1. Convex Hulls
Given a set of points (x1,y1),(x2,y2),…,(xn,yn), the convex hull is the smallest convex polygon containing all the points.

2. Convex Hulls
Given a set of points (x1,y1),(x2,y2),…,(xn,yn), the convex hull is the smallest convex polygon containing all the points.

3. Convex Hulls
Gift-wrapping algorithm running in time O(n2), or, more precisely, O(nk) where k is the number of vertices on the hull.
An O(n log n) algorithm ?

4. Convex Hulls
The Graham Scan algorithm
Running time ?

5. Convex Hulls
A divide-and-conquer algorithm ?
Running time ?

6. Closest Pair of Points Input: n points in the plane (x1,y1),…,(xn,yn)
Output: distance of the closest pair of points, i.e., mini,j (i j) distance of (xi,yi) and (xj,yj)

7. Closest Pair of Points Input: n points in the plane (x1,y1),…,(xn,yn)
Output: distance of the closest pair of points, i.e., mini,j (i j) distance of (xi,yi) and (xj,yj)

8. Closest Pair of Points Input: n points in the plane (x1,y1),…,(xn,yn)
Output: distance of the closest pair of points, i.e., mini,j (i j) distance of (xi,yi) and (xj,yj)