Today’s Material

1. Today’s Material • Computational Geometry Problems • Closest Pair Problem • Convex Hull • Jarvis’s March, Graham’s scan • Farthest Point Problem

2. Closest pair problem • Given a set of points on the plane, find the two points whose distance from each other is the maximum • What’s the brute-force solution? • An efficient divide-and-conquer solution?

3. Convex Hull (CH) • Convex Hull (CH) of a set Q of points is the smallest convex polygon P, for which each point in Q is either on the boundary of P or in its interior p10 p12 p11 p6 p9 p7 p8 p5 p4 p3 p2 p1 p0

4. Convex Hull – Jarvis’s March p10 p12 p11 p6 p9 p7 p8 p5 p4 p3 p2 p1 p0 • Take the horizontal line going through p0, and gift wrap it around the remaining points by turning the wrap to the left. • The first point touched is the next point on CH • In this example, it is p1

5. Convex Hull – Jarvis’s March p10 p12 p11 p6 p9 p7 p8 p5 p4 p3 p2 p1 p0 • Continue gift-wrapping from p1. Next point is p3

6. Convex Hull – Jarvis’s March p10 p12 p11 p6 p9 p7 p8 p5 p4 p3 p2 p1 p0 • Continue gift-wrapping from p3. Next point is p10

7. Convex Hull – Jarvis’s March p10 p12 p11 p6 p9 p7 p8 p5 p4 p3 p2 p1 p0 • Continue gift-wrapping from p10. Next point is p12

8. Convex Hull – Jarvis’s March p10 p12 p11 p6 p9 p7 p8 p5 p4 p3 p2 p1 p0 • Continue gift-wrapping from p12. Next point is p0

9. Convex Hull – Jarvis’s March p10 p12 p11 p6 p9 p7 p8 p5 p4 p3 p2 p1 p0 • Continue gift-wrapping from p12. Next point is p0. We are done now.

10. Jarvis’s March – Running Time • O(n*h), where h is the number of points on the convex hull • If h is O(logn), then we get a running time of O(nlogn) • If h is O(n), then we a quadratic running time of O(n2) • Can we make sure that the running time is always O(nlogn)? • Graham’s scan

11. Graham’s Scan • Let p0 be the point in Q with the min y-coordinate or the leftmost such point in the case of a tie • Let <p1, p2, .., pm> be the remaining points in Q sorted by polar angle in counter-clockwise order around p0. If more than one point has the same angle, remove all but the one that is farthest from p0 p10 p11 p6 p9 p7 p12 p8 p5 p4 p3 p2 p1 p0

12. Graham’s Scan • Maintain a stack S of candidate points • Each point of the input set Q is pushed once onto the stack, and the points that are not vertices of CH(Q) are eventually popped off the stack • When the algorithm terminates, S contains exactly the vertices of CH(Q) in counter-clockwise order of their appearance on the boundary

13. Graham’s Scan - Algorithm Push(p0, S); Push(p1, S); Push(p2, S); for i=3 to m do while the angle formed by points NEXT_TO_TOP(S), TOP(S) and pi makes a non-left turn do Pop(S); endwhile Push(pi, S); endfor return S;

14. Convex Hull – Initial State p10 p11 p6 p9 p7 p12 p8 p5 p4 p3 p2 p1 p2 p0 p1 p0 Stack

15. Convex Hull – p1p2p3 p10 p11 p6 p9 p7 p12 p8 p5 p4 p3 p2 p1 p2 p0 p1 p0 Stack p1p2p3 is a right turn. Pop p2, push p3

16. Convex Hull – p1p3p4 p10 p11 p6 p9 p7 p12 p8 p5 p4 p3 p2 p1 p3 p0 p1 p0 Stack p1p3p4 is a left turn. Push p4

17. Convex Hull – p3p4p5 p10 p11 p6 p9 p7 p12 p8 p5 p4 p3 p2 p1 p4 p3 p0 p1 p0 Stack p3p4p5 is a right turn. Pop p4, push p5

18. Convex Hull – p3p5p6 p10 p11 p6 p9 p7 p12 p8 p5 p4 p3 p2 p1 p5 p3 p0 p1 p0 Stack p3p5p6 is a left turn. Push p6

19. Convex Hull – p5p6p7 p10 p11 p6 p9 p7 p12 p8 p5 p4 p3 p2 p6 p1 p5 p3 p0 p1 p0 Stack p5p6p7 is a left turn. Push p7

20. Convex Hull – p6p7p8 p10 p11 p6 p9 p7 p12 p8 p5 p4 p3 p7 p2 p6 p1 p5 p3 p0 p1 p0 Stack p6p7p8 is a left turn. Push p8

21. Convex Hull – p7p8p9 p10 p11 p6 p9 p7 p12 p5 p8 p4 p3 p8 p7 p2 p6 p1 p5 p3 p0 p1 p0 Stack p7p8p9 is a right turn. Pop p8

22. Convex Hull – p6p7p9 p10 p11 p6 p9 p7 p12 p5 p8 p4 p3 p7 p2 p6 p1 p5 p3 p0 p1 p0 Stack p6p7p9 is a right turn. Pop p7, push p9

23. Convex Hull – p6p9p10 p10 p11 p6 p12 p9 p5 p7 p8 p4 p3 p9 p2 p6 p1 p5 p3 p0 p1 p0 Stack p6p9p10 is a right turn. Pop p9, p6, p5 and p6, push p10

24. Convex Hull – p3p10p11 p10 p12 p6 p11 p9 p7 p8 p5 p4 p3 p2 p1 p10 p3 p0 p1 p0 Stack p3p10p11 is a left turn. Push p11

25. Convex Hull – p10p11p12 p10 p12 p6 p11 p9 p7 p8 p5 p4 p3 p2 p11 p1 p10 p3 p0 p1 p0 Stack p10p11p12 is a right turn. Pop p11, push p12

26. Convex Hull – Final p10 p12 p6 p11 p9 p7 p8 p5 p4 p3 p2 p12 p1 p10 p3 p0 p1 p0 Stack

27. Graham’s Scan – Running Time • Computing and sorting the polar angles – O(nlogn) • The remaining scan – O(n) • Total: O(nlogn)

28. Farthest pair problem • Given a set of points on the plane, find the two points whose distance from each other is the maximum • It is shown than the farthest points must be on the Convex Hull of the points • The idea is to compute the convex hull in O(nlogn), and then find the pair that is farthest, which can be done in O(n) • So, we have a total running time of O(nlogn)