Created 2005-06-24 Modified 2007-01-01
Chelton Evans
Convex Hull Algorithms
Divide and Conquer
Algorithm
Discussion
Sublinear Time
Point Deletion
References
Partition s into two sets of points a and b as equally as possible.
Recursively find the convex hulls of a and b.
Merge the two hulls together to from hull(s).
Shamos[1] described algorithms and variations, some of which related to angle sorting which is a pain.
However Shamos described merging of two convex hulls. He referred to the property that the hull is actually already sorted in an angular sense because it is convex.
I would like to point out that this is not only an angle property. You can transverse the boundary. If two convex objects intersect the intersection is inside.
The transversal goes from visibility to the intersection where the other boundary is not visible. Continuing the iteration until the other intersection is reached and the other convex object is again visible.
The point is that the "order" property is not restricted to angles, but boundary transversal too. Since Shamos referred to algorithms that were built using the angle perspective, it should be clear that these could have equally been built using boundary transversal instead.
Comparing the basic divide-and-conquer of hulls and merge with quick-hull you will notice that its worst case is because it evenly divides the set in two before rejoining. Since quickhull has no guarantee of set division its complexity in the worst case is quadratic.
If the convex hull merge can be done in sublinear time then every things cosha.
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I suggest keeping not just the hull but the tessellation. So a transversal through the tessellations a and b could be used to find and merge two convex hulls.
Another strategy is to use point deletion. Maintain a mesh with the hull points only. I suggest a balanced mesh approach. See Convex Meshes with Hull or Boundary Points Only.
A really ambitious strategy is to use an on line Inclusion in a Convex Polygon algorithm for constant time where the function is adaptive. As new points are added the function is updated to reflect the change. It inserts the new entry and as all the entries are already sorted, all higher entries have their indexes added by one. I imagine this to be a really big job as I would guess to start looking at approximations, the heavy side function and how a function could be altered to do this.
If such a function could be built it would have dramatic impact on the problem giving constant access time.
Its beauty is that it breaks the theoretical restriction. By exploiting a finite process, so the function will only be valid for some finite number of points. Still this is powerful stuff as its not easy to break the barrier.