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Algorithms and Analysis COSC 2123/1285

创建日期：2024-04-11 15:36:54

所属分类：计算机computer science

标签： COSC 2123

Algorithms and Analysis

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Algorithms and Analysis

COSC 2123/1285
Assignment 2: Algorithm Design & Complexity Analysis
Assessment Type Individual Assignment. Submit online via Canvas → Assign-
ments → Assignment 2. Clarifications/updates/FAQ can be
found in Canvas Announcements and Discussion → Assign-
ment 2 General Discussion.
Marks 40
IMPORTANT NOTES
• If you are asked to develop/design an algorithm, you need to describe it in
plain English first, say a paragraph, and then provide an unambiguous pseudo
code. The description must include enough details to understand how the algo-
rithm 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.
• 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.
• Marks are given based on correctness, conciseness (with page limits), and clar-
ity of your answers. If the marker thinks that the answer is completely not under-
standable, 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 are applied to ALL problems in this assignment. Over-length an-
swers 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 con-
trol. Please do NOT include the problem statements in your submission,
because this may increase Turnitin’s similarity scores significantly.
• 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":
2
Problem 1 (3 marks - at most 1/2 page). Determine if the following statements are true
or false AND provide a formal proof using either limits or the definitions of the big-O,
big-Omega, and big-Theta notations. For instance, to prove that f (n) ∈O(g(n)) or f (n) ∉
O(g(n)), using the definitions of big-O, we need to demonstrate the existence of a constant
c and a sufficient large n0 such that f (n)≤ cg(n) for all n≥ n0, or showing that there are
no such values. Using limits, for instance, we need to show that limn→∞ f (n)/g(n) <∞
for f (n) ∈ O(g(n)), or showing that limn→∞ f (n)/g(n) =∞ for f (n) ∉ O(g(n)). Note that
there will be no marks if only true/false answer is given.
a) [1 mark]
p
4n2+2n ∈Θ(n).
b) [1 mark] n100 ∈O(en).
c) [1 mark] loge(n2) ∈Ω(
p
n).
Problem 2 (3 marks - at most 2 sentences). Arrange the following functions in an in-
creasing order of their growth rates.
f1(n)= 1000n,
f2(n)= (logn)2,
f3(n)= n!,
f4(n)= n logn,
f5(n)= 2n.
Problem 3 (4 marks - at most 1 page). In the following questions we assume that m
grows more slowly than logn.
a) [2 marks] Design an in-place algorithm (description + pseudo code + complexity
analysis) to find m smallest elements (possibly repeated) in an array of real num-
bers of size n with time complexity O(nm). Note that no extra data structures
are allowed.
a) [2 marks] Design another algorithm (description + pseudo code complexity analysis)
to find m smallest elements (possibly repeated) in an array of real numbers of size
n with time complexity O(n logm). Extra data structures are allowed.
Problem 4 (2 marks - at most 1/2 page). Design a non-recursive algorithm (algorithm
description + pseudo code) that takes as input the root of a binary search tree and
return the maximum key stored in the tree.
3
Problem 5 (5 marks - at most 1.5 pages). Let A be an array of size n contains a list of
records in which the field "gender" has value either M or F. Suppose that n is even for
simplicity.
a) [2 marks] Design an in-place recursive decrease-and-conquer algorithm (description
+ pseudo code) of complexity O(n2) that swaps the elements of the array so that
their genders alternate between M and F as much as possible. If there are more
M than F, then all the uncoupled M are grouped at the end of the array. Similarly,
if there are more F than M, then all the uncoupled F are grouped at the end of the
array. For example, if the list is
(ID1,F), (ID2,F), (ID3,M), (ID4,F), (ID5,M), (ID6,M), (ID7,F), (ID8,F),
then the output could be
(ID5,M), (ID7,F), (ID3,M), (ID8,F), (ID6,M), (ID1,F), (ID2,F), (ID4,F).
Note that inefficient algorithm, in particular, algorithms with complexity Ω(n3)
will attract a zero mark.
a) [2 marks] Determine the recurrence relation for the number of comparisons C(n)
required by your algorithm and solve it using the backward substitution method.
What is the complexity class of C(n) in big-O notation?
b) [1 marks] Determine the recurrence relation for the number of swaps S(n) required
by your algorithm and solve it using the backward substitution method. What is
the complexity class of S(n) in big-Theta notation?
4
Problem 6 (9 marks - at most 2 pages). Two years after graduation, you are hired by
RMIT’s School of Computing Technologies as a software engineer. Your first task is to
develop a Course Management software that can answer common students’ questions
automatically. One of the common questions, frequently asked by students who don’t
want to complete the whole program but are only interested in a few courses, is about
the number of minimum credits one needs to accumulate before taking a certain course.
The input is a curriculum which consists of n courses, indexed by integers from 0 to
n−1. Each course i has ci credit points and also a list Pi of prerequisites (PRQs). Let
m be the number of pairs (i, j), 0≤ i 6= j ≤ n−1, where i is a PRQ of j. Your task is to
design two algorithms (description + pseudo code + complexity analysis) that
return the total number of credits Ti a student has to accumulate before each course
i = 0,1, . . . ,n−1, can be taken, under two different scenarios in Part a) and Part b). Your
algorithm must output the minimum number of credits required as PRQs for every
course (see Fig. 1).
i 0 1 · · · n−1
Ti ? ? · · · ?
Figure 1: This is the table of the accumulated credits Ti required before taking each
course i, 0≤ i ≤ n−1, output by your algorithms.
a) [3 marks] Scenario 1: the PRQs are nonequivalent, that is, each course can be taken
only if ALL PRQs have been completed. Apply your algorithm to the toy example
in Fig. 2 and complete the output table (see Fig. 2).
b) [6 marks] Scenario 2: the PRQs are equivalent, that is, if Pi 6=∅, then the course
i can be taken as long as ONE of the PRQs in Pi has been completed. Design a
dynamic programming algorithm to achieve your task. Apply your algorithm
to the toy example in Fig. 2 and complete the output table (see Fig. 2). Note that
you should also explain explicitly the recurrence formula and the backtrace
(given i, list the sequence of courses that a student needs to take before enrolling
in course i with minimum sum of credits).
Note that to achieve full marks, the algorithms should have complexity O(nm+
n2) in Part a) and O(n+m) in part b) or better. Inefficient algorithms, e.g. with
complexity exponential in n or m, will attract a zero mark.
5
Figure 2: A toy example for Problem 6 is given with input and required output. An arrow
i→ j implies that Course i is a PRQ of Course j. The number next to each course is its
number of credits. For instance, Algorithms & Analysis has 12 credits and has Further
Programming and Advanced Programming Techniques as prerequisites.
Problem 7 (7 marks - at most 2 pages including the drawing of the tree). The year is
2040, and you are now on a voyage to a newly discovered Earth-like planet, named DST,
which is the habitat to many mysterious and previously unknown species. After years
of digging and researching, your biologist and palaeontologist colleagues have collected
a large number of DNA sequences from all living or dead animals they encountered, and
now ask for your help in building an evolutionary tree, which represents the ancestral
relationships among all animals.
The root of the evolutionary tree is the predicted ancestor of all species. Each node in
the tree represents the DNA sequence (or rather, a DNA strand consisting of six bases A,
C, G, T, H, J) of one species. The distance between any two strands Su and Sv is d(Su,Sv)
defined as the minimum number of base deletions, insertions, and substitutions required
to turn Su into Sv. The tree must satisfy the following two evolution rules (see Fig. 3).
1. (Rule 1) Each species evolves away from the ancestors: if u is an ancestor of v, then
d(Su,Sv) ≥ d(Sx,Sy) for every pair of parent and child (x, y) along the path of the
tree from u down to v (including u and v).
2. (Rule 2) Sibling species evolve away from each other: if v and w are not parent and
child and their closest common ancestor is u, then d(Sv,Sw) ≥ d(Sx,Sy) for every
pair of parent and child (x, y) along the paths of the tree from u downto v and from
u down to w (including u, v, and w themselves).
6
Figure 3: This is a toy example that illustrates the rules in the evolutionary tree.
a) [3 marks] Given a root r, develop an efficient algorithm that builds an evolutionary
tree rooted at r and satisfying Rule 1 and Rule 2. Include an algorithm description
and an unambiguous pseudo code. Provide a complexity analysis of your algorithm
with respect to the input size n and `, where ` is the maximum length of each DNA
sequence. Note that incorrect or inefficient algorithms will not receive a full mark,
in particular, exponential-time complexity algorithms may attract a zero mark.
b) [1 marks] Prove that your algorithm produces an evolutionary tree satisfying Rule
1 and Rule 2. Hint: if you are on the right track, the proof should take a short
paragraph only.
c) [3 marks] Implement your algorithm and run it on the dataset provided in Fig. 4
(see also Fig. 5). Suppose that beefalo is the predicted ancestor of all other
species. Submit the code together with a drawing of the (rooted) evolutionary tree
for this example (with nodes labelled by names or images of the creatures).
7
Species DNA sequence
Beefalo AACTHJJTGG
Bearger TTHJJGTAT
Bunnyman AACTHJJGG
Buzzard CAAHGHTG
Catcoon AHCTHHJJTGG
Deerclops HJCTHJJGT
Dragonfly AHCTHHJGJTGGC
Elder Mandrake CATHJHTJGT
Eyeplant CTHJJGT
Grass Gekko CTJHJT
Hound JJHHJJTGH
Lavae AHHJGJTGGC
MacTusk HCTHJJTGGJH
Merm JHTHHJJTGA
Moleworm AACTHJJ
Pengull AHCTHJJTGG
Tallbird CJHHJJHGA
Tentacle JJHHGH
Terrorbeak AHATHHJJTAC
Volt Goat AHCTHJJTG
Figure 4: A dataset that contains the names/DNAs of 20 species encountered on DST.
Figure 5: Newly discovered species, illustrated by Klei Entertainment.
8
Problem 8 (7 marks - at most 3 pages). Research a problem of your own interest in any
field (science, engineering, technology) that can be solved by a computer algorithm.
a) [5 marks] Write a 1- to 2-page report on an existing algorithm that is among
the most efficient ones solving that particular problem and include in your report:
(1) the problem statement, (2) why is the problem important/interesting, (3) the
algorithm description, (4) a pseudo code, and (5) a complexity analysis. Note that
the problem/algorithm should NOT be among those already discussed in the pre-
recorded lectures. You should include a few (1-5) references that you used when
researching the problem/algorithm, but the writing should be your own. A simi-
larity score of 25% and above between your report and any existing source may
indicate plagiarism. Marks will be decided based on the correctness, clarity,
and the sophistication of the problem/algorithm discussed. If the report is not
well written or the problem/algorithm is trivial or straightforward, you may not
receive a full mark.
b) [2 marks] Propose your own improvement of that algorithm, either in space or
time complexity. It should be your own idea and not from any existing source.
You should state clearly at first what the improvement is, and then explain how
you do it. You may receive from 0 to 2 marks depending on the significance of the
improvement.
9
1 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). You may also submit the code in Problem 7.
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 sub-
missions. 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.

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BCPM0008 ECN 620 QBUS6830 ECSE-415 ECON5102 MGMT20005 MAT 653 ECO 480 COMP 5/600 CSSE 5600 ME 571 7CCMMS61 CS 550 COMP2865 MATH3075 STATS 506 QTM 220 ISYS2120 FIT5106 STAT 153 MKTG3112 ECON2102 ELE2018 COMP3031 MSIN0010 SCC461 MATH237 CS246 CSC3065 CSE 101 ICTWEB411 COMP9313 ECOS 2025 FN620 MATH11111 MATH 3018 KE1068 FINA 6112 C++ MISM 6202 MSIN0105 CSC8631 MECO6935 COMP4121 ACCT2011 DPBS1150 EEL 3701C F19PL OPM 661 CSE 310 CS918 BIOL30001 ACCT3563 BANK3600 MET CS622 MATH367 ACCT5907 Continuing COMP5048 COMP61021 CS3334 GSOE9210 CSCI-570 COMP 5322 HW 2 ALY-6020 IEOR 4706 DATA 200 EBUS504 CGE12412 COMP4108 7SSMM800 ECMM171P MATE50002 CS225 RS 6003 BH2274 ELEC3111 MAT6669 ECON-UA 227 PHIL 1012 7SSMM603 TB028A P6 ECS708 CSC 427 CS6476 CSE 250A A1A2 ITS61504 EECS 376 model HW6 UV0010 ECON 6074 MG4F7 CMP-7030Y INFO3008 BFIP013 ECON 5068 MATH6002 CS3483 PSTAT 131 C31FM CSOR W4246 AAEC 5307 DATA ELEC362 COMSM0047 ECN6006 MA20224 BST351 MATH 323 MATH323 AA2021 COMP3811 COMP2013 IB9X60 INVP001 EMATM0051 COMP220 BLAW1002 E20 TCP8001 ECON61001 EART40005 ECONM1022 MATH 365 MATH 367 ELEC6218 COMP 5812M EDUC 5061M ELEC ENG 1101 equation MATH6174 ST334 C4 EC9570 A33648 EBUS603 ECON47815 stata WFMS0001 PEM102 INF6060 ionically ELECTRICAL MATH 375 MMW 12 MKTG860 ECON60401 ALY6010 SOST20131 Introduction COM3524 MME2 GGR305 JNB 538 EECS101 COMM5030 IEOR E4601 2B CSC336 EN2315 LAW 432 CSC263 MATH2001 OLET1143 CSCB63 SIT772 COMM1170 STAT 441 MS930 INFT3050 GLBH0030 Stat 120B MAST10007 MGEC61 Machine GGR 252H1S ECON7030 Financial Marketing MGMT30019 ECON5120 SDSC 8009 CSC165H1S DSCI 551 3081W ACM41000 ECA 5304 MATH 1064 ECSE444 DATA 604 COMP7105 MIE1615 ECON4004 ENV200H1S EC481 MKT 306 EC1B1 MATH08051 ALY6140 CS260C CSE30 MSIN0226 CS4103 CSC3064 AM16 COMP11212 CSE4101 PSTAT 174 Math 380W CNT 5106C STAT 310 FIN2028 CS5014 STAT 425 COMP 4102A ELEC2140 MANF9544 QRM 9 QRM 8 ECON 457 COMPSCI 4091 MATH 523 DNSC-4211 PEAS 6000 MSIN0180 COMP2119 A2A HT 2022 IB93F0 MAS316/414 ACCT2522 MSIN0181 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