UMass Lowell Computer Science Advanced Algorithms Computational Geometry Prof. Karen Daniels Spring, 2007 Lecture 3 Chapter 3: 2D Convex Hulls Friday, 2/9/07

Chapter 3 2D Convex Hulls Definitions Gift Wrapping Graham Scan QuickHullIncrementalDivide-and-Conquer Lower Bound in  (nlgn)

Convexity & Convex Hulls  A convex combination of points x 1,..., x k is a sum of the form  1 x  k x k where ä Convex hull of a set of points is the set of all convex combinations of points in the set. nonconvex polygon convex hull of a point set source: O’Rourke, Computational Geometry in C source: textbook Cormen et al. We will construct boundary of convex hull.

Naive Algorithms for Extreme Points Algorithm: INTERIOR POINTS for each i do for each j = i do for each j = i do for each k = j = i do for each k = j = i do for each L = k = j = i do for each L = k = j = i do if p L in triangle(p i, p j, p k ) then p L is nonextreme O(n 4 ) Algorithm: EXTREME EDGES for each i do for each j = i do for each j = i do for each k = j = i do for each k = j = i do if p k is not (left or on) (p i, p j ) if p k is not (left or on) (p i, p j ) then (p i, p j ) is not extreme O(n 3 ) source: O’Rourke, Computational Geometry in C

Algorithms: 2D Gift Wrapping ä Use one extreme edge as an anchor for finding the next  O(n 2 ) Algorithm: GIFT WRAPPING i 0 index of the lowest point i i 0 repeat for each j = i for each j = i Compute counterclockwise angle  from previous hull edge Compute counterclockwise angle  from previous hull edge k index of point with smallest  k index of point with smallest  Output (p i, p k ) as a hull edge Output (p i, p k ) as a hull edge i k i k until i = i 0 source: O’Rourke, Computational Geometry in C

Gift Wrapping source: textbook Cormen et al Output Sensitivity: O(n 2 ) run-time is actually O(nh) where h is the number of vertices of the convex hull.

Algorithms: 3D Gift Wrapping CxHull Animations: O(n 2 ) time [output sensitive: O(nF) for F faces on hull]

Algorithms: 2D QuickHull ä Concentrate on points close to hull boundary ä Named for similarity to Quicksort a b O(n 2 ) Algorithm: QUICK HULL function QuickHull(a,b,S) if S = 0 return() if S = 0 return() else else c index of point with max distance from ab c index of point with max distance from ab A points strictly right of (a,c) A points strictly right of (a,c) B points strictly right of (c,b) B points strictly right of (c,b) return QuickHull(a,c,A) + (c) + QuickHull(c,b,B) return QuickHull(a,c,A) + (c) + QuickHull(c,b,B) source: O’Rourke, Computational Geometry in C finds one of upper or lower hull c A

Algorithms: 3D QuickHull CxHull Animations:

Algorithms for Qhull: Convex Hull boundary is intersection of hyperplanes, so worst-case combinatorial size (not necessarily running time) complexity is in:

Graham’s Algorithm ä Points sorted angularly provide “star-shaped” starting point ä Prevent “dents” as you go via convexity testing O(nlgn) Algorithm: GRAHAM SCAN, Version B Find rightmost lowest point; label it p 0. Sort all other points angularly about p 0. In case of tie, delete point(s) closer to p 0. In case of tie, delete point(s) closer to p 0. Stack S (p 1, p 0 ) = (p t, p t-1 ); t indexes top i 2 while i < n do if p i is strictly left of p t-1 p t then Push(p i, S) and set i i +1 else Pop(S)  source: O’Rourke, Computational Geometry in C “multipop” p0p0p0p0

Graham Scan source: textbook Cormen et al.

Graham Scan source: textbook Cormen et al. 33.7

Graham Scan source: textbook Cormen et al. 33.7

Graham Scan source: textbook Cormen et al.

Graham Scan source: textbook Cormen et al.

Algorithms: 2D Incremental ä Add points, one at a time ä update hull for each new point ä Key step becomes adding a single point to an existing hull. ä Find 2 tangents ä Results of 2 consecutive LEFT tests differ ä Idea can be extended to 3D. O(n 2 ) Algorithm: INCREMENTAL ALGORITHM Let H 2 ConvexHull{p 0, p 1, p 2 } for k 3 to n - 1 do H k ConvexHull{ H k-1 U p k } H k ConvexHull{ H k-1 U p k } can be improved to O(nlgn) source: O’Rourke, Computational Geometry in C

Algorithms: 3D Incremental CxHull Animations: O(n 2 ) time

Algorithms: 2D Divide-and-Conquer ä Divide-and-Conquer in a geometric setting ä O(n) merge step is the challenge ä Find upper and lower tangents ä Lower tangent: find rightmost pt of A & leftmost pt of B; then “walk it downwards” ä Idea can be extended to 3D. Algorithm: DIVIDE-and-CONQUER Sort points by x coordinate Divide points into 2 sets A and B: A contains left n/2 points B contains right n/2 points Compute ConvexHull(A) and ConvexHull(B) recursively Merge ConvexHull(A) and ConvexHull(B) O(nlgn) A B source: O’Rourke, Computational Geometry in C

Algorithms: 3D Divide and Conquer CxHull Animations: O(n log n) time !

Lower Bound of O(nlgn)  Worst-case time to find convex hull of n points in algebraic decision tree model is in  (nlgn) ä Proof uses sorting reduction: ä Given unsorted list of n numbers: (x 1,x 2,…, x n ) ä Form unsorted set of points: (x i, x i 2 ) for each x i ä Convex hull of points produces sorted list! ä Parabola: every point is on convex hull ä Reduction is O(n) (which is in o(nlgn))  Finding convex hull of n points is therefore at least as hard as sorting n points, so worst-case time is in  (nlgn) Parabola for sorting 2,1,3 source: O’Rourke, Computational Geometry in C