Due: 12/08/2005 (Last Class)

- [1.] Prove the following statement:
For any point set P, the Delaunay triangulation DT(P)
is an optimal solution to the following set of
optimization problems.
- [a.] Minimize the largest circum-circle of all triangles in a triangulation of P.
- [b.] Minimize the largest enclosing circle of all triangles in a triangulation of P, where a circle C encloses a triangle t, if t is contained in C.
- [c.] Maximize the sum of the radii of inscribed-circles of all triangles in a triangulation of P. (Hint: if you think this the last one is too hard, you should try to find about it on line)

- [2] Let S = {1,2,3,...,n}. Suppose you build a binary search
tree of S By first uniformly randomly permuting S,
then incrementally adding elements from S according this
permutation into an initially empty tree. Use the backward
analysis to show that your randomized process builds the
binary search tree for S in expected O(nlog n) time.
- [3] Suppose X = {1,2,...,n} is a set of n numbers.
If we choose a random element x from X , then x is a
(1/4)-median with probability 1/2.
Now suppose we choose a sample of S of k random elements from
X and let s be the median of S , then what is the probability
that s is a c-median of X , where 0 < c < 1/2.
You can write a program to run an experiment first to
set up your conjecture before you prove it.
Applying approximation to make your expression as simple as possible.
- [4] Similar to the "nearest-neighbor" Voronoi Diagram, we can also
defin so called the "farthest-neighbor" Voronoi diagram of a
point set P. Show that
this new diagram has the following properties:
- A non-empty Voronoi region is associated with a pont q in P if and only if q is an extreme point of P.
- Every Voronoi region is a convex unbounded region.
- Assume there exist no four points co-circlear, then the "farthest-neighbor" diagram is a 3-degree tree.
- the projection of upper convex hull under the lifting
mapping (x,y) -> (x, y, x
^{2}+ y^{2}) back onto xy-plane gives the DUAL "fathest-neighbor" Voronoi diagram.