Homework 4: Half-Spaces & Randomization

NOTE: This week we will only have book problems!

Book Problems

Prepare a PDF file named hw4_halfspaces_randomization.pdf with your answers to the book problems below. You may use Latex, Google Docs (export/save as PDF), MS Word (export/save as PDF), and/or legibly handwrite on paper and scan to PDF.

From "Computational Geometry: Algorithms and Applications", de Berg, Cheong, van Kreveld, and Overmars.

You may discuss these problems with your classmates, but you must write up your solutions individually. Note the names of anyone you talked to or collaborated with in your hw4_halfspaces_randomization.pdf writeup.

Please do not search for solutions or notes published by the authors or any instructor or notes or solutions shared by other students on the internet. Any use of these types of materials is considered a violation of Academic Integrity for this course.

4.1
In this chapter we studied the casting problem for molds of one piece. A sphere cannot be manufactured in this manner, but it can be manufactured if we use a two-piece mold. Give an example of an object that cannot be manufactured with a two-piece mold, but that can be manufactured with a three-piece mold.

4.10
Let H be a set of at least three half-planes with a non-empty intersection such that not all bounding lines are parallel. We call a half-plane h ∈ H redundant if it does not contribute an edge to ∩ H. Prove that for any redundant half-plane h ∈ H there are two half-planes h', h'' ∈ H such that h' ∩ h'' ⊂ h. Give an O(n log n) time algorithm to compute all redundant half-planes.

4.13
In the text we gave a linear time algorithm for computing a random permutation. The algorithm needed a random number generator that can produce a random integer between 1 and n in constant time. Now assume we have a restricted random number generator available that can only generate a random bit (0 or 1) in constant time. How can we generate a random permutation with this restricted random number generator? What is the running time of your procedure?