Computational Geometry 2D Convex Hulls. Joseph S. B. Mitchell Stony Brook University

Transcription

1 Computational Geometry 2D Convex Hulls Joseph S. B. Mitchell Stony Brook University

2 Comparing O(n), O(n log n), O(n 2 ) n n log n n² ³ Interactive Processing n log n algorithms n² algorithms n = 1000 yes? n = ? no 2

3 Convexity p p Set X is convex if p,q X pq X Point p X is an extreme point if there exists a line (hyperplane) through p such that all other points of X lie strictly to one side q convex q non-convex Extreme points in red q r p Fact: If X=S is a finite set of points in 2D, then CH(X) is a convex polygon whose vertices (extreme points) are points of S. 3

4 Fundamental Problem: 2D Convex Hulls Input: n points S = (p 1, p 2,, p n ) More generally: CH(polygons) p 2 p 8 p 7 p 3 p 4 p 5 p 1 p 6 Output: (9,6,4,2,7,8,5) p 9 Output: A boundary representation, e.g., ordered list of vertices (extreme points), of the convex hull, CH(S), of S (convex polygon) 4

5 Equivalent Definitions of Convex Hull, CH(X) {all convex combinations of d+1 points of X } [Caratheodory s Thm] (in any dimension d) T T X, T convex H X H, Hhalfspace Set-theoretic smallest convex set containing X. In 2D: min-area (or min-perimeter) enclosing convex body containing X In 2D: abc a, b, c X 5

6 2D Convex Hull Algorithms O(n 4 ) simple, brute force (but finite!) O(n 3 ) still simple, brute force O(nh) simple, output-sensitive h = output size (# vertices) O(n log n) worst-case optimal (as fcn of n) O(n log h) ultimate time bound (as fcn of n,h) Randomized, expected O(n log n) Lower bound: (n log n) From SORTING: Simple, elegant y= x 2 (x i,x i 2 ) Note: Even if the output of CH is not required to be an ordered list of vertices (e.g., just the # of vertices), (n log n) holds x 6

7 Primitive Computation Left tests: sign of a cross product (determinant), which determines the orientation of 3 points Time O(1) ( constant ) Left( a, b, c ) = TRUE ab ac > 0 a c b c is left of ab 7

8 SuperStupidCH: O(n 4 ) Fact: If s pqr, then s is not a vertex of CH(S) q p q p r p,q O(n 3 ) s p,q,r: If s pqr then mark s as NON-vertex s p r O(n) Output: Vertices of CH(S) Can sort (O(n log n) ) to get ordered 8

9 StupidCH: O(n 3 ) Fact: If all points of S lie strictly to one side of the line pq or lie in between p and q, then pq is an edge of CH(S). p q p O(n 2 ) r p,q: If r red then mark pq as NON-edge (ccw) Output: Edges of CH(S) O(n) Can sort (O(n log n) ) to get ordered 9 Caution!! Numerical errors require care to avoid crash/infinite loop! q r p Applet by Snoeyink

10 O(nh) : Gift-Wrapping Idea: Use one edge to help find the next edge. Jarvis March r O(n) per step h steps q Total: O(nh) Output: Vertices of CH(S) Demo applet of Jarvis march p Key observation: Output-sensitive! 10

11 O(n log n) : Graham Scan Idea: Sorting helps! Start with v lowest (min-y), a known vertex O(n) Sort S by angle about v lowest Graham scan: O(n log n) O(n) Maintain a stack representing (left-turning) CH so far If p i is left of last edge of stack, then PUSH Else, POP stack until it is left, charging work to popped points Demo applet v lowest CH so far 12

12 O(n log n) : Divide and Conquer Split S into S left and S right, sizes n/2 Recursively compute CH(S left ), CH(S right ) Merge the two hulls by finding upper/lower bridges in O(n), by wobbly stick Time O(n) Time 2T(n/2) Time O(n) S left S right Time: T(n) 2T(n/2) + O(n) T(n) = O(n log n) Demo applet 13

13 QuickHull Applet (programmed by Jeff So) Applet (by Lambert) 14

14 QuickHull QuickHull(a,b,S) to compute upperhull(s) If S=, return () Else return (QuickHull(a,c,A), c, QuickHull(c,b,B)) c c = point furthest from ab Discard points in abc Worst-case: O(n 2 ) Avg: O(n) Works well in higher dimensions too! Qhull website a B A S Qhull, Brad Barber (used within MATLAB) b 15

15 O(n log n) : Randomized Incremental Add points in random order Keep current Q i = CH(v 1,v 2,,v i ) Add v i+1 : If v i+1 Q i, do nothing Else insert v i+1 by finding tangencies, updating the hull Expected cost of insertion: O(log n) 16

16 Each uninserted v j Q i (j>i+1) points to the bucket (cone) containing it; each cone points to the list of uninserted v j Q i within it (with its conflict edge) Add v i+1 Q i : Start from conflict edge e, and walk cw/ccw to establish new tangencies from v i+1, charging walk to deleted vertices Rebucket points in affected cones (update lists for each bucket) Total work: O(n) + rebucketing work E(rebucket cost for v j at step i) = O(1) P(rebucket) O(1) (2/i ) = O(1/i ) e v i+1 v j E(total rebucket cost for v j ) = O(1/i ) = O(log n) Total expected work = O(n log n) Backwards Analysis: v j was just rebucketed iff the last point inserted was one of the 2 endpoints of the (current) 17 conflict edge, e, for v j

17 More Demos Various 2D and 3D algorithms in an applet 18

18 More CG Applets Applets of algorithms in O Rourke s book Applets from Jack Snoeyink 19