# CAS CS 532 Computational Geometry

## 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, x2 + y2) back onto xy-plane gives the DUAL "fathest-neighbor" Voronoi diagram.