Largest and Smallest Convex Hulls for Imprecise Points

1
Largest and Smallest Convex Hulls for Imprecise
Points

• Maarten Löffler Marc van Kreveld

Center for Geometry, Imaging and Virtual
Environments Utrecht University
2
Introduction

• Many geometric problems take a point set as input
• Theoretic algorithms assume points to be known
  exactly
• In practice, points are imprecise
• Obtained by measuring the real world
• Result from inexact computation

3
Imprecision Regions
y
2
(1.7, 2.1)
1
0
x
0
1
2
4
Exact Convex Hull

• Given point set
• Smallest convex set containing all points
• Computable in O(nlogn) time
• Solved long ago

5
Imprecise Convex Hull

• Given a set of imprecise points
• What is the convex hull?
• Many possible
• Exact bounds on the area

6
Exact Bounds on the Area

• A set of points
• Area of hull?
• One possibility
• Upper bound
• Lower bound
• Largest hull
• Smallest hull

0
10
20
7
Large Class of Problems

• Model
• Circle
• Square
• Line segment
• Other Shapes
• Measure
• Area
• Perimeter

• Restrictions
• Same size
• Same orientation
• Disjoint
• Goal
• Largest
• Smallest

8
Results
9
Largest Area for Squares

• All vertices of the convex hull must be corners
  of their squares
• Otherwise, we can move them to increase area

10
Smallest Area for Squares

• Up to 4 vertices need not be on vertices of their
  squares
• Must be the extreme points

11
Largest Area Line Segment

• Parallel line segments
• Each segment has two potential points on the hull
• Dynamic programming approach
• For each pair of an upper and a lower endpoint,
  compute the optimal subsolution

12
Algorithm in Action
13
Time Complexity

• O(n2) pairs of endpoints
• Each takes linear time to compute
• O(n3) time in total

14
Smallest Area Line Segments

• Parallel vertical line segments
• The upper chain is the upper half of the convex
  hull of all lower endpoints of the segments
• Symmetrically the lower chain
• The chains enclose the greatest common
  substructure

15
Upper and Lower Chains
16
Algorithm

• Leftmost and rightmost points can move over their
  line segments
• Connect them to their tangent points on the
  chains
• Move them to their optimal positions
• Independent

17
Optimal Solution
18
Time Complexity

• Compute the chains in O(nlogn) time
• Move to their optimal positions in O(n) time
• Total O(nlogn) time

19
Conclusions

• Smallest convex hull easier than largest convex
  hull
• Fewer restrictions
• Better time bounds
• Area easier for largest, perimeter easier for
  smallest convex hull