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 x y (1.7, 2.1) 012 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 01020

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 goalmeasuremodelrestrictionstime largestarealine segmentsparallelO(n3)O(n3) largestareasquaresdisjointO(n7)O(n7) largestareasquaresunitO(n 5 ) largestareasquaresdisjoint, unitO(n3)O(n3) largestperimeterline segmentsparallelO(n5)O(n5) largestperimetersquaresdisjointO(n 10 ) smallestarealine segmentsparallelO(nlogn) smallestareasquaresO(n2)O(n2) smallestperimeterline segmentsparallelO(nlogn) smallestperimetersquaresO(nlogn)

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(n 2 ) pairs of endpoints Each takes linear time to compute O(n 3 ) 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