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MA407 Algorithms and Computation

创建日期：2024-06-08 15:58:35

所属分类：数学mathematics

标签： MA407

Algorithms and Computation

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MA407 Algorithms and Computation

Assessed Coursework
This set of exercises is Assessed Coursework, constituting 25% of the final mark for this
course. It is due on Wednesday, 10 January 2024 at 4:00pm (GMT).
The work should be yours only. You may not work in groups or even discuss these problems
with anyone else. You have to upload your work on Moodle at the Gradescope links that will
be provided. By submitting your solutions, you would be confirming that you have read and
understood the School’s rules on Plagiarism and Assessment Offences, which can be found here
and here.
Your submission for this coursework should consist of two files: A Python file CompactList.py
for Question 1 and SumFreeLists.pdf, for your solution to Question 2. The contents of your work
must remain anonymous, so do not include your name or your student number in the
files. Do not submit anything directly to your lecturer or class teacher. Instead, please put your
candidate number in both files, for example, in a comment at the top of your Python code.
The deadline is sharp. Late work carries an automatic penalty of 5 deducted marks (out
of 100) for every 24-hour period that the coursework is late. Submission of answers to this
coursework is mandatory. Without any submission, your mark for this course is incomplete, and
you may not be eligible for the award of a degree.
Use of Python Facilities
In the Python implementations of the algorithms, you are not allowed to use any Python library
functions beyond what might be explicitly stated in the questions. The aim of this assessment
is not to measure your proficiency in using Python facilities to solve problems. The questions
test your algorithm design and analysis skills, as well as your ability to implement algorithms
in Python.
Question 1: Compact List Data Structure
In this task, you have to implement a data structure called a compact list. Compact lists
are compact ways of representing large sets of integers, where the set contains long runs of
consecutive integers. For example, the set of integers
{−1000,−999,−998, . . . ,−10} ∪ {100, 101, 102, . . . , 200} ∪ {1001, 1002, 1003, . . . , 1999}
is represented by the compact list ⟨−1000,−9, 100, 201, 1001, 2000⟩. The meaning of this list is
that the numbers start at −1000, then the first number that is not in the list is −9, then the
first number after −9 that is again in the set is 100, then the first number after 100 that is not
in the set is 201, then the first number after 201 that is in the set is 1001, and the first number
after 1001 that is not in the set is 2000. Since the list ends here, no number from 2000 and
onwards is in the set.
As another example, the set {−5,−4,−3, 0, 1, 2, 7, 8, 9, 10} is represented by the compact list
⟨−5,−2, 0, 3, 7, 11⟩.
Compact lists can also have an odd length. For example, the compact list ⟨5, 70, 80⟩ rep-
resents the set {5, 6, 7, . . . , 69, 80, . . . }. Thus, we understand a compact list of odd length as
representing a set which after a certain value (in this case 80) contains all integers from that
value on. In practice, such as when we implement compact lists for integers, we let the set go
up to a maximum value, which will fix as a variable MAX. We also set MIN as the minimum value
of an integer included in a set. In other words, the sets that we represent as compact lists will
be subsets of the universal set {MIN, MIN+1,. . . ,MAX}.
1
In general, a compact list of integers represents a set of integers as a list ⟨x0, x1, x2, . . . ⟩ of
integers such that x0 < x1 < x2 < . . . , with the meaning that all integers from x0 to x1 − 1,
as well as all from x2 to x3 − 1, and so on, are included in the set. Thus, the included integers
start at x0, then the first integer larger than x0 not to be included is x1, then the next integer
after x1 to be included is x2 and so on. The empty set is represented as an empty list (of length
0). If the compact list has an odd length, it contains all values from the last entry in the list
up to the maximum value MAX. As a final example, the set of all integers is represented by the
one-element compact list ⟨MIN⟩: that is, it consists of the single number MIN.
You have to implement a class called CompactList which implements various set operations,
as described below. Before we describe the various functionalities, we quickly review the basic
set operations and relations, to establish notation. Given sets A and B,
their union is the set A ∪ B defined as the set of all elements that are in A or in B or in
both A and B,
their intersection is the set A ∩ B defined as the set of all elements that are in A and in
B,
their difference is the set A \B defined as the set of all elements that are in A but not in
B.
The set A is a subset of set B, written A ⊆ B, if each element of A is also an element of B. The
sets A and B are considered to be equal if they have the same elements.
If all the sets that we consider are contained in some universal set U , then the complement
of set A is the set Ac = U \ A. Below, whenever we mention union, intersection, difference or
complement, we always refer to the sets represented by compact lists of integers. For example,
the union of the set A = {10, . . . , 19} represented by the compact list ⟨10, 20⟩, and the set
B = {30, . . . , 39} represented by the compact list ⟨30, 40⟩, is the set A ∪ B represented by the
compact list ⟨10, 20, 30, 40⟩. As another example, the intersection of the set A with the set
C = {15, . . . , 24}, represented by the compact list ⟨15, 25⟩, is the set A∩C = {15, 16, 17, 18, 19}
represented by the compact list ⟨15, 20⟩.
For the purposes of our set operations, the universal set is the set {MIN,MIN+1,. . . ,MAX} of
integers, and so we take complements with respect to this set. For example, the complement of
set A is the set {MIN, . . . , 9} ∪ {20, . . . }, represented by the compact list ⟨MIN, 10, 20⟩.
The following list (a )–(m ) describes the functionalities that you have to implement. You
can use the template file CompactList.py on Moodle.
(a ) Constructor for a compact list from a Python list: __init__(alist)
Creates a compact list to represent the set of integers given by the Python list alist.
As an example, CompactList([8,3,5,4]) initialises the compact list ⟨3, 6, 8, 9⟩.
(b ) Instance method: cardinality()
Returns the number of elements in the set represented by this compact list.
The worst-case time should be O(n) where n is the length of the compact list.
(c ) Instance method: insert(value)
Inserts value into the set represented by this compact list if value is not already contained
in the set.
The worst-case time should be O(n) where n is the length of the compact list.
(d ) Instance method: delete(value)
Removes value from the set represented by this compact list if value is in the set, otherwise
does nothing.
The worst-case time should be O(n) where n is the length of the compact list.
(e ) Instance method: contains(value)
Returns True if value is an element of the set represented by this compact list, otherwise
returns False.
The worst-case time should be O(log n), where n is the length of this compact list.
2
(f ) Instance method: subsetOf(cl)
Returns True if the set represented by this compact list is a subset of the set represented
by cl, otherwise returns False.
The worst-case time should be O(m+ n) where m is the length of this compact list and n
is the length of cl.
(g ) Instance method: equals(cl)
Returns True if the set represented by this compact list has the same elements as the set
represented by cl, otherwise False.
The worst-case time should be O(m+ n) where m is the length of this compact list and n
is the length of cl.
(h ) Instance method: isEmpty()
Returns True if the compact list represents the empty set, False if not.
The worst-case running time should be O(1).
(i ) Instance method: complement()
Returns the complement of the set represented by this compact list, also represented as
a compact list, where the complement is with respect to the universal set of integers
determined by MIN and MAX.
The worst-case time should be O(n) where n is the length of the compact list.
(j ) Instance method: union(cl)
Returns the union of the set represented by this compact list with the set represented by
the compact list cl. The method does not alter the current compact list or cl.
The worst-case time should be O(m+ n) where m is the length of this compact list and n
is the length of cl.
(k ) Instance method: intersection(cl)
Returns the intersection of the set represented by this compact list with the set represented
by the compact list cl. The method does not alter the current compact list or cl.
The worst-case time should be O(m+ n) where m is the length of this compact list and n
is the length of cl.
(l ) Instance method: difference(cl)
Returns the set difference this\cl of the set represented by this compact list with the set
represented by cl removed. The method does not alter the current compact list or cl.
The worst-case time should be O(m+ n) where m is the length of this compact list and n
is the length of cl.
(m ) Instance method: __str__()
Returns a string representation of this compact list ⟨x_0, x_1, x_2, ...⟩ in the form
[x_0,x_1 - 1] U [x_2,x_3 - 1] U ...
(that is, as a union of closed intervals of integers).
If this compact list is empty, it returns empty.
Hints:
1. Finding the union or the intersection of two compact lists is similar to the Merge() method
that we saw when we looked at Merge Sort.
2. Once you have implemented union(cl) and complement() and you have made sure that they
work, then it is very simple to implement intersection(cl), difference(cl), subsetOf(cl),
equals(cl), as well as insert(value) and delete(value), by writing these set operations
and relations in terms of union(cl) and complement(). For example, A ∩ B = (Ac ∪ Bc)c
and A \B = A ∩Bc.
Remember to explain all your code with comments, when needed, and to indent properly. Do
not use any Python modules or libraries, except the standard functionalities of Python lists.
[60 points]
3
Question 2: Sum-Free Lists
Given three sets A, B, and C, we would like to determine whether there exists a triple (a, b, c)
such that a ∈ A, b ∈ B, c ∈ C, and a+ b+ c = 0.
The following Python function receives three lists representing these sets as input and claims
to output such a triple as a list [a,b,c], if it exists, or None otherwise.

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