Largest and Smallest Convex Hulls for Imprecise Points Maarten Löffler Marc van Kreveld Center for Geometry, Imaging and Virtual Environments Utrecht University

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

Imprecision Regions x y (1.7, 2.1)

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

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

Exact Bounds on the Area A set of points Area of hull? One possibility Upper bound Lower bound Largest hull Smallest hull 01020

Large Class of Problems Model –Circle –Square –Line segment –Other Shapes Measure –Area –Perimeter Restrictions –Same size –Same orientation –Disjoint Goal –Largest –Smallest

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)

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

Smallest Area for Squares Up to 4 vertices need not be on vertices of their squares Must be the extreme points

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

Algorithm in Action

Time Complexity O(n 2 ) pairs of endpoints Each takes linear time to compute O(n 3 ) time in total

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

Upper and Lower Chains

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

Optimal Solution

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

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