Java程序辅导

COMP3600/6466 Algorithms Assignment 2
COMP3600/6466 — Algorithms
Convenor & lecturer: Hanna Kurniawati
E-mail: comp 3600 6466@anu.edu.au
Assignment 2
Due: Monday, 21 September 2020 23:59 Canberra Time
Grace Period Ends: Tuesday, 22 September 2020 13:00 Canberra Time
Late Penalty: 100%
Notes:
• In this assignment, you need to submit: .
– A written/typed answer to the questions as a single .pdf file, named A2-studentID.pdf.
– Source codes as a single file, named A2-studentID.cpp file (or suitable extension for the programming
language you use). In this assignment, you can use C/C++/Java. However, support will be provided only
for C/C++. Moreover, please pay attention to the following:
∗ To ensure we can run your program with no problem, you need to use the template provided during
tutorials and test your program on the provided test cases (the template and test cases will be uploaded
to the class website on 28 August 2020).
∗ We will compile and run your program from command prompt in Linux using the command:
> g++ -std=c++11 A2-studentID.cpp -o A2-studentID
> ./A2-studentID input1.txt > output1.txt
– You need to compress all files (the .pdf and .cpp) into a single .zip file named A2-studentID.zip and save
this .zip file as draft in Wattle before the due date .
• We provide 13 hours grace period. This means, there will be no penalty if you submit OR save as draft before
the grace period ends. However, we will NOT accept assignment after the grace period ends.
• Assignment marking:
– The total mark you can get in this assignment is 100 points.
– Whenever explanation/argument/reasoning/derivation is requested, the explanation/derivation is worth
80% of the points you can get for the question.
– This assignment will contribute 20% to your overall mark.
• Discussion with your colleagues are allowed and encouraged. However, you still need to work on the assign-
ment on your own AND write the names you discussed this assignment with.
[25 pts] Part A
1. [10 pt] Please find a tight asymptotic bound for the following recurrences and provide an explanation.
(a) [5 pt] T (n) = 2T (n4 ) +
√
n where T (n) = c for n ≤ 2 and c is a constant positive integer.
We can use Master’s Theorem. In this case, a = 2, b = 4, logb a = log4 2 =
1
2 , f (n) =
√
n. This
means case-2 of Master’s Theorem holds, and therefore T (n) = Θ(
√
n log(n))
(b) [5 pt] T (n) = 2T (n4 − 100) +
√
n where T (n) = c for n ≤ 2 and c is a constant positive integer.
The answer is the same as the answer of 1(a), i.e., T (n) = Θ(
√
n log(n)). The reason is for large n, the
‘-100’ is dominated by n4 , and therefore in asymptotic analysis, the ‘-100’ can be ignored, which leaves us
to the exact same question as 1(a).
2. [5 pt] Suppose n 6-sided fair dice are rolled independently. What is the expected sum of the outcome of these
dice? Please provide the derivation.
Let Xi(i ∈ [1, n]) be the random variables representing outcome of rolling the ith die. And suppose S = ∑ni=1 Xi.
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COMP3600/6466 Algorithms Assignment 2
Then, E[S ] = E[
∑n
i=1 Xi] =
∑n
i=1 E[Xi].
The expectation E[Xi] =
∑6
i=1
1
6 i =
21
6 = 3.5.
Hence, E[S ] = 3.5n.
3. [10 pt] For each statement below, please identify if the following sorting algorithm is stable, in-place, or both
stable and in-place, and provide a short explanation to support your answer. Sometimes, these sorting algorithms
can have various modifications. In this assignment, the following algorithms refer to the pseudo-codes discussed
in lectures.
(a) [2.5 pt] Insertion Sort: Stable, in-place.
(b) [2.5 pt] Merge Sort: Stable, not in-place.
(c) [2.5 pt] Rand Quick Sort: Not stable, in-place.
(d) [2.5 pt] Counting Sort: Stable, not in-place.
[40 pts] Part B
4. [10 pt] Your house-mate, Mr S, works as a file clerk in a law firm. As a file clerk, he needs to find files requested
by the lawyers fast. Mr S knows that all files in the firm are marked with a unique ID, where each ID is a positive
integer number in [0,m]. Furthermore, Mr S notices that all requests for files come in the form of a short range
of IDs. However, the placement of the files with respect to their IDs are rather random, which makes it difficult
for Mr S to find the right files fast. Tired with the increasing demand of requests for files, Mr S asked for your
help. Given the above information, you believe you can design a method to re-arrange the files in Θ(n+m) time
(n is the number of documents), so that Mr S can retrieve the files based on a range of IDs in O(1) time. To be
more specific, you can assume the range of IDs to be of the form [a · · · b], where 0 ≤ a < b ≤ m and b − a ≤ c
for a positive integer constant c. To help Mr S, please provide the following:
(a) [5 pt] The method for arranging the files along with an explanation that the run-time complexity of this
method is indeed Θ(n + m) time. You can assume moving a single file takes constant time.
You can pre-sort the files based on ID using counting sort in Θ(n + m) time.
(b) [5 pt] The method for retrieving the files given a range of IDs in O(1) time. Please also provide an expla-
nation of the time complexity.
After pre-sorting, we can maintain an array that maps ID to location, such that retrieving the book with ID
a only involves retrieving its location from a random access array, which takes constant time. Since the
files are already ordered, Mr S only needs to retrieve as many as (b − a) files, starting from the location of
the file with ID a, which again will take constant time. Hence, the overall retrieval time is O(1).
5. [15 pt]
Algorithm 1 Search(A sorted array of integers A, An integer v)
1: Let s = 1
2: Let e = A.length
3: while s ≤ e do
4: m =
⌊
s+e
2
⌋
5: if A[m] == v then
6: Return m
7: else if A[m] < v then
8: Let s = m + 1
9: else
10: Let e = m − 1
11: Return unsuccessful.
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COMP3600/6466 Algorithms Assignment 2
Assuming that all elements in A are distinct, the value v is in A and can be any element of A with equal
probability, and n = 2k for a positive integer k, please derive the following in regards to Algorithm 1.
(a) [5 pt] An asymptotic worst-case upper bound. Please make your bound as tight as possible.
The worst case happen when the search is unsuccessful. In such a case, the while-loop in line 3 to 10
will be executed for log e×. Since, each line other than line 3 takes Θ(1) time, the algorithm takes a total
of Θ(log e) time.
(b) [10 pt] An asymptotic average-case upper bound. Please make your bound as tight as possible.
Hint: It might help to think in terms of recurrence tree and consider the number of elements that can be
identified at different levels of the tree
For unsuccessful case, the average-case and worst-case are the same, i.e., Θ(log e).
For successful cases, the computation is a bit more involved. Let’s assume v is uniformly distributed
among the elements in A and suppose S is the random variable indicating the number of iterations (line
3-10) until we reach m where A[m] == v. Wlog, let’s assume e = 2k+1 − 1. Then, E[S ] = ∑ki=1 i · Nie ,
where Ni is the number of elements in A that requires i iterations to find. To compute Ni, we can view the
division of A in line 3-10 as a binary tree whose root is the entire array A and its children are the sub-arrays
A[1, · · · ,m − 1] and A[m + 1, e], where each sub-array is recursively divided into two sub-arrays in the
same manner. At each node, if v == A[m], then v is found and the iteration stops. Therefore, in this
tree, Ni is equivalent to the number of nodes at depth-(i-1). Hence, E[S ] =
∑k
i=1 i · Nie =
∑k
i=1 i · 2
i−1
e =
1
2e
∑k
i=1 i · 2i = 12e
(
(k − 1)2k+1 + 2
)
≤ 12ee log e = 12 log e = O(log e).
6. [15 pt] A new on-line game, called ”Guess What the Monkey is Typing”, has just been opened. In this game, a
monkey types a random string of characters with length at most n. Let’s denote this string as X. For simplicity,
the monkey uses a simple keyboard that consists of only 6 characters: Q, W, E, R, T, Y. You can also assume
that each character typed by the monkey is independent of the other characters and they are picked uniformly
at random from the 6 characters.
After the monkey finished typing, a player tries to guess the string typed by the monkey. The player can
gain additional information about X by privately asking the host of the competition whether a string of his/her
choosing, say S , is a prefix of X. A player can ask this question as many times as he/she wants, but must pay
$1 for each question. If the player guesses X correctly, he/she will get $4n, and therefore collect a total of
$P = $(4n − k) where k is the number of questions asked before the player guesses X correctly.
Given the above information, please answer the following questions:
(a) [10 pt] If small win means the player receives a profit P, where $0.3n < $P < $0.6n, is there a randomised
algorithm such that in the expected worst-case scenario, the player receives a small win? If there is, please
provide the algorithm and explanation on the expected worst-case profit. If not, please explain why such
an algorithm does not exist.
Yes when n > 17. The contestant can start by asking a prefix of length 1 (i.e., the first character) by
sampling uniformly at random from the 6 characters. Once the first character is known, this character is
fixed and he/she proceeds to ask a prefix of length 2 (i.e., the first and second character), where the second
character is sampled uniformly at random from the 6 characters. This process is repeated until none of the
6 characters can be added to the prefix, in which case the string X is found.
Suppose Ti represents the number of tries to identify the ith character of X. Then E[Ti] =
∑6
i=1
1+2+3+4+5+6
6 =
21
6 = 3.5. Since we need to iterate (n + 1)× if X is the longest possible, the expected worst case cost is
3.5(n + 1), which means P = $(4n − 3.5(n + 1)) = $(0.5n − 3.5). If we compare with the requirement
0.3n < P < 0.6n, 0.5n − 3.5 < 0.6n and for 0.5n − 3.5 > 0.3n for n > 17.
(b) [5 pt] If large win means the player receives a profit $P ≥ $0.6n, is there a randomised algorithm such
that in the expected worst-case scenario, the player receives a large win? If there is, please provide the
algorithm and explanation on the expected worst-case profit . If not, please explain why such an algorithm
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COMP3600/6466 Algorithms Assignment 2
does not exist.
Yes, if we were given the length of X exactly (rather than just an upper bound). In this scenario, we
can use a slight modification of the algorithm in (a): We only check for at most 5 of the 6 characters
because if the fifth trial sill returns a False, we know the correct character is the one we have not asked.
Suppose Ti represents the number of tries to identify the ith character of X. Then E[Ti] =
∑6
i=1
1+2+3+4+5+5
6 =
20
6 = 3.3, which means P = $(4n − 3.3n) = $0.7n ≥ $0.6n.
Note: Expected worst case means the average time the algorithm takes on the worst input of a given size. The
expectation is computed over the probabilities due to randomisation in the algorithm.
[35 pts] Part C
7. To prepare for the upcoming bushfire season, the government of the Coast of the Southern (CS) Region is setting
up an emergency helipad. This helipad will allow helicopters to land safely and quickly to evacuate residents
from surrounding towns. Since the creation of a helipad also requires road constructions from the towns to
the helipad, the government of CS would like to place the helipad, such that the the total distance between the
towns and the helipad is minimised. An illustration is in Fig. 1.
Figure 1: An illustration of a helipad
and the towns served.
To be more precise, let’s define the position of the helipad as the X-Y
coordinate of its center, i.e., (xh, yh). If the helipad serves n towns and
each town-i is specified as the X-Y coordinate of the town center, i.e.,
(xi, yi), then (xh, yh) must be set such that D =
∑n
i=1 |xi − xh| + |yi − yh| is
minimised. For simplicity, you can assume the X and Y coordinates are
integers in [INT MIN, INT MAX] and n < INT MAX, where INT MIN
and INT MAX are the minimum and maximum values for int data type
in C/C++.
Since there is a desire to replicate this type of development in multiple regions, the federal government hires
your company to develop a program that can compute the desired position of the helipad (as per the mentioned
specification) given the locations of the towns it serves. To this end, please:
(a) [5 pt] Formulate the problem of finding the right position of the helipad as one of finding medians of the
X coordinates and the Y coordinates. Please also prove that such a formulation will minimise D.
xh = median(x1, · · · , xn) and yh = median(y1, · · · , yn).
To minimize D, notice that D =
∑n
i=1 |xi − xh| + |yi − yh| =
∑n
i=1 |xi − xh| +
∑n
i= j |y j − yh|, which means,
we can minimize D by minimizing the total distance in the X and Y dimension separately. Now, using
contradiction, we can show that the median of the X coordinates will minimize the total distance in X
dimension. Suppose we move xh as much as k units to the right. Then, for those x values to the right of
xh, then although the total X-distance between the cities to the right of xh and the helipad will reduce, this
reduction will be nullified by the increase in X-distance of the cities to the left of xh. Similar argument
can be made for the Y dimension too. Therefore, setting xh and yh to be the median of each of the X and
Y coordinates separately would suffice.
(b) [10 pt] Design a randomized algorithm with an expected running time of Θ(n) to find the position (xh, yh)
that minimises D. Your algorithm can only use at most a constant number of extra storage outside the
array that contains the positions of the towns the helipad serves. Please also provide an explanation that
the expected time complexity of the algorithm you propose is indeed Θ(n).
You can assume that the positions of the towns are independent and uniformly distributed, positions of the
helipad and towns can overlap, and all X coordinates are distinct and all Y coordinates are also distinct.
Hint: You might want to think of modifying RandQuickSort.
Use quick select.
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COMP3600/6466 Algorithms Assignment 2
(c) [10 pt] Please implement your algorithm.
Input to the Program: The program will accept two argument, which is the name of the input file. The
input file contains n+ 1 lines, where n is the number of towns served by the helipad. The first line consists
of one integer number, n. Each line in the next n lines consists of two integer numbers, separated by a
white space. They represent the x and y coordinate of each town. Example:
3
1 2
3 1
4 5
The above input means the helipad is serving three towns, centered at (1, 2), (3, 1), and (4, 5).
Output of the Program: A single line in the standard output, containing two integer numbers separated
by a white space. The first number is xh and the second number is yh. The output of the example input is:
3 2
Program Marking: If your program compiles and runs, you will get 1 point. We will then run your
program on 6 test cases: 2 cases would have up to 500 towns, 2 cases would have 501 – 500,000 towns,
and 2 cases would have 500,001 – 250,000,000 towns. For each test case, your program will be given(
15·#towns
100,000 + 0.15
)
ms CPU time to find a solution. The time limit will be rounded up to 2 decimal digit. You
can assume your program will have access to at most 8GB RAM. It will be run as a single thread process
on a computer with Intel i7 3.4GHz processor. For each test case that your program solves correctly within
the given time and memory limit, you will get 1.5 points.
Tips: Be careful not to use arrays allocated in the program stack. Also, although we will provide several
test cases (available in the class website from 28 August 2020), obviously, your program will be tested on
different test cases than the examples provided.
(d) [10 pt] Experimental analysis of the time complexity of your algorithm and comparison against the theo-
retical analysis (7b). You will need to have sufficient data to fit a function well and will need to generate
your own test cases for this purpose.
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