## Submission Instructions

Handin your solutions using handin. You can write your solutions by hand and scan the pages or take pictures of them with your phone; or use a word processing package to typeset your solutions.
Whatever you do, you should produce pdf files assign2Q1.pdf, assign2Q2.pdf, etc. containing your solutions for Question 1, Question 2, etc.
To handin: Copy your solution files to the directory /cs420/a2 in your home directory on a CS undergraduate machine. (You may have to create this directory using mkdir /cs420/a2.)
Then run handin cs420 a2 from your home directory.

We will grade a subset of these questions of size at least two. It’s a good idea to do all of the questions because (1) you don’t know which ones we’ll grade and (2) answering these questions is good practice for the exams. We will email feedback to the email address associated with the account you used to handin the assignment.

## Questions

Try to answer these on your own but if you work with someone or use an outside source you must acknowledge them in your write-up.

• The depth of a point p in a multi-set of points P in the plane (there might be duplicate points) is 0 if p is on the boundary of the convex hull of P , otherwise the depth of p is one plus the depth of p in the set P without the points on the boundary of the convex hull of P . Point depth is indicated in the figure.
Prove that any algorithm that calculates the maximum three duplicate points depth of a point in P takes (n log n) time, where n = |P|, in the algebraic decision tree model of computation.
You may use the fact that solving Element Uniqueness takes (n log n) time in the algebraic decision tree model.
• Voronoi diagram. Let S = {s1 , s2 , …. , sn} be a set of n points in the plane. A Voronoi diagram partitions the plane into Voronoi regions R1 , R2 , … , Rn where Ri is the set of points in the plane whose closest site is si , that is, Ri = {p | d(p, si) ≤ d(p, sj) for all j}.
Prove that every Voronoi region is convex. Remember that a set R is convex if for any two points a and b in R, the line segment connecting a and b is in R; and the intersection of convex sets is convex.
• Delaunay Triangulation The Delaunay triangulation of a set S of points in the plane is a partition of the convex hull of S into triangles whose vertices are the points of S such that the circumcircle of any triangle in the partition contains no points of S in its interior. The following figure shows the Delaunay triangulation, with bold lines, and the Voronoi diagram of a set of points. As you can see, the Delaunay triangulation is closely related to the Voronoi diagram.
Suppose the points are colored white and black. Prove that the closest pair of points of different colors in S is an edge of a triangle in the Delaunay triangulation. (This should be a very short proof. You may use the fact that pq for p, q S is an edge of a Delaunay triangle for S if and only if there exists a circle through p and q that contains no points in S.)
• (from Exercise 5.4.5.2 from “Computational Geometry in C” by J. O’Rourke) One-dimensional
Voronoi diagrams. Let S = s1 [ s2 [ [ sn be a sequence of real numbers. We think of these as Voronoi sites on the x-axis. A one-dimensional Voronoi diagram of S is a sequence of real numbers (m1, m2 , … , mn1) such that mi is the midpoint of si si+1.
Suppose you are given a sequence M = (m1 , m2 , … , mn1) of real numbers. Describe an efficient algorithm that finds a sequence of n sites for which M is the Voronoi diagram, or reports that there is no such sequence.
• Kuba’s March is an algorithm to calculate the convex hull of a set of points P in the plane. Here it is:

(a) Explain why Kuba’s March takes O(nh) time where n = |P| and h is the number of points on the convex hull of P .
(b) As in Chan’s algorithm, describe how to find the convex hull of dn/he convex hulls, each of size at most h using Graham’s Scan and Kuba’s March in O(n log h) time.

• Consider the set Z of inputs x1 , x2 , . . . , xn (where xi R for all i) that are NO-inputs of Element Uniqueness (i.e., xi = xj for some i 6= j). How many connected components does Z have? Explain why.
• Let I = {(x1, y1), (x2, y2), … , (xn, yn)} be a set of n intervals on the positive real line: 0 xi yi . Interval (xi , yi) is contained in interval (xj , yj) if xj xi and yi yj . We are interested in calculating the number of pairs of intervals in I such that one interval is contained in the other.
• (a) Describe an algorithm that solves this problem in O(n log n) time in the linear decision tree model of computation. Hint: There is a divide and conquer solution when the intervals are viewed as points in the plane. (Please use English and pseudo-code to describe your algorithm. You do not need to describe the algorithm using a linear decision tree. In fact, a linear decision tree only works for a particular input size, while your algorithm should work for any input size.)
• (b) Prove that any algorithm (in the linear decision tree model) that solves this problem takes (n log n) time. Hint: Use a reduction.