Published: 29 Nov 2024 | Reading Time: 6 min read
Polynomial addition in a data structure is necessary for performing tasks such as manipulating algebraic formulas, fitting curves in engineering programs, interpreting signals in communication, and more.
A simple example of a polynomial is P(x) = 4x²+3x+5, where x is the variable, the coefficients are [4, 3, 5], and the exponents are [2,1,0].
Adding two polynomials in algebra involves adding terms with the same exponent and adding the constants to give a result. In programming, before we can add polynomials, it's essential to understand how polynomials are stored in terms of arrays, linked lists, or other structures.
This article explores how polynomials are represented and looks into the algorithm for polynomial addition in data structure. With examples, we can grasp the fundamentals and learn how polynomial addition is applied in real-world scenarios.
Polynomials can be represented as arrays or linked lists depending on the specific requirements of the program.
The array is a straightforward representation of a polynomial in a data structure. Here, the polynomial is stored using an array where each index corresponds to the power of the variable, and the value of that index represents the coefficient of the variable.
For example, if the polynomial is 4x⁶+3x+5, the array would look like:
| Index | 0 | 1 | 2 | 3 | 4 | 5 | 6 | |-------|---|----|----|----|----|----|----|----| | Coefficient | 5 | -2 | 0 | 0 | 0 | 0 | 4 |
Explanation:
As it is evident, arrays work well for dense polynomials where most terms have non-zero coefficients. However, sparse polynomials like 4x¹⁰⁰+5, where many coefficients are zero, can waste a lot of space.
With a linked list representation, each term of a polynomial is stored as a node in a linked list. Each node contains three elements:
Node Structure:
| Coefficient | Exponent | Address of the next node |
|---|
Considering the same example as above, the polynomial 4x⁶+3x+5 can be represented as a linked list where each node holds the coefficient and exponent as a pair, and the pointer links to the following term in sequence. Linked list representation works very well for sparse polynomials as there is no need to store zero coefficients.
The polynomial addition in data structure occurs in a manner similar to that of a math class. It involves adding two like polynomials by combining their coefficients.
Example: Consider adding 4x²+3x+5 and 2x²+3x+5. The resulting polynomial would be 6x²+6x+10.
Now let's take a look at how it can be done with arrays and linked lists with the help of examples:
Let's take two different polynomials as examples of the algorithm for polynomial addition in data structure:
Polynomial A: 3x+5x²+7x³
Polynomial B: 7+ 3x+ 4x²
Steps to add the polynomials:
Polynomial A Representation:
A = 3x+5x²+7x³ = 0x⁰+3x¹+5x²+7x³
| Index | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| Coefficient | 0 | 3 | 5 | 7 |
Polynomial B Representation:
B = 7x⁰+ 3x¹ + 4x² = 7x⁰+ 3x¹ + 4x²+ 0x³
| Index | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| Coefficient | 7 | 3 | 4 | 0 |
Resultant Polynomial C:
C = 7x⁰+ 6x¹ + 9x²+ 7x³
| Index | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| Coefficient | 7 | 6 | 9 | 7 |
Now, let's consider an example of polynomial addition in a data structure algorithm using linked lists.
Polynomial A: 12x⁴ + 2x² + 10
Polynomial B: 9x³ + 8x² + x
Storing each as a linked list, the polynomials are represented with nodes containing coefficient-exponent pairs linked together. The addition process involves traversing both lists and combining terms with matching exponents.
Here is an example of polynomial addition using linked list in a data structure in C:
// C program to add two polynomials
#include <stdio.h>
#include <stdlib.h>
// Define the Node structure for representing polynomial terms
struct Node {
int coeff;
int pow;
struct Node* next;
};
// Function to create a new node
struct Node* createNode(int c, int p) {
struct Node* newNode = (struct Node*)malloc(sizeof(struct Node));
newNode->coeff = c;
newNode->pow = p;
newNode->next = NULL;
return newNode;
}
// Function to add two polynomials
struct Node* addPolynomial(struct Node* head1, struct Node* head2) {
// If any list is empty, return the other list
if (head1 == NULL) return head2;
if (head2 == NULL) return head1;
// Compare powers and recursively process the lists
if (head1->pow > head2->pow) {
struct Node* nextPtr = addPolynomial(head1->next, head2);
head1->next = nextPtr;
return head1;
} else if (head1->pow < head2->pow) {
struct Node* nextPtr = addPolynomial(head1, head2->next);
head2->next = nextPtr;
return head2;
} else {
// Add coefficients if powers are equal
struct Node* nextPtr = addPolynomial(head1->next, head2->next);
head1->coeff += head2->coeff;
head1->next = nextPtr;
return head1;
}
}
// Function to print the polynomial
void printList(struct Node* head) {
struct Node* curr = head;
while (curr != NULL) {
printf("%d,%d ", curr->coeff, curr->pow);
curr = curr->next;
}
printf("\n");
}
// Main function
int main() {
// Create 1st polynomial: 5x^2 + 4x^1 + 2x^0
struct Node* head1 = createNode(5, 2);
head1->next = createNode(4, 1);
head1->next->next = createNode(2, 0);
// Create 2nd polynomial: -5x^1 - 5x^0
struct Node* head2 = createNode(-5, 1);
head2->next = createNode(-5, 0);
// Add the two polynomials
struct Node* head = addPolynomial(head1, head2);
// Print the result
printList(head);
return 0;
}
5,2 -1,1 -3,0
This output represents the polynomial: 5x² - 1x¹ - 3x⁰
The application of polynomial addition in data structure is numerous in the fields of science and engineering. Here are a few examples:
Polynomials can be used to model physical phenomena by simulating systems and solving equations in engineering. They are used to approximate complex functions by writing them down as simpler expressions.
Polynomials are essential for creating curves, shapes, and transformations that are used for rendering 3D objects. For example, Bezier curves rely on polynomial equations.
Techniques like integration and differentiation often use polynomials to approximate solutions for complex equations. Polynomials in algorithms are used to simplify solving these differential equations.
Polynomials are used in machine learning for regression models, relationships between variables, and identifying dataset trends. It uses simple linear models to capture non-linear relationships.
Digital signals are often represented as polynomials for filtering, encoding, and analyzing data. Polynomials make it easier to manipulate and interpret signal behavior.
Polynomial operations are one of the most basic topics taught in computer science. Having got a snapshot of what polynomial addition in data structure involves, you are now in a position to handle more programming exercises that will enhance your experience. To continue learning more and mastering the application of the concepts at a professional level, you can enroll in NxtWave's certification program to Learn to Code.
Two commonly used data structures for polynomial addition are arrays and linked lists. Arrays are effective for dense polynomials, and linked lists handle sparse polynomials more effectively.
You have to first traverse both lists simultaneously and compare the exponents of terms. Then, you have to combine terms with matching exponents by adding their coefficients. Later, you add the values of unmatched terms directly to the resultant list in sorted order.
In data structures, polynomials are more formalized mathematical constructs that involve variables or coefficients whereby terms are usually stored (and even manipulated) as arrays or linked lists.
Polynomials can be represented using an array by mapping each coefficient to the index that corresponds to its exponent. For example, the constant term of the polynomial is stored at the first index. The coefficient of the first power of the variable is stored at the second index, and so on.
Linked lists are preferred for sparse polynomials over arrays because they only store terms with non-zero coefficients. Therefore, it avoids the need to allocate memory for zero terms and makes it efficient when working with large polynomials with many missing terms.
Source: NxtWave - CCBP Blog
Original URL: https://www.ccbp.in/blog/articles/polynomial-addition-in-data-structure