CPSC 413 Fall 2019 — Assignment 3
Due no later than Friday, November 8, 2019, 10:30am
This assignment consists of (1) one problem and (1) one bonus programming problem.
The bonus programming problem will be posted as a separate part.
Full answer (100%) is 10 points plus 6 bonus points.
Collaboration and plagiarism
You are welcome to work and discuss the assignment with other students enrolled in this course (i.e.,
CPSC 413 — Fall 2019). You must clearly state whom your collaborators are, if any, for each
problem on the assignment.
Verbal collaboration is allowed. Written collaboration is dis-allowed. No collaboration can use
any form of written material. While discussing this assignment with other enrolled students, the
following aids are strictly forbidden: papers, notes, notebooks, textbooks, webpages, email, twitter,
facebook, sms, messages, whiteboards, blackboards, etcetera. All written work that you submit must
be your own sole work. Anything else will be considered plagiarism. All collaborations must take
place on an equal footing, where all parties contribute equally to the discussion. Any conversation
must be ended if all parties are not contributing equally.
The use of published literature is allowed. If you use any published literature (texts, articles, websites,
etc) to complete your assignment, you must quote your sources. I suggest that you develop your own
solutions however, without the use of any published materials. You will be asked to answer similar
questions on the exams for this course and during the exams no such sources will be available. Note
that no collaborations or sources are allowed for the bonus programming problem.
You may read about the regulations on plagiarism in the calendar here:
http://www.ucalgary.ca/pubs/calendar/current/k-2.html. If you have any doubt whether a
collaboration is allowed or not, ask the lecturer before entering the collaboration.
Cover page
The second last page in the assignment is a cover page. Put your name and TA lab on the cover page
and submit your assignment with the cover page as the first page of your submission. Use a single
staple in the upper left corner to staple all of your sheets of paper together. Assignments diverging
from these guidelines, including being submitted without the cover page, or submitted with incorrectly
filled cover page, will be deducted 1 or more points.
Submission
You may turn in your assignment using the drop boxes on the ground floor in the math science
building (MS), or by giving your assignment in person to one of the TAs or the instructor. You must
submit your assignment on or before Friday, November 8, 2019, 10:30am. Note that 10:30am is in the
morning. No late submissions will be accepted. The deadline is firm.
On the assessment of your work
The assessment will include an evaluation of the writing and communication style. We want the
writing to be succinct, precise, unambiguous, and written with clarity. When applying and analyzing
algorithms, we want to use examples and counter-examples that are among the simplest and most
transparent instances that convey our message. We want the order and structure of our text to be
logical. You can find information and guidelines about writing about algorithms on the D2L Discussion
Board under the topic “Assignments”.
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Problems
1. String Addition Problem (10 points)
We are given a string X = x1 x2 · · · xn of length n over the alphabet {a, b, c, d} of size 4. We
can add any two symbols together according to the following table.
⊕
a
b
c
d
a
b
b
a
b
b
c
b
a
c
c
d
c
c
d
d
b
a
b
c
For instance a ⊕ b = c, b ⊕ a = b, and b ⊕ d = a. Our addition operation is not commutative
since for instance a⊕b 6= b⊕a. Our addition operation is neither associative since for instance
(a ⊕ b) ⊕ c 6= a ⊕ (b ⊕ c).
We want to know if there is a way that we can add the characters in X together so that the
resulting value is c. On the input abccbdc, the answer is “Yes” because
a((bc)c) (bd)c = a(cc) (bd)c = ac (bd)c = d (bd)c = d ac = d d = c.
Note that we are not allowed to swap elements in the string X.
String Addition Problem
Input:
a string X = x1 x2 . . . xn of length n over the alphabet {a, b, c, d}.
Output: “Yes” if it is possible to parenthesize the string X in such a way that the string
evaluates to c.
“No” otherwise.
1. Give a polynomial-time (in n) dynamic programming algorithm for the String Addition
Problem problem. State your algorithm clearly and succinctly in the following order:
(a) Very clearly state the recurrences you are using. Justify your choice of recurrences.
(b) Explain the base cases of your recurrences.
(c) Explain how you compute the output of your algorithm from your recurrences.
(d) Show how you implement your recursive algorithm in polynomial time using a twodimensional array A. Clearly state what each entry A[i, j] contains.
(e) Argue why your algorithm is correct. (Or, if it is not correct, explain why and what
may be missing.)
(f) Briefly explain why your algorithm runs in time polynomial in n.
(g) Using the back side of the cover page, for the input abccbdc, show the complete
array A that is generated by your own algorithm.
It suffices that your algorithm outputs the correct answer “Yes” or “No”. Your algorithm
does not need to output the parenthesization.
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Cover page
CPSC 413 Assignment 3 Fall 2019
Name:
Lab section:
T01
Mon 15:00
T02
Tue 11:00
T03
Tue 14:00
T04
Mon 17:00
T05
Mon 11:00
My sources and my collaborators, if any, on the problems were:
Problem 1:
Bonus programming problem:
No collaboration permitted
On the input abccbdc, my dynamic programming algorithm produces the following array A:
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