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COMP3121/9101Assignment

创建日期：2024-06-06 18:11:11

所属分类：计算机computer science

标签： COMP3121

COMP3121/9101Assignment

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COMP3121/9101Assignment

Due Friday 10th of November at 6pm Sydney time (week 9)
In this assignment we will apply dynamic programming. There are three problems each worth 20
marks, for a total of 60 marks. Partial credit will be awarded for progress towards a solution. We’ll
award one mark for a response of “one sympathy mark please” for a whole question, but not for
parts of a question.
Any requests for clarification of the assignment questions should be submitted using the Ed forum.
We will maintain a FAQ thread for this assignment.
For each question requiring you to design an algorithm, you must justify the correctness of your
algorithm. If a time bound is specified in the question, you also must argue that your algorithm
meets this time bound. The required time bound always applies to the worst case unless otherwise
specified.
You must submit your response to each question as a separate PDF document on Moodle. You
can submit as many times as you like. Only the last submission will be marked.
• we will release a LATEX template for each assignment question.
Other typesetting systems that support mathematical notation (such as Microsoft Word) are also
acceptable.
Your assignment submissions must be your own work.
• You may make reference to published course material (e.g. lecture slides, tutorial solutions)
without providing a formal citation. The same applies to material from COMP2521/9024.
• You may make reference to either of the recommended textbooks with a citation in any
format.
• You may reproduce general material from external sources in your own words, along with a
citation in any format. ‘General’ here excludes material directly concerning the assignment
question. For example, you can use material which gives more detail on certain properties of
a data structure, but you cannot use material which directly answers the particular question
asked in the assignment.
• You may discuss the assignment problems privately with other students. If you do so, you
must acknowledge the other students by name and zID in a citation.
• However, you must write your submissions entirely by yourself.
– Do not share your written work with anyone except COMP3121/9101 staff, and do not
store it in a publicly accessible repository.
– The only exception here is UNSW Smarthinking, which is the university’s official writing
support service.
Please read the Frequently Asked Questions document, which contains extensive information about
these assignments, including:
• how to get help with assignment problems, and what level of help course staff can give you
• extensions, Special Consideration and late submissions
• an overview of our marking procedures and marking guidelines
• how to appeal your mark, should you wish to do so.
1
COMP3121/9101 23T2 — Assignment 3 (UNSW Sydney)
Question 1 OurExperience
You are collecting survey results from k students over n days. Each day, all k students answer
the survey by providing a real number between 0 and 10 representing their stress level. For each
student, their survey responses over the n days are given in an array Si[1..n].
You want to combine all k survey responses to report to your boss how the students are feeling.
You are going to make a report which uses one of the responses from each day, and you want to
pick-and-choose responses so that students appear to be emotionally stable. You have designed a
heuristic to do so:
Choose a sequence of survey responses R[1..n], where each R[i] is the response from
one of the students on day i, minimising the change between each response in the
sequence. That is, choose a sequence R where each R[i] is one of {S1[i], S2[i], . . . , Sk[i]}
minimising the fluctuation
f =
n∑
i=2
|R[i− 1]−R[i]| .
For example, with n = 4 and k = 3, if you measure the results
S1 = [2, 5, 8, 9.9], S2 = [5, 2.5, 1, 4], S3 = [10, 3, π
2, 7],
then the optimal sequence is R = [5, 5, 8, 7] with
f = |5− 5|+ |5− 8|+ |8− 7| = 4,
obtained by taking the measurements from sensors 2, 1, 1, 3.
1.1 [6 marks] Consider this attempt to solve the problem with a greedy algorithm:
Start with S1[1], then for each subsequent day choose the survey response closest to the
previous one. Repeat by starting at each S2[1], S3[1], . . . , Sk[1], and of these, choose
the sequence with the minimum fluctuation.
Show that this algorithm does not solve the problem correctly by giving a counterexample. You
must provide n, k, the k sequences of survey responses, the answer (R and f) produced by the
greedy algorithm, and the correct answer. Your example must have n ≤ 4 and k ≤ 3.
1.2 [14 marks] Design an O(nk2) algorithm to find the minimal fluctuation of an optimal
sequence R, and the sequence of survey responses used to achieve this fluctuation. If there are
multiple optimal sequences, your algorithm can find any one of them.
Answers that determine only the minimal fluctuation (without finding the sequence of re-
sponses) will receive at most 10 marks for this part.
2
COMP3121/9101 23T2 — Assignment 3 (UNSW Sydney)
Question 2 Crappy Bird
Crappy Bird is a game where you control a bird starting at (1, 1) in a grid with m rows of n
columns (where m ≥ 2). Some columns j (1 ≤ j ≤ n) have pipes from the bottom of the screen to
row PipeUp(j), and some have pipes from the top of the screen to row PipeDown(j). If a column
j has no pipe from the bottom then PipeUp(j) = 0, and if it has no pipe from the top then
PipeDown(j) = m + 1. The goal is to reach row n while hitting as few pipes as possible. If you
exit the screen by flying too low or too high, you lose the game immediately!
Each step of the game, the bird moves one column to the right. Additionally, if you tap the screen
the bird moves one row up, otherwise it falls one row down. That is, if the bird is at (x, y) at the
some step, then it can be at (x+ 1, y + 1) or (x+ 1, y − 1) in the next step.
For example, this is a valid path which hits two pipes. The PipeUp and PipeDown values are shown
below and above the grid respectively.
0 0 2 0 0 2 3 0 0 0
6 6 6 6 6 5 6 4 3 6
A better path is achieved by going down from (6, 4) instead of up, hitting only one pipe.
2.1 [10 marks] Design an algorithm that runs in O(mn) time, and determines the fewest pipes
the bird can run into without exiting the screen.
2.2 [7 marks] The developers have realised that the game’s implementation of gravity is entirely
unrealistic! To fix it, they have patched the game to include acceleration for downwards move-
ment. When the bird begins falling (after a sequence of upwards movements), it first falls 1 unit
downwards. If it falls again in the next step, it falls 2 units, then 3, 4, and so on. That is, the kth
consecutive time the bird falls, it moves from (x, y) to (x+ 1, y − k).
3
COMP3121/9101 23T2 — Assignment 3 (UNSW Sydney)
For example, the bird may take this path:
Note that no pipes are hit, as at each step the bird is in a cell that does not coincide with a pipe.
The movement between positions is not considered.
Design an algorithm that runs in O(m2n) time, and determines the fewest pipes the bird can run
into.
2.3 [3 marks] Describe how the algorithm for part 2.2 can be improved to O(nm
√
m). You
do not need to restate the entire algorithm - just describe and justify how you would change the
algorithm to achieve this.
You may choose the skip this part and instead give an O(nm
√
m) algorithm for part 2. If
you do, your answer to 2.2 will be marked for both parts.
4
COMP3121/9101 23T2 — Assignment 3 (UNSW Sydney)
Question 3 Dance!
In the arcade game Dance Dance Revolution (DDR), players stand on a stage and hit arrows as
they scroll across the screen. More specifically, a sequence of n arrows ( , , , ) will scroll across
the screen, and as each arrow hits the top of the screen, the player must stand on the corresponding
arrow on the stage.
We play a variant of DDR, aptly named Don’t Dance Revolution (DDR2), where the goal is to
play the game like DDR but move as little as possible. The game plays like DDR except when an
arrow reaches the top of the screen and the player already has a foot on the correct arrow, then
the player is awarded one lazy point. If neither foot is on the correct arrow, then the player must
move exactly one foot from its current location to the correct arrow on the platform.
Unfortunately, the game is a bit unforgiving: any wrong move will cause the player to lose the
game and all of their lazy points. Wrong moves include:
• Failing to step on the correct arrow.
• Moving more than one foot at any given time.
• Moving either foot when the player is already stepping on the correct arrow.
You are given a sequence A of n arrows. Assume that your left foot starts on and your right
foot starts on , and you have memorised the entire sequence of arrows.
For example, consider the following sequence: . We can earn up to two lazy points as
follows:
L R
Start game!
L
R
L
R
Lazy point!
L
R
L
R
Lazy point!
3.1 [4 marks] Show that it is always possible to earn at least ⌊n/4⌋ lazy points during a round
of DDR2.
3.2 [16 marks] Design an O(n) algorithm to determine the maximum number of lazy points you
can earn in a round of DDR2.

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MAT136H1 GY 6183 COMP3170 AGRI90089 PHYSICS 1002 ECON10003 IEOR E4004 ACS61012 MATH 40550 MKTG2113 TELE4653 ENSC1004 ECE 232E MK726 MTRX2700 SHEET 3 CSCA08 COMP30027 Math 2XX3 statistic MMAN3200 BISM7213 ECMM109 ECON 2022 FNCE 20005 GR5067 CSCB09 ME 3017 CS 5100 PHYS 2903 COMS 4771 COMP222 COMP 1046 CSC148H1 ISYS2038 FINS2615 OT5206 COMP2048 A6 MECH5770M MA826/20 CMPT 726 STA457 INFS6071 INFS588 integral ECON 216 IRDR 0017 MA323 PA 15213 PSYC2001 ENV222 Statistics ACCT5930 ACCT12 LAND2121 ME 5405 COMM1180 TELE3113 A7 ECE 568 ACCT2019 INFS1200 ENGG1340 CS 1400 ASST3 FINM7409 SWEN30006 INFO2222 FIN20150 SOSS1000 CSSE7231 FINS5513 COMS3200 CECN 702 ECON4060H MN3515 HW2 PSYA02H3 ECON 2112 COMP90051 CS 1652 MA30059 AF1605 MKTG7501 ECO-6003B COM4521 CTM ECON3350 COMP 3804 ELEC9762 BISM 7255 ST22 MATH20014 MNE3122 COMP343 MATH5165 ELE7029 ENGR30002 SEEM3500 ECON7350 RISK5002 FINM7405 COMP90059 MOS 3320B STAT 431 BISM7221 MATH3014 MTH783P EBA 35302 ACS133 BUS2206 IE5600 ECOS 3997 EDST2003 FIN 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