Input for Jarvis algorithm so that is faster than Graham's (convex hull)

Question

Jarvis: This algorithm requires O(nh) time in the worst case for n input points with h extreme points.

Graham: O(nlogn) in the worst case scenario.

Source the ref of CGAL, from where I use the two algorithms.

That means that Jarvis can be faster for a dataset (let's say in 2 dimensions), when h is less than logn. However I would like to see it in action, but I fail in finding a dataset for this purpose. Does anybody know?

Googling yield this link, which actually supports what I am claiming above.

Answer 1

I just did similar stuff a while ago, so I'm posting answer even though there is an accepted answer, just for the numbers...

Using CGAL's implementation with 10^6 points and 3 points on the hull, Graham takes ~150ms and Jarvis ~87ms, see setup (blue square are all the other points):

3 points on the hull:

points| Jarvis | Graham
10^7  | 850ms  | 1820ms
10^6  | 87ms   | 150ms
10^5  | 10ms   | 15ms

5 points on the hull:

points| Jarvis  | Graham
10^7  | 1500ms  | 1820ms
10^6  | 139ms   | 150ms

6 points on the hull:

points| Jarvis  | Graham
10^7  | 2560ms  | 1820ms
10^6  | 170ms   | 150ms

But besides these few special cases, Graham is much faster than Jarvis.

Answer 2

Let's assume that we have a big traingle and a lot of points inside it. The number of points on the hull(that is, h) is 3. If the number of points inside is really big, then h = 3 is less than log n. Jarvis would be much faster here.