Algorithms and Analysis
COSC 2123/1285
Assignment 2: Algorithm Design & Complexity Analysis
Assessment Type Individual Assignment. Submit online via Canvas → Assignments → Assignment 2. Clarifications/updates/FAQs can be
found in Ed Forum → Assignment 2: General Discussions.
Due Dates Deadline 1 Week 11, October 4, 23:59, for Problems 1-3,
Deadline 2 Week 12, October 16, 23:59, for Problems 4-5
Marks 40
IMPORTANT NOTES
• If you are asked to design an algorithm, you need to describe it in plain English first, say a paragraph, and then provide an unambiguous pseudo code,
unless specified otherwise. The description must include enough details to understand how the algorithm runs and what the complexity is roughly. All algorithm
descriptions and pseudo codes required in this assignment are at most half a page
in length. Worst-case complexity is assumed unless specified otherwise.
• Standard array operations such as sorting, linear search, binary search, sum,
max/min elements, as well as algorithms discussed in the pre-recorded lectures
can be used straight away (but make sure to include the input and output if you
are using them as a library). However, if some modification is needed, you have to
provide a full description. If you are not clear whether certain algorithms/operations are standard or not, post it to Ed Discussion Forum or drop Hoang a Team
message.
• Marks are given based on correctness, conciseness (with page limits), and clarity of your answers. If the marker thinks that the answer is completely not understandable, a zero mark might be given. If correct, ambiguous solutions may still
receive a deduction of 0.5 mark for the lack of clarity.
• Page limits apply to ALL problems in this assignment. Over-length answers may
attract mark deduction (0.5 per question). We do this to (1) make sure you develop
a concise solution and (2) to keep the reading/marking time under control. Please
do NOT include the problem statements in your submission because this
may increase Turnitin’s similarity scores significantly.
• This is an individual assignment. While you are encouraged to seek clarifications
for questions on Ed Forum, please do NOT discuss solutions or post hints leading
to solutions.
• In the submission (your PDF file), you will be required to certify that the submitted
solution represents your own work only by agreeing to the following statement:
I certify that this is all my own original work. If I took any parts from
elsewhere, then they were non-essential parts of the assignment, and they
are clearly attributed in my submission. I will show that I agree to this
honour code by typing “Yes":
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1 Part I: Fundamental
Problem 1 (8 marks, 1 page). Consider the algorithm mystery() whose input consists
of an array A of n integers, two nonnegative integers `,u satisfying 0 ≤ ` ≤ u ≤ n−1, and
an integer k. We assume that n is a power of 2.
Algorithm mystery(A[0,...,(n−1)],`,u,k)
if ` == u then
if A[`] == k then
return 1;
else
return 0;
end if
else
m = b(`+ u −1)/2c;
return mystery(A,`,m,k)+mystery(A,m+1,u,k);
end if
a) [2 marks] What does mystery(A[0..(n −1)],0,n −1,k) compute (0.5 mark)? Justify
your answer (1.5 marks).
b) [1 mark] What is the algorithmic paradigm that the algorithm belongs to?
c) [2 marks] Write the recurrence relation for C(n), the number of additions required
by mystery(A,0,n−1,k).
d) [2 marks] Solve the above recurrence relation by the backward substitution method
to obtain an explicit formula for C(n) in n.
e) [1 mark] Write the complexity class that C(n) belongs to using the Big-Θ notation.
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Problem 2 (8 marks, 1.5 pages). Let A be an array of n integers.
a) [2 marks] Describe a brute-force algorithm that finds the minimum difference between two distinct elements of the array, where the difference between a and b
is defined to be |a − b| [1 mark]. Analyse the time complexity (worst-case) of the
algorithm using the big-O notation [1 mark]. Pseudocode/example demonstration
are NOT required. Example: A = [3,−6,1,−3,20,6,−9,−15], output is 2 = 3-1.
b) [2 marks] Design a transform-and-conquer algorithm that finds the minimum difference between two distinct elements of the array with worst-case time complexity
O(nlog(n)): description [1 mark], complexity analysis [1 mark]. Pseudocode/example demonstration are NOT required. If your algorithm only has average-case
complexity O(nlog(n)) then a 0.5 mark deduction applies.
c) [4 marks] Given that A is already sorted in a non-decreasing order, design an algorithm with worst-case time complexity O(n) that outputs the absolute values
of the elements of A in an increasing order with no duplications: description and
pseudocode [2 marks], complexity analysis [1 mark], example demonstration on
the provided A [1 mark]. If your algorithm only has average-case complexity O(n)
then 2 marks will be deducted. Example: for A = [−15,−9,−6,−3,1,3,6,20], the
output is B = [1,3,6,9,15,20].
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Problem 3. [10 marks, 1.5 pages] (Dijkstra’s algorithm + min-heap) Given a graph
as in Fig. 1, we are interested in finding the shortest paths from the source a to all
other vertices using the Dijkstra’s algorithm and a min-heap as a priority queue. Note
that a min-heap is the same as a max-heap, except that the key stored at a parent node
is required to be smaller than or equal to the keys stored at its two child nodes. In the
context of the Dijkstra’s algorithm, a node in the min-heap tree has the format v(pv,dv),
where dv is the length of the current shortest path from the source to v and pv is the
second to last node along that part (right before v). For example, c(a,1) is one such node.
We treat dv as the key of Node v in the heap, where v ∈ {a,b, c,d, e, f , g,h}.
Figure 1: An input graph for the Dijkstra’s algorithm. Edge weights are given as integers
next to the edges. For example, the weight of the edge (a,b) is 7.
a) [1 mark] The min-heap after a(a,0) is removed is given in Fig. 2. The next node to
be removed from the heap is c(a,1). Draw the heap after c(a,1) has been removed
and the tree has been heapified, assuming that ∞ ≥ ∞ (note: no need to swap if
both parent and children are ∞). No intermediate steps are required.
c(a, 1)
b(a, 7)
h(−, ∞)
e(−, ∞) f(−, ∞) g(−, ∞)
d(a, 5)
Figure 2: The min-heap (priority queue) after a(a,0) has been removed.
b) [2 marks] Draw the heap(s) after each neighbour of c has been updated and the
tree has been heapified (see the pseudocode in the lecture Slide 30, Week 9). If there
are multiple updates then draw multiple heaps, each of which is obtained after one
update. Note that neighbours are updated in the alphabetical order, e.g., d must
be updated before e. No intermediate steps, i.e., swaps, are required. Follow the
discussion on Ed Forum for how to update a node in a heap.
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S: vertices whose shortest
paths have been known
Priority queue of remaining vertices
1 a(a,0) b(a,7), c(a,1),d(a,5), e(−,∞), f (−,∞), g(−,∞),h(−,∞)
2 a(a,0), c(a,1)
Table 1: Complete this table for Part c).
c) [5 marks] Complete Table 1 with correct answers. You are required to follow strictly
the steps in the Dijkstra’s algorithm taught in the lecture of Week 9.
d) [2 marks] Fill Table 2 with the shortest paths AND the corresponding distances
from a to ALL other vertices in the format a →? →? → v | dv, for instance, a → c | 1.
Shortest Paths Distances
Table 2: Complete this table for Part d).
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2 Part II: Advanced
Problem 4 (9 marks, 1.5 pages). A database of n records, each of which has size m bits
(we assume that m is large) is maintained at two different servers X and Y. As sometimes
a record may get updated at one server and not at the other, the servers X and Y have
to sync their (possibly different) databases x = (x1,..., xn) and y = (y1,..., yn) from time
to time to make sure they are the same. At the synchronization time, the two servers
exchange some data in one or several rounds of communications.
A trivial synchronisation algorithm is as follows: X sends x via the internet to Y,
which verifies every record of x against that of its database y; for each 1 ≤ i ≤ n, if xi = yi
then Y does nothing, otherwise, it will either update yi ← xi
if xi
is newer or send (i, yi)
to X if its record is newer; upon receiving a pair (i, yi), X updates xi ← yi
. We assume
there is an efficient mechanism (which is not our focus) to decide which version of the
record is newer (e.g., using a naming convention like ver-1.2.1 is newer than ver-1.2.0).
While this solution works, it requires a huge bandwidth because at least n records have
to be sent across the network.
Our goal is to design a more efficient protocol/algorithms using a tool called cryptographic hash function. A cryptographic hash function h(·) takes as input any piece of
data x of any size and returns a fixed-size string h(x) (e.g., h(x) is of size 256 bits for
SHA-256). Moreover, different from a non-cryptographic hash function (Week 11), they
are collision-resistant, that is, it’s hard to find two different inputs that hash to the same
output. As a consequence, in practice, we can assume that if a 6= b then h(a) 6= h(b). For
simplicity, we assume that the following assumptions hold.
• At most one item is different between the two databases, that is, either xi = yi for
all i = 1,...,n, or there exists an index i
∗
such that xi
∗ 6= yi
∗ and xi = yi for all i 6= i
• The computation complexity is measured by the number of hashes required to
be computed by both X and Y and also by the amount of data that are hashed. For
instance, the trivial protocol described above requires no hash computation, and
hence, the computation complexity is zero.
• The communication complexity is measured by the number of hashes and the
number of records communicated between X and Y. For instance, the trivial protocol sends no hashes and n records, which is very expensive (a record is much larger
than a hash).
Your task is to design a synchronisation algorithm that requires a small communication complexity (most important) and also a reasonable computation complexity.
More specifically, you need to include the following in your answer.
a) (Overview - 2 mark) An overview of how your algorithm works, why it solves the
problem, and what are the communication and computation complexities.
b) (Detailed description - 3 marks) A detailed description of the main steps of the
algorithm, describing what X and Y do to sync the two databases. You could also
write your description using a pseudocode. The goal is to guarantee that other people can understand clearly how the algorithm works. The format of the pseudocode
is not important.
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c) (Demonstration - 2 marks) A demonstration of how the algorithm works on a
small example, e.g., when n = 8.
d) (Analysis - 2 marks) State the communication and computation complexities of
your algorithm and explain why it has these complexities. The input size is (n,m).
The given marks are maximum possible for the parts. Your actual marks depend on
how efficient your algorithm is (regarding the communication complexity (major criteria) AND the computation complexity) and also how well it is presented. A correct and
efficient but poorly written solution could still attract a low mark. An incorrect solution
will get a zero mark.
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Problem 5 (5 marks, 2 pages). Research a well-known problem of your own interest in
any field (science, engineering, technology) that can be solved by a computer algorithm.
Write a 1- to 2-page report on a popular algorithm that solves that particular problem and include in your report: (1) a problem description and why it is important and/or
interesting, (2) the algorithm description, (3) a pseudocode, (4) a demonstration on a toy
example, and (5) a complexity analysis.
You could include a few (1-5) references that you used when researching the problem/algorithm, but the writing should be your own. A similarity score of 25% and above
between your report and any existing source may indicate plagiarism. The report should
be typed in a text editor, e.g., words or Latex, and not handwritten. Marks will be
decided based on the correctness, clarity, and the sophistication of the problem/algorithm discussed. A report that is not well written or about a trivial/straightforward
problem/algorithm will receive a low mark.
Note that the problem/algorithm should NOT be among those already discussed in
the pre-recorded lectures/workshops. If you present a problem/algorithm that has been
discussed in the lectures/workshops, you will get a zero mark for Problem 5.
You could start from our textbook or check the following list from Wiki for a start.
https://en.wikipedia.org/wiki/List_of_algorithms
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3 Submission
The final submission (via Canvas) will consist of:
• Your solutions to all questions in a PDF file of font size 12pt and your agreement
to the honour code (see the first page).
Late Submission Penalty: Late submissions will incur a 10% penalty on the total
marks of the corresponding assessment task per day or part of day late, i.e, 4 marks per
day. Submissions that are late by 5 days or more are not accepted and will be awarded
zero, unless Special Consideration has been granted. Granted Special Considerations
with new due date set after the results have been released (typically 2 weeks after the
deadline) will automatically result in an equivalent assessment in the form of a
practical test, assessing the same knowledge and skills of the assignment (location and
time to be arranged by the coordinator). Please ensure your submission is correct and
up-to-date, re-submissions after the due date and time will be considered as late submissions. The core teaching servers and Canvas can be slow, so please do double check
ensure you have your assignments done and submitted a little before the submission
deadline to avoid submitting late.
Assessment declaration: By submitting this assessment, you agree to the assessment declaration - https://www.rmit.edu.au/students/student-essentials/ assessment-andexams/assessment/assessment-declaration
4 Plagiarism Policy
University Policy on Academic Honesty and Plagiarism: You are reminded that all submitted work in this subject is to be the work of you alone. It should not be shared with
other students. Multiple automated similarity checking software will be used to compare
submissions. It is University policy that cheating by students in any form is not permitted, and that work submitted for assessment purposes must be the independent work of
the student(s) concerned. Plagiarism of any form will result in zero marks being given
for this assessment, and can result in disciplinary action.
For more details, please see the policy at
https://www.rmit.edu.au/students/student-essentials/assessment-and-results/
academic-integrity.
5 Getting Help
There are multiple venues to get help. We will hold separate Q&A sessions exclusively
for Assignment 2. We encourage you to check and participate in the Ed Discussion Forum, on which we have a pinned discussion thread for this assignment. Although we
encourage participation in the forums, please refrain from posting solutions or suggestions leading to solutions.