9/4/08: Convex hulls in 2D for n points. (Chapter 1 of BKOS)

Photos of chalk boards

Convex hulls of points in the plane. Definitions of convex combinations, convex set, convex hull, properties of convex hulls. Extreme points.
Naive algorithms:

(0). SuperDuperStupidCH takes exponential time (enumerate all subsets of points and all orderings of them; test each to see if it is correct),

(1). SuperStupidCH takes time O(n^4),

(2). StupidCH takes time O(n^3).

Importance of two key issues in CG: (a) Degeneracies, and (b) Robustness (from evaluating geometric predicates using inexact, fixed precision arithmetic).

(3). NEXT TIME: Output-sensitive algorithm for "gift wrapping" (Jarvis march): O(nh) (worst case O(n^2)). Note that the expected size of the hull, E(h), is only O(log n) for n points uniformly distributed in a square.

NEXT TIME: (4). Graham Scan algorithm takes time O(n log n), exploiting sorting.