Homework Assignment 5
Due: Mar. 2 before class starts
Assignment
-
(50 points) What is the best, average and worst-case asymptotic time
complexity for performing
n
(correct)remove()
operations on a map that initially contains2n
entries and implemented with:- Unordered list
- Direct address table
- Sorted search table
- Hash table with chaining
- Hash table with linear probing
- Hash table with double hashing
-
(50 points) Suppose that each row of an
n × n
arrayA
consists of 1's and 0's such that, in any row ofA
, all the 1's come before any 0's in that row. AssumingA
is already in memory, describe a method running inO(n log n)
time (notO(n2)
time) for counting the number of 1's inA
.- Provide and explain pseudocode
- Statement and proof of time complexity
- Statement and proof of memory complexity
-
Bonus.
(10 points) The quadratic probing strategy has a clustering problem that
relates to the way it looks for open slots when a collision occurs. Namely,
when a collision occurs at bucket
h(k)
, we checkA[(h(k) + i2) mod N]
, fori = 1,2,...,N - 1
.- Show that
i2 mod N
will assume at most(N + 1)/2
distinct values, forN
prime, asi
ranges from1
toN - 1
. As a part of this justification, note thati2 mod N = (N - i)2 mod N
for alli
. - A better strategy is to choose a prime
N
, such thatN mod 4 = 3
and then to check the bucketsA[(h(k) ± i2) mod N]
asi
ranges from1
to(N - 1)/2
, alternating between plus and minus. Show that this alternate version is guaranteed to check every bucket inA
.
- Show that
General Instructions, Turning in assignments, and Grading
General Instructions
- All homework assignments must be typeset using LaTex. Using another program (e.g., Microsoft Word) or handwriting assignments will result in grade of 0. Here is an example homework problem worked out to see the expected format and the tex file that generated it to help you start.
- If you are unfamiliar with LaTex, here is a short tutorial and the tex file that generated it. This tutorial shows how to write common mathematics, how to write pseudocode, how to cite sources, how to include images, and how to include tables.
- Use pseudocode and good mathematical style to describe algorithms and data structures. Do not specify Java code unless otherwise stated. Described more in example homework document above.
- Complete every part of every problem!
- Follow turn-in instructions precisely.
- Failure to complete any of these steps will result in a significant loss of points.
Turn in Instructions
Each assignment will be turned in in class (hard copy). Assignments are due BEFORE, let me repeat, before class starts. This does not mean five minutes after class starts. Details:
- The first page of your hard copy must be a signed coverpage.
- Next put the problems in order as described in the description. This may seem silly but you would be surprised sometimes.
- If you do not know how to print you may have to consult University of Richmond webpages to learn how to use campus printers. If possible, please print 2-sided.
- Staple all pages together.
- Turn in packet before class begins.
- I reserve the right to assign a 0 to any assignment failing to comply with these instructions. Even for something as small as a missing staple.
Points
- Each assignment is graded out of 100 points (not including bonus).
- Criteria and point distribution (This distribution is a general guideline
that may change depending on the specific problem - exceptions for example
include but are not limited to proof-only problems.)
- Approximately 50% on a problem is devoted to the algorithm. This includes items like describing a correct and efficient algorithm, using proper and clear pseudocode, and describing your algorithm clearly.
- Approximately 25% on a problem is devoted to time complexity. This includes stating a lemma or theorem of your result and providing a proof and justification of the result.
- Approximately 25% on a problem is devoted to memory complexity. This includes stating a lemma or theorem of your result and providing a proof and justification of the result.
- If the problem is simply a proof, 100% is split between a proper statement (lemma or theorem) of the result and proving a clear and concise proof of the result.
- You can receive points off for not following formatting guidelines.
- If there are any discrepencies in grades please see the instructor during his office hours or by appointment (do not discuss with the lab assistants or graders).