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

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

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

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Imprecise Convex Hull

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

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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
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Large Class of Problems

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

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

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Results
goal measure model restrictions time
largest area line segments parallel O(n3)
largest area squares disjoint O(n7)
largest area squares unit O(n5)
largest area squares disjoint, unit O(n3)
largest perimeter line segments parallel O(n5)
largest perimeter squares disjoint O(n10)
smallest area line segments parallel O(nlogn)
smallest area squares O(n2)
smallest perimeter line segments parallel O(nlogn)
smallest perimeter squares O(nlogn)
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Largest Area for Squares

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

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Smallest Area for Squares

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

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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

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Algorithm in Action
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Time Complexity

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

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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

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Upper and Lower Chains
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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

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Optimal Solution
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Time Complexity

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

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Conclusions

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