Programming Assignment 1 : Polynomials via Linked Lists Solution

Description

• Objectives

In this assignment you are expected to implement a doubly linked list data structure, in which each node will have pointers to both the previous and the next node. The data structure will also include a head and a tail pointer that points to the rst and the last nodes in the list. The details of the structure is explained further in the following sections. You will use this specialized linked list structure to represent polynomials and implement some functions on them.

Keywords: Linked List, Polynomial

• Polynomials (Mathematical Background)

Most of the information about polynomials in this section is from the Wikipedia page \Polyno-mial” at the following url https://en.wikipedia.org/wiki/Polynomial.

In mathematics, a polynomial is an expression consisting of variables (also called indeterminates) and coe cients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. An example of a polynomial of a single indetermi-nate, x, is x² 4x + 7. An example in three variables is x³ + 2xyz² yz + 1. The x occurring in a polynomial is commonly called either a variable or an indeterminate. When the polynomial is considered as an expression, x is a xed symbol which does not have any value (its value is “indeterminate”). However, when one considers the function de ned by the polynomial, then x represents the argument of the function, and is therefore called a “variable”. These two words are used interchangeably. A polynomial P in the indeterminate x is commonly denoted either as P or as P (x).

A polynomial in a single indeterminate x can always be written (or rewritten) in the form

a_nxⁿ + a_{n 1}x^{n 1} + ::: + a₂x² + a₁x¹ + a₀x⁰

where a₀; :::; a_n are constants and x is the indeterminate. A polynomial can either be zero or can be written as the sum of a nite number of non-zero terms. Each term consists of the product of a number|called the coe cient of the term|and a nite number of indeterminates, raised to nonnegative integer powers.

The exponent on an indeterminate in a term is called the degree of that indeterminate in that term; the degree of the term is the sum of the degrees of the indeterminates in that term, and the degree of a polynomial is the largest degree of any one term with nonzero coe cient. A term with no indeterminates and a polynomial with no indeterminates are called, respectively, a constant term and a constant polynomial. The degree of a constant term and of a nonzero constant polynomial is 0. The degree of the zero polynomial, 0, (which has no terms at all) is generally treated as not de ned. For example: 5x²y is a term. The coe cient is 5, the indeterminates are x and y, the degree of x is two, while the degree of y is one. The degree of the entire term is the sum of the degrees of each indeterminate in it, so in this example the degree is 2 + 1 = 3.

Forming a sum of several terms produces a polynomial. For example, the following is a polyno-mial: 3x² 5x + 4. It consists of three terms: the rst is degree two, the second is degree one, and the third is degree zero. The rst term has coe cient 3, indeterminate x, and exponent 2. In the second term, the coe cient is 5. The third term is a constant. Because the degree of a non-zero polynomial is the largest degree of any one term, this polynomial has degree two.

Two expressions that may be transformed, one to the other, by applying the usual properties of commutativity, associativity and distributivity of addition and multiplication are considered as de ning the same polynomial. Two terms with the same indeterminates raised to the same powers are called “similar terms” or “like terms”, and they can be combined, using the distribu-tive law, into a single term whose coe cient is the sum of the coe cients of the terms that were combined. It may happen that this makes the coe cient 0.

A real polynomial is a polynomial with real coe cients. Similarly, an integer polynomial is a polynomial with integer coe cients. A polynomial in one indeterminate is called a univariate polynomial, a polynomial in more than one indeterminate is called a multivariate polynomial.

• Linked List Implementation (80 pts)

The linked list data structure used in this assignment is implemented as the class template LinkedList with the template argument T, which is used as the type of the data stored in the nodes. The node of the linked list is implemented as the class template Node with the template argument T, which is the type of the data stored in nodes. Node class is the basic building block of the LinkedList class. LinkedList class has two Node pointers (head and tail) in its private data eld, which point to the rst and the last nodes of the linked list.

The LinkedList class has its de nition and implementation in LinkedList.hpp le and the Node class has its in Node.hpp le.

3.1 Node

Node class represents nodes that constitute linked lists. A Node keeps two pointers (namely prev and next) to its previous and next nodes in the list, and the data variable of type T (namely element) to hold the data. The class has three constructors (including a copy constructor), and the overloaded output operator. They are already implemented for you. You should not change anything in le Node.hpp.

3.2 LinkedList

LinkedList class implements a doubly linked list data structure with the head and the tail pointers. Previously, data members of LinkedList class have been brie y described. Their use will be elaborated in the context of utility functions discussed in the following subsections. You must provide implementations for the following public interface methods that have been declared under indicated portions of LinkedList.hpp le.

3.2.1 LinkedList();

This is the default constructor. You should make necessary initializations in this function.

3.2.2 LinkedList(const LinkedList &obj);

This is the copy constructor. You should make necessary initializations, create new nodes by copying the nodes in given obj and insert new nodes into the linked list.

3.2.3 LinkedList();

This is the destructor. You should deallocate all the memory that you were allocated before.

3.2.4 Node<T> *getFirstNode() const;

This function should return a pointer to the rst node in the linked list. If the linked list is empty, it should return NULL.

3.2.5 int getNumberOfNodes();

This function should return an integer that is the number of nodes in the linked list.

3.2.6 bool isEmpty();

This function should return true if the linked list is empty (i.e. there exists no nodes in the linked list). If it is not empty, it should return false.

3.2.7 void insertAtTheFront(const T &data);

You should create a new node with given data and insert it at the front of the linked list as the rst node. Do not forget to make necessary pointer and head-tail modi cations.

3.2.8 void insertAtTheEnd(const T &data);

You should create a new node with given data and insert it at the end of the linked list as the last node. Do not forget to make necessary pointer and head-tail modi cations.

3.2.9 void insertAfterGivenNode(const T &data, Node<T> *prev);

You should create a new node with given data and insert it after the given node prev as its next node. Do not forget to make necessary pointer and head-tail modi cations. If the given node prev is not in the linked list, do nothing.

3.2.10 void removeNode(Node<T> *node);

You should remove the given node node from the linked list. Do not forget to make necessary pointer and head-tail modi cations. If the given node node is not in the linked list, do nothing.

3.2.11 void removeAllNodes();

You should remove all nodes in the linked list.

3.2.12 Node<T> *findNode(const T &data);

You should search for the node in the linked list with the data same with the given data and return a pointer to that node. You can use the operator== to compare two T objects. If there exists no such node in the linked list, you should return NULL.

3.2.13 void swapNodes(Node<T> *node1, Node<T> *node2);

You should swap the two given nodes node1 and node2. You are not allowed to just swap the data in the nodes. Also, you are not allowed to create new nodes in this function. Do not forget to make necessary pointer and head-tail modi cations. If either of the given nodes is not in the linked list, do nothing.

3.2.14 void printAllNodes();

You should print all nodes in the linked list. Each node must be printed on a single line. You should directly use the overloaded operator<< of Node class. If the linked list is empty, you should do nothing.

3.2.15 LinkedList &operator=(const LinkedList &rhs);

This is the overloaded assignment operator. You should remove all nodes in the linked list and then create new nodes by copying the nodes in given rhs and insert new nodes into the linked list.

• Polynomial Implementation (20 pts)

Some mathematical details of the polynomials were given in section 2. For this assignment, we will have some assumptions for the polynomials that will be worked on. First of all, the polyno-mials in this assignment will be univariate polynomials (i.e. there will be only one indeterminate, x). You will not be asked for multivariate polynomials. Second, the polynomials will be integer polynomials (i.e. all coe cients of the terms will be integers). You will not be asked for real polynomials. Third, you will not be asked for any ordering between terms of polynomials with respect to the powers of the indeterminates. And the last assumption is that you will not be given zero terms or zero polynomials.

The polynomials in this assignment is implemented as the class Polynomial. Polynomial class has a LinkedList object (namely terms) with the type Term in its private data eld. Term class represents the terms of polynomials. It is the type of the data stored in nodes of the linked list.

The Polynomial class has its de nition in Polynomial.hpp le and its implementation in Poly-nomial.cpp le. The Term class has its de nition in Term.hpp le and its implementation in Term.cpp le.

4.1 Term

Term class represents terms that constitute polynomials. A Term keeps the coe cient and ex-ponent variables of type int (namely coefficient and exponent) to hold the data related with the term. The class has three constructors (including a copy constructor), getter and setter functions, the overloaded output operator, and the overloaded equality check operator. They are already implemented for you. You should not change anything in les Term.hpp and Term.cpp.

4.2 Polynomial

In Polynomial class, there is a single member variable named as terms, which is a LinkedList object. Information of all terms will be stored in this linked list and no other information will be stored. All member functions should utilize this linked list to operate as described in the following subsections. Default constructor and the constructor that takes a polynomial expression string and populates the terms to the linked list have already been implemented for you. Do not change those implementation. In polynomial.cpp le, you need to provide implementations for following functions declared under polynomial.hpp header to complete the assignment. You should not change anything in le Polynomial.hpp.

4.2.1 Polynomial();

This is the default constructor. It is already implemented. It does nothing.

4.2.2 Polynomial(std::string expression);

This is the constructor that takes a polynomial expression string and populates the terms to the linked list. It is already implemented. You can track the required format for the expression strings from the code. A sample string is \(3)x² + ( 1)x² + (4)x⁵ + (7)x³“

4.2.3 void simplifyByExponents();

You should simplify the polynomial by combining the \similar terms”. Two terms are similar if they have same determinates raised to the same powers. Combine them similar terms into a single term whose coe cient is the sum of the coe cients of the terms that were combined. While combining two similar nodes in the linked list, update the coe cient of the rst node (the one that comes rst) and delete the second node (the other one) the rest of the nodes. If the combined coe cient is 0, then delete both nodes the rst node too.

4.2.4 void printPolynomial();

This function prints the polynomial as a polynomial expression to the standard output. It is already implemented. It uses the same format with the already implemented constructor above.

4.2.5 Polynomial operator+(const Polynomial &rhs) const;

This is the overloaded + operator. You should create a new Polynomial object with the terms of the rst polynomial with the nodes of second polynomial (rhs) appended.

• Driver Programs

To enable you to test your LinkedList and Polynomial implementations, two driver programs, main linkedlist.cpp and main polynomial.cpp are provided. Their expected outputs are also provided in output linkedlist.txt and output polynomial.txt les, respectively.

• Regulations

• 1. Programming Language: You will use C++.

• 2. Standard Template Library is not allowed.

• 3. External libraries other than those already included are not allowed.

• 4. Those who do the operations (insert, remove, nd, print) without utilizing the linked list will receive 0 grade.

• 5. Those who modify already implemented functions and those who insert other data variables or public functions and those who change the prototype of given functions will receive 0 grade.

• 6. Those who use STL vector or compile-time arrays or variable-size arrays (not existing in ANSI C++) will receive 0 grade. Options used for g++ are “-ansi -Wall -pedantic-errors -O0”. They are already included in the provided Make le.

• 7. You can add private member functions whenever it is explicitly allowed.

• 8. Late Submission: Each assignment will have a xed duration of 10 days and every student has a total of 5 days for late submissions during which no points will be deducted. Your assignment will not be accepted if you used up all of the 5 late days.

• 9. Cheating: We have zero tolerance policy for cheating. In case of cheating, all parts involved (source(s) and receiver(s)) get zero. People involved in cheating will be punished according to the university regulations. Remember that students of this course are bounded to code of honor and its violation is subject to severe punishment.

• 10. Newsgroup: You must follow the Forum (odtuclass.metu.edu.tr) for discussions and pos-sible updates on a daily basis.

• Submission

Submission will be done via CengClass (cengclass.ceng.metu.edu.tr). Do not write a main function in any of your source les.

A test environment will be ready in CengClass.

{ You can submit your source les to CengClass and test your work with a subset of evaluation inputs and outputs.

{ Additional test cases will be used for evaluation of your nal grade. So, your actual grades may be di erent than the ones you get in CengClass.

{ Only the last submission before the deadline will be graded.

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