Polynomial Addition in Data Structure

Polynomials can be represented as arrays or linked lists depending on the specific requirements of the program.

Array Representation 

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:

Here, the coefficient 5 is in slot 0 of the array, which represents 5 x⁰

The coefficient -2 is in slot 1 of the array, which represents -2 x¹

The coefficient 4 is in slot 6 of the array and represents 4 x⁶

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.

Link List Representation 

With a linked list representation, each term of a polynomial is stored as a node in a linked list. Each node contains three elements:

1. The coefficient of the term.
2. The exponent of the term.
3. Address of the next node in the list.

Considering the same example as above, the polynomial 4x⁶+3x+5 can be written as

Here, 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. For 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 link lists with the help of examples:

Polynomial Addition Using Array 

Let’s take two different polynomials as examples of the algorithm for polynomial addition in data structure:

A = 3x+5x²+7x³

B = 7+ 3x  + 4x² 

The steps to add the polynomials are as follows:

• Start by determining the polynomial with the highest degree, as the resulting polynomial will have the same degree.
• Store each coefficient in the array index corresponding to its exponent.
• Add the coefficients from the same index positions in both arrays and save the sum in the corresponding index of the resulting array.

A = 3x+5x²+7x³

A = 0x⁰+3x¹+5x²+7x³

B = 7x⁰+ 3x¹  + 4x² 

B = 7x⁰+ 3x¹  + 4x²+ 0x³

The summation,

C= 7x⁰+ 6x¹  + 9x²+ 7x³

Addition of Polynomials Represented as Linked Lists 

Now, let’s consider an example of polynomial addition in a data structure algorithm using linked lists.

We have to add the polynomials 12x⁴ + 2x² + 10 and 9x³ + 8x² + x          

 Storing each as a linked list,

A = 12x⁴ + 2x² + 10

B = 9x³ + 8x² + x

Resultant linked list:

Here is an example of polynomial addition using linked list in a data structure in c:

Input:

// 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;
}

Output:

5,2 -1,1 -3,0 

The application of polynomial addition in data structure is numerous in the fields of science and engineering. Here are a few examples:

1. Scientific Computations: 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.
2. Computer Graphics: Polynomials are essential for creating curves, shapes, and transformations that are used for rendering 3D objects. For example, bezier curves rely on polynomial equations.
3. Numerical Methods in Calculus: 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.
4. Machine Learning: 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.‍
5. Signal Processing: 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 our certification program to Learn to Code.

1.Which data structure is used for polynomial addition?

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.

2.How to perform polynomial addition using a linked list?

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.

3.What are polynomials in data structures?

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.

4.How can polynomials be represented using an array?

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.

5.Why are linked lists preferred for sparse polynomials?

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.