CS 4102, Algorithms
Problems Set #3, Spring 2010
Updated: Feb. 19, 2010, 5:15pm
Homework 3 is a "short" homework, worth less the HWs 1 and 2.
There are fewer problems to turn in, and most are simpler. Also,
you can work with up to three people in a group. The main goal
for HW3 is to make you look at the kind of problems from the most
recent part of the course that might be on Exam 1.
To turn in:
Turn in solutions to these problems. (Below they're marked
with this image: 
)
- Problem Rec2
(was part of and counts for HW2)
- Algorithms textbook: p. 253, Problem 13
- Any two of these three: Sort-LB-1, Invers-1, QS-Med3-1
Important: Put the answer to each question on a separate sheet of
paper (or one problem on front and another
on the back). This is to make grading easier for us.
Collaboration Rules for This Assignment:
- There are many problems here. There are five to submit. For
the ones that are not required
to be submitted, you can collaborate or discuss in any way with as many
people as you wish.
- For the required problems that you submit:
- You may submit your work on your own or with up to two partners.
If you work with a partner, you must pledge that all partners did
approximately equal amounts of work on the required problems.
- If you work alone, you can discuss how to solve the problems with one
other group or one other single person. List the names of these
people on your submission.
- If you don't understand the question or what it is asking, you can get
clarification from other students only
if this does not involve saying how
the problem might be solved in any way.
If in doubt, don't discuss and see the TA or the instructor.
- Fill out the cover sheet on the website and submit
it with your homework.
Problems from Algorithms
textbook
(Solutions for problems marked "S" begin on page 656)
- Section 6.1, pp. 241f. Problems 6, 14, 16S, 17, 19
- Section 6.2, pp. 252f. Problems 1S, 2, 3, 13 , 17S,
18, 19
- Section 6.3, pp. 256f. Problem 4S, 5
(Posted as Part of HW2) Problem Rec2:
(The grade will count as part of HW2.)
Use the either the Main Theorem or Master
Theorem to give the order-class for each of the
following recurrences. If you can't use either one of the theorems for
a problem, just write "not applicable"
and explain why.
(A) T(n) = 2T(n/2) + n
(B) T(n) = 2T(n/4) + n lg n
(C) T(n) = T(n/2) + lg n
Submit any two of the next three problems
-- you chose!
Problem Sort-LB-1:
Using a decision tree argument, state what the lower bound for the
number of comparisons would be for an array of size 5. (This is the
first part of Problem 5 on p. 256.)
Problem Invers-1:
Describe a divide
and conquer algorithm that counts the number of inversions in a list in
o(n2) operations, and give the order-class (i.e. the
BigTheta class) of your solution.
Big Hint: To do this, take mergesort and modify it or add code to
it in the merge algorithm. You will need to count something in merge
(not mergesort) that contributes to the overall sum of the total number
of inversions. Think about how merge works! Look at simple
example data.
Note: By "describe", I mean that you do not have to describe the
complete code or pseudo-code if you can clearly explain how merge would
be modified.
Problem QS-Med3-1:
Let's say you implement Quicksort with the median-of-three method for
choosing the pivot. Write the recurrence relation for the
worst-case performance for Quicksort with this enhancement.
Assume you are using a partition algorithm that requires n-1 comparisons for a list of size
n. Don't forget to
include the cost (if any) of doing the median-of-three. Note: You
do not have to reduce the
recurrence to closed form.