4, 2007
  • Unknown location

Imprecise Points

  • Known region of

possible locations

  • Regions are simple

geometric objects

  • Disc
  • Square
  • Rectangle
  • Convex polygon
slide-11
SLIDE 11

5-2 Geometric Problems with Imprecise Input Points Maarten L¨

  • ffler and Marc van Kreveld, July 4, 2007
  • Unknown location

Imprecise Points

  • Known region of

possible locations

  • Regions are simple

geometric objects

  • Disc
  • Square
  • Rectangle
  • Convex polygon
slide-12
SLIDE 12

5-3 Geometric Problems with Imprecise Input Points Maarten L¨

  • ffler and Marc van Kreveld, July 4, 2007
  • Unknown location

Imprecise Points

  • Known region of

possible locations

  • Regions are simple

geometric objects

  • Disc
  • Square
  • Rectangle
  • Convex polygon
slide-13
SLIDE 13

5-4 Geometric Problems with Imprecise Input Points Maarten L¨

  • ffler and Marc van Kreveld, July 4, 2007
  • Unknown location

Imprecise Points

  • Known region of

possible locations

  • Regions are simple

geometric objects

  • Disc
  • Square
  • Rectangle
  • Convex polygon
slide-14
SLIDE 14

5-5 Geometric Problems with Imprecise Input Points Maarten L¨

  • ffler and Marc van Kreveld, July 4, 2007
  • Unknown location

Imprecise Points

  • Known region of

possible locations

  • Regions are simple

geometric objects

  • Disc
  • Square
  • Rectangle
  • Convex polygon
slide-15
SLIDE 15

6-1 Geometric Problems with Imprecise Input Points Maarten L¨

  • ffler and Marc van Kreveld, July 4, 2007
  • Multiple possibilities
  • True structure is

unknown

  • We want to capture

the imprecision in the

  • utput

Imprecision in Geometric Structures

  • Given a set L of n

imprecise points

  • Consider the same

geometric structures

slide-16
SLIDE 16

6-2 Geometric Problems with Imprecise Input Points Maarten L¨

  • ffler and Marc van Kreveld, July 4, 2007
  • Multiple possibilities
  • True structure is

unknown

  • We want to capture

the imprecision in the

  • utput

Imprecision in Geometric Structures

  • Given a set L of n

imprecise points

  • Consider the same

geometric structures

slide-17
SLIDE 17

6-3 Geometric Problems with Imprecise Input Points Maarten L¨

  • ffler and Marc van Kreveld, July 4, 2007
  • Multiple possibilities
  • True structure is

unknown

  • We want to capture

the imprecision in the

  • utput

Imprecision in Geometric Structures

  • Given a set L of n

imprecise points

  • Consider the same

geometric structures

slide-18
SLIDE 18

6-4 Geometric Problems with Imprecise Input Points Maarten L¨

  • ffler and Marc van Kreveld, July 4, 2007
  • Multiple possibilities
  • True structure is

unknown

  • We want to capture

the imprecision in the

  • utput

Imprecision in Geometric Structures

  • Given a set L of n

imprecise points

  • Consider the same

geometric structures

slide-19
SLIDE 19

7-1 Geometric Problems with Imprecise Input Points Maarten L¨

  • ffler and Marc van Kreveld, July 4, 2007
  • Measure function

µ : F(R2) → R

Bounds on Measures

  • Largest and smallest

possible values of µ

  • Output imprecision
slide-20
SLIDE 20

7-2 Geometric Problems with Imprecise Input Points Maarten L¨

  • ffler and Marc van Kreveld, July 4, 2007
  • Measure function

µ : F(R2) → R

Bounds on Measures

  • Largest and smallest

possible values of µ

  • Bounding box area
  • For example
  • Output imprecision
slide-21
SLIDE 21

7-3 Geometric Problems with Imprecise Input Points Maarten L¨

  • ffler and Marc van Kreveld, July 4, 2007
  • Measure function

µ : F(R2) → R

Bounds on Measures

  • Largest and smallest

possible values of µ

  • Bounding box area
  • For example
  • Output imprecision
slide-22
SLIDE 22

7-4 Geometric Problems with Imprecise Input Points Maarten L¨

  • ffler and Marc van Kreveld, July 4, 2007
  • Measure function

µ : F(R2) → R

Bounds on Measures

  • Largest and smallest

possible values of µ

  • Bounding box area
  • For example
  • Output imprecision
slide-23
SLIDE 23

7-5 Geometric Problems with Imprecise Input Points Maarten L¨

  • ffler and Marc van Kreveld, July 4, 2007
  • Measure function

µ : F(R2) → R

Bounds on Measures

  • Largest and smallest

possible values of µ

  • Bounding box area
  • Diameter
  • For example
  • Output imprecision
slide-24
SLIDE 24

7-6 Geometric Problems with Imprecise Input Points Maarten L¨

  • ffler and Marc van Kreveld, July 4, 2007
  • Measure function

µ : F(R2) → R

Bounds on Measures

  • Largest and smallest

possible values of µ

  • Bounding box area
  • Diameter
  • For example
  • Output imprecision
slide-25
SLIDE 25

7-7 Geometric Problems with Imprecise Input Points Maarten L¨

  • ffler and Marc van Kreveld, July 4, 2007
  • Measure function

µ : F(R2) → R

Bounds on Measures

  • Largest and smallest

possible values of µ

  • Bounding box area
  • Diameter
  • For example
  • Output imprecision
slide-26
SLIDE 26

7-8 Geometric Problems with Imprecise Input Points Maarten L¨

  • ffler and Marc van Kreveld, July 4, 2007
  • Measure function

µ : F(R2) → R

Bounds on Measures

  • Largest and smallest

possible values of µ

  • Bounding box area
  • Diameter
  • Convex hull perimeter
  • For example
  • Output imprecision
slide-27
SLIDE 27

7-9 Geometric Problems with Imprecise Input Points Maarten L¨

  • ffler and Marc van Kreveld, July 4, 2007
  • Measure function

µ : F(R2) → R

Bounds on Measures

  • Largest and smallest

possible values of µ

  • Bounding box area
  • Diameter
  • Convex hull perimeter
  • For example
  • Output imprecision
slide-28
SLIDE 28

7-10 Geometric Problems with Imprecise Input Points Maarten L¨

  • ffler and Marc van Kreveld, July 4, 2007
  • Measure function

µ : F(R2) → R

Bounds on Measures

  • Largest and smallest

possible values of µ

  • Bounding box area
  • Diameter
  • Convex hull perimeter
  • For example
  • Output imprecision
slide-29
SLIDE 29

7-11 Geometric Problems with Imprecise Input Points Maarten L¨

  • ffler and Marc van Kreveld, July 4, 2007
  • Measure function

µ : F(R2) → R

Bounds on Measures

  • Largest and smallest

possible values of µ

  • Bounding box area
  • Diameter
  • Convex hull perimeter
  • MST weight
  • For example
  • Output imprecision
slide-30
SLIDE 30

7-12 Geometric Problems with Imprecise Input Points Maarten L¨

  • ffler and Marc van Kreveld, July 4, 2007
  • Measure function

µ : F(R2) → R

Bounds on Measures

  • Largest and smallest

possible values of µ

  • Bounding box area
  • Diameter
  • Convex hull perimeter
  • MST weight
  • For example
  • Output imprecision
slide-31
SLIDE 31

7-13 Geometric Problems with Imprecise Input Points Maarten L¨

  • ffler and Marc van Kreveld, July 4, 2007
  • Measure function

µ : F(R2) → R

Bounds on Measures

  • Largest and smallest

possible values of µ

  • Bounding box area
  • Diameter
  • Convex hull perimeter
  • MST weight
  • For example
  • Output imprecision
slide-32
SLIDE 32

8-1 Geometric Problems with Imprecise Input Points Maarten L¨

  • ffler and Marc van Kreveld, July 4, 2007
  • Bounding box

Results

  • Largest area or perimeter, squares or discs
  • Smallest area or perimeter, squares
  • Smallest area or perimeter, discs
  • Smallest enclosing circle

O(n) O(n) O(n2)

  • Largest or smallest radius, squares or discs

O(n)

  • Convex hull
  • Largest area, disjoint squares

O(n7)

  • Largest perimeter, disjoint squares

O(n10)

  • Smallest area, squares

O(n2)

  • Smallest perimeter, squares

O(n log n)

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SLIDE 33

8-2 Geometric Problems with Imprecise Input Points Maarten L¨

  • ffler and Marc van Kreveld, July 4, 2007
  • Diameter

Results

  • Largest diameter, squares or discs
  • Smallest diameter, squares
  • Smallest diameter, discs
  • Width

O(n log n) O(n log n) O(ncε−1)

  • Smallest width, squares or discs

O(n log n)

  • Closest pair
  • Largest width, line segments
  • Largest width, squares or discs

NP-hard ?

  • Smallest distance, squares or discs

O(n log n)

  • Largest distance, squares or discs

NP-hard

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SLIDE 34

8-3 Geometric Problems with Imprecise Input Points Maarten L¨

  • ffler and Marc van Kreveld, July 4, 2007
  • Minimum spanning tree

Results

  • Largest weight, squares or discs
  • Tours

?

  • Shortest tour, sequence of squares

O(n)

  • Longest tour, sequence of squares
  • Existence of simple tour, any sequence
  • Smallest weight, squares or discs

NP-hard O(n) NP-hard

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SLIDE 35

9-1 Geometric Problems with Imprecise Input Points Maarten L¨

  • ffler and Marc van Kreveld, July 4, 2007
  • Diameter of imprecise points, square model

Diameter

  • Largest diameter
  • Place two points as far away as possible
  • Relatively easy
  • Optimal O(n log n) algorithm
  • Smallest diameter
  • Place all points as close together as possible
  • Much harder
  • O(n2) algorithm
  • Optimal O(n log n) algorithm
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SLIDE 36

10-1 Geometric Problems with Imprecise Input Points Maarten L¨

  • ffler and Marc van Kreveld, July 4, 2007

Largest Diameter

  • Compute the

diameter of all corners

  • If the computed

points belong to different regions, we are happy

  • Otherwise, they are

diagonally opposite corners of one big square S

slide-37
SLIDE 37

10-2 Geometric Problems with Imprecise Input Points Maarten L¨

  • ffler and Marc van Kreveld, July 4, 2007

Largest Diameter

  • Compute the

diameter of all corners

  • If the computed

points belong to different regions, we are happy

  • Otherwise, they are

diagonally opposite corners of one big square S

slide-38
SLIDE 38

10-3 Geometric Problems with Imprecise Input Points Maarten L¨

  • ffler and Marc van Kreveld, July 4, 2007

Largest Diameter

  • Compute the

diameter of all corners

  • If the computed

points belong to different regions, we are happy

  • Otherwise, they are

diagonally opposite corners of one big square S

slide-39
SLIDE 39

10-4 Geometric Problems with Imprecise Input Points Maarten L¨

  • ffler and Marc van Kreveld, July 4, 2007

Largest Diameter

  • Compute the

diameter of all corners

  • If the computed

points belong to different regions, we are happy

  • Otherwise, they are

diagonally opposite corners of one big square S

slide-40
SLIDE 40

10-5 Geometric Problems with Imprecise Input Points Maarten L¨

  • ffler and Marc van Kreveld, July 4, 2007

Largest Diameter

  • Compute the

diameter of all corners

  • If the computed

points belong to different regions, we are happy

  • Otherwise, they are

diagonally opposite corners of one big square S

slide-41
SLIDE 41

10-6 Geometric Problems with Imprecise Input Points Maarten L¨

  • ffler and Marc van Kreveld, July 4, 2007

Largest Diameter

  • Compute the

diameter of all corners

  • If the computed

points belong to different regions, we are happy

  • Otherwise, they are

diagonally opposite corners of one big square S

slide-42
SLIDE 42

10-7 Geometric Problems with Imprecise Input Points Maarten L¨

  • ffler and Marc van Kreveld, July 4, 2007

Largest Diameter

  • Compute the

diameter of all corners

  • If the computed

points belong to different regions, we are happy

  • Otherwise, they are

diagonally opposite corners of one big square S

slide-43
SLIDE 43

10-8 Geometric Problems with Imprecise Input Points Maarten L¨

  • ffler and Marc van Kreveld, July 4, 2007

Largest Diameter

  • Compute the

diameter of all corners

  • If the computed

points belong to different regions, we are happy

  • Otherwise, they are

diagonally opposite corners of one big square S

slide-44
SLIDE 44

10-9 Geometric Problems with Imprecise Input Points Maarten L¨

  • ffler and Marc van Kreveld, July 4, 2007

Largest Diameter

  • Compute the

diameter of all corners

  • If the computed

points belong to different regions, we are happy

  • Otherwise, they are

diagonally opposite corners of one big square S

slide-45
SLIDE 45

10-10 Geometric Problems with Imprecise Input Points Maarten L¨

  • ffler and Marc van Kreveld, July 4, 2007

Largest Diameter

  • Compute the

diameter of all corners

  • If the computed

points belong to different regions, we are happy

  • Otherwise, they are

diagonally opposite corners of one big square S

slide-46
SLIDE 46

11-1 Geometric Problems with Imprecise Input Points Maarten L¨

  • ffler and Marc van Kreveld, July 4, 2007

Largest Diameter

  • Diameter determined

by two corners of S

  • Two possible cases
  • Or it does not
  • Either S contributes

S

  • Try all corner pairs
  • Only a linear number
  • Compute diameter of

remaining corners

  • Points cannot belong

to one square again

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SLIDE 47

12-1 Geometric Problems with Imprecise Input Points Maarten L¨

  • ffler and Marc van Kreveld, July 4, 2007

Smallest Diameter

  • Diameter could be

determined by more pairs simultaneously

  • Bends only occur at

axis-extreme squares

  • Reflection angles

must be less than 90◦

  • Consequence:

at most two bends

  • The pairs form a star
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SLIDE 48

13-1 Geometric Problems with Imprecise Input Points Maarten L¨

  • ffler and Marc van Kreveld, July 4, 2007
  • The optimal star has at most two bends
  • Compute the star for every subset of:
  • Two axis-extreme squares
  • And two other squares
  • The largest among these is the optimal star
  • We now know the optimal diameter d and the

star that defines it

  • We still need to place points in all regions
  • There are O(n2) stars to be computed

O(n2) Algorithm

slide-49
SLIDE 49

14-1 Geometric Problems with Imprecise Input Points Maarten L¨

  • ffler and Marc van Kreveld, July 4, 2007

O(n2) Algorithm

  • We know d
  • Let R be axis-extreme
  • Valid placements in R

are at most d away from any other square

R

  • This is sufficient
  • Place axis-extreme

points validly and at most d away from each other

slide-50
SLIDE 50

14-2 Geometric Problems with Imprecise Input Points Maarten L¨

  • ffler and Marc van Kreveld, July 4, 2007

O(n2) Algorithm

  • We know d
  • Let R be axis-extreme

d

  • Valid placements in R

are at most d away from any other square

R

  • This is sufficient
  • Place axis-extreme

points validly and at most d away from each other

slide-51
SLIDE 51

14-3 Geometric Problems with Imprecise Input Points Maarten L¨

  • ffler and Marc van Kreveld, July 4, 2007

O(n2) Algorithm

  • We know d
  • Let R be axis-extreme

d

  • Valid placements in R

are at most d away from any other square

R

  • This is sufficient
  • Place axis-extreme

points validly and at most d away from each other

slide-52
SLIDE 52

14-4 Geometric Problems with Imprecise Input Points Maarten L¨

  • ffler and Marc van Kreveld, July 4, 2007

O(n2) Algorithm

  • We know d
  • Let R be axis-extreme
  • Valid placements in R

are at most d away from any other square

R d

  • This is sufficient
  • Place axis-extreme

points validly and at most d away from each other

slide-53
SLIDE 53

14-5 Geometric Problems with Imprecise Input Points Maarten L¨

  • ffler and Marc van Kreveld, July 4, 2007

O(n2) Algorithm

  • We know d
  • Let R be axis-extreme
  • Valid placements in R

are at most d away from any other square

R d

  • This is sufficient
  • Place axis-extreme

points validly and at most d away from each other

slide-54
SLIDE 54

14-6 Geometric Problems with Imprecise Input Points Maarten L¨

  • ffler and Marc van Kreveld, July 4, 2007

O(n2) Algorithm

  • We know d
  • Let R be axis-extreme
  • Valid placements in R

are at most d away from any other square

R d d

  • This is sufficient
  • Place axis-extreme

points validly and at most d away from each other

slide-55
SLIDE 55

14-7 Geometric Problems with Imprecise Input Points Maarten L¨

  • ffler and Marc van Kreveld, July 4, 2007

O(n2) Algorithm

  • We know d
  • Let R be axis-extreme
  • Valid placements in R

are at most d away from any other square

R d d

  • This is sufficient
  • Place axis-extreme

points validly and at most d away from each other

slide-56
SLIDE 56

14-8 Geometric Problems with Imprecise Input Points Maarten L¨

  • ffler and Marc van Kreveld, July 4, 2007

O(n2) Algorithm

  • We know d
  • Let R be axis-extreme
  • Valid placements in R

are at most d away from any other square

R d

  • This is sufficient
  • Place axis-extreme

points validly and at most d away from each other

slide-57
SLIDE 57

14-9 Geometric Problems with Imprecise Input Points Maarten L¨

  • ffler and Marc van Kreveld, July 4, 2007

O(n2) Algorithm

  • We know d
  • Let R be axis-extreme
  • Valid placements in R

are at most d away from any other square

R d

  • This is sufficient
  • Place axis-extreme

points validly and at most d away from each other

slide-58
SLIDE 58

14-10 Geometric Problems with Imprecise Input Points Maarten L¨

  • ffler and Marc van Kreveld, July 4, 2007

O(n2) Algorithm

  • We know d
  • Let R be axis-extreme
  • Valid placements in R

are at most d away from any other square

R

  • This is sufficient
  • Place axis-extreme

points validly and at most d away from each other

slide-59
SLIDE 59

15-1 Geometric Problems with Imprecise Input Points Maarten L¨

  • ffler and Marc van Kreveld, July 4, 2007

O(n2) Algorithm

  • We know the

axis-extreme points

  • Move rest of points

towards the middle

  • Optimal for given

axis-extreme points

  • Therefore, the

resulting point set must have diameter d

  • Solution of d exists
slide-60
SLIDE 60

15-2 Geometric Problems with Imprecise Input Points Maarten L¨

  • ffler and Marc van Kreveld, July 4, 2007

O(n2) Algorithm

  • We know the

axis-extreme points

  • Move rest of points

towards the middle

  • Optimal for given

axis-extreme points

  • Therefore, the

resulting point set must have diameter d

  • Solution of d exists
slide-61
SLIDE 61

15-3 Geometric Problems with Imprecise Input Points Maarten L¨

  • ffler and Marc van Kreveld, July 4, 2007

O(n2) Algorithm

  • We know the

axis-extreme points

  • Move rest of points

towards the middle

  • Optimal for given

axis-extreme points

  • Therefore, the

resulting point set must have diameter d

  • Solution of d exists
slide-62
SLIDE 62

15-4 Geometric Problems with Imprecise Input Points Maarten L¨

  • ffler and Marc van Kreveld, July 4, 2007

O(n2) Algorithm

  • We know the

axis-extreme points

  • Move rest of points

towards the middle

  • Optimal for given

axis-extreme points

  • Therefore, the

resulting point set must have diameter d

  • Solution of d exists
slide-63
SLIDE 63

16-1 Geometric Problems with Imprecise Input Points Maarten L¨

  • ffler and Marc van Kreveld, July 4, 2007

Concluding Remarks

  • Largest and smallest diameter for squares can

be computed efficiently

  • Simple geometric problems become really

interesting when input points are imprecise

  • Open problems
  • Largest width problem
  • Several variants of the convex hull
  • Third dimension?
slide-64
SLIDE 64

17-1 Geometric Problems with Imprecise Input Points Maarten L¨

  • ffler and Marc van Kreveld, July 4, 2007

Questions?