Polynomial representation using Linked Lists- Evaluation -Addition and Multiplication

Representing a polynomial using a linked list is a common approach in computer science and mathematics. Each node in the linked list corresponds to a term in the polynomial, with the coefficients and exponents stored as data in each node.

Here is the node that is having a coefficient, an exponent, and a pointer (next) to the next node. Let us define the structure for this.

struct Node {
int coefficient;
int exponent;
struct Node *next;
}
This structure has a coefficient and exponent of type int and a pointer to the next node of type Node*. If the coefficient is in decimal, then you can take float also.

Example:

P(x) = 4x^3 + 9x^2 + 6^x + 7


If suppose the value of ‘x’ is 1 then,

P (x) = (4 * 1^3) + (9 * 1^2) + (6 * 1) + 7=26

The time complexity for adding a term (polynomial representation) and evaluating the polynomial is both O(n), where n is the number of terms in the polynomial.
The space complexity is O(n) as well, where n is the number of terms, due to the linked list structure.

C program Implementation- Polynomial Representation and Evaluation
/**********************************************************************************/
#include <stdio.h>
#include <stdlib.h>
#include <math.h>
struct Node {
    int coeff;
    int exp;
    struct Node * next;
}* poly = NULL;
void create() {
    struct Node * t, * last = NULL;
    int num, i;
    printf("Enter number of terms: ");
    scanf("%d", & num);
    printf("Enter each term with coeff and exp:\n");
    for (i = 0; i < num; i++) {
        t = (struct Node * ) malloc(sizeof(struct Node));
        scanf("%d%d", & t -> coeff, & t -> exp);
        t -> next = NULL;
        if (poly == NULL) {
            poly = last = t;
        } else {
            last -> next = t;
            last = t;
        }
    }
}
void Display(struct Node * p) {
    
    while (p!=NULL) {
        printf("%dx^%d ", p -> coeff, p -> exp);
        if(p->next!=NULL) printf("+");
        p = p -> next;
    }
    printf("\n");
}
long Eval(struct Node * p, int x) {
    long val = 0;
    while (p) {
        val += p -> coeff * pow(x, p -> exp);
        p = p -> next;
    }
    
    return val;
}
int main() {
    int x;
    create();
    Display(poly);
    
    printf("Enter value of x: ");
    scanf("%d", &x);
    
    printf("The value P(%d)=%ld\n",x, Eval(poly, x));
    return 0;
}

Polynomial Addition

To add two polynomials that are represented as a linked list, we will add the coefficients of variables with the degree.

We will traverse both the list and at any step, we will compare the degree of current nodes in both lists:
  • We will add their coefficients if their degree is the same and append them to the resultant list.
  • Otherwise, we will append the node with a greater node in the resultant list.
Algorithm
  • Create a new linked list, result to store the resultant list.
  • Traverse both lists until one of them is null.
  • If any list is null insert the remaining node of another list in the resultant list.
  • Otherwise compare the degree of both nodes, a (first list’s node) and b (second list’s node). Here three cases are possible:
    • If the degree of a and b is equal, we insert a new node in the resultant list with the coefficient equal to the sum of coefficients of a and b and the same degree.
    • If the degree of a is greater than b, we insert a new node in the resultant list with the coefficient and degree equal to that of a.
    • If the degree of b is greater than a, we insert a new node in the resultant list with the coefficient and degree equal to that of b
Time Complexity: 
O(m+n) is the time complexity for the polynomial addition using linked list in C, m and n being the size of the linked lists.

Space Complexity: 
O(m+n) is the space complexity for the polynomial addition using linked list in C, as in the worst case we need to allocate memory for each node in both lists.

C Program Implementation- Polynomial Addition using Linked List
/************************************************************************/
#include <stdio.h>
#include <stdlib.h>
struct Node {
    int coef;
    int expo;
    struct Node* next;
};
typedef struct Node Node;
void insert(Node** poly, int coef, int expo) {
    Node* temp = (Node*) malloc(sizeof(Node));
    temp->coef = coef;
    temp->expo = expo;
    temp->next = NULL;
    
    if (*poly == NULL) {
        *poly = temp;
        return;
    }
    
    Node* current = *poly;
    
    while (current->next != NULL) {
        current = current->next;
    }
    
    current->next = temp;
}
void print(Node* poly) {
    if (poly == NULL) {
        printf("0\n");
        return;
    }
    
    Node* current = poly;
    
    while (current != NULL) {
        printf("%dx^%d", current->coef, current->expo);
        if (current->next != NULL) {
            printf(" + ");
        }
        current = current->next;
    }
    
    printf("\n");
}
Node* add(Node* poly1, Node* poly2) {
    Node* result = NULL;
    
    while (poly1 != NULL && poly2 != NULL) {
        if (poly1->expo == poly2->expo) {
            insert(&result, poly1->coef + poly2->coef, poly1->expo);
            poly1 = poly1->next;
            poly2 = poly2->next;
        } else if (poly1->expo > poly2->expo) {
            insert(&result, poly1->coef, poly1->expo);
            poly1 = poly1->next;
        } else {
            insert(&result, poly2->coef, poly2->expo);
            poly2 = poly2->next;
        }
    }
    
    while (poly1 != NULL) {
        insert(&result, poly1->coef, poly1->expo);
        poly1 = poly1->next;
    }
    
    while (poly2 != NULL) {
        insert(&result, poly2->coef, poly2->expo);
        poly2 = poly2->next;
    }
    
    return result;
}
int main() {
    Node* poly1 = NULL;
    insert(&poly1, 5, 5); //5 x^5+3x^4+1x^2+1x^0
    insert(&poly1, 3, 4);
    insert(&poly1, 1, 2);
    insert(&poly1, 1, 0);
    Node* poly2 = NULL;
    insert(&poly2, 4, 4); // 4x^4+2x^2+1x^1
    insert(&poly2, 2, 2);
    insert(&poly2, 1, 1);
    printf("First polynomial: ");
    print(poly1);
    printf("Second polynomial: ");
    print(poly2);
    Node* result = add(poly1, poly2);
    printf("Result: ");    
    print(result);   // 5x^5 + 7x^4 + 3x^2 + 1x^1 + 1x^0
    return 0;
}

Multiplication
In multiplication we have to multiply first term of the  term of polynomial1 with all the terms of the polynomial2.
After multiplying each term the result can be added to a new polynomial

Algorithm: MultiplyPolynomials(poly1, poly2) 
Input: Two polynomials poly1 and poly2 represented as linked lists
Output: Resultant polynomial after multiplying poly1 and poly2

Initialize Result Polynomial:
    Create an empty linked list to represent the result polynomial. Let's call it resultPoly.

For each term term1 in poly1: 
    a. For each term term2 in poly2:
        Multiply the coefficients of term1 and term2 to get the new  coefficient.
        Add the exponents of term1 and term2 to get the new exponent.

Check for Existing Term in resultPoly:
    Traverse resultPoly to check if a term with the same exponent already exists:
        If found, update its coefficient by adding the new coefficient.
       If not found, insert a new term with the calculated coefficient and exponent at the correct position           to maintain the descending order of exponents.

Return resultPoly as the product of poly1 and poly2.

Time Complexity of polynomial multiplication using linked list : The time complexity of polynomial multiplication using linked list is O(n*m), where n is the total number of nodes in the first polynomial and m is the number of nodes in the second polynomial.

Space Complexity of polynomial multiplication using linked list: The space complexity of polynomial multiplication using linked list is O(n+m), we need to store all the multiplied values in the node.

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