700513. Polynomial Addition
Here is a detailed Markdown blog post for the Polynomial Addition problem:
Polynomial Addition
Summary
The Polynomial Addition problem involves adding two polynomials represented as linked lists. Each node in the linked list represents a term in the polynomial, with the value and exponent of each term stored in the node. The task is to write an algorithm that adds these two polynomials by iterating through their terms and combining like terms.
Detailed Explanation
To solve this problem, we'll first initialize a new linked list to store the result of the addition. Then, we'll iterate through the terms of both input polynomials simultaneously. For each term, we'll check if it's already present in the result polynomial or if it's a new term that needs to be added.
Here's a step-by-step breakdown of the algorithm:
- Initialize an empty linked list
resultto store the sum of the two polynomials. - Iterate through the terms of both input polynomials simultaneously using three pointers:
ifor the first polynomial,jfor the second polynomial, andkfor the result polynomial. - For each term, check if it's already present in the result polynomial by comparing the exponents. If the exponents are equal, add the values of the two terms and update the value of the corresponding node in the result polynomial.
- If the exponents are not equal, create a new node in the result polynomial with the term's value and exponent.
- Continue iterating through the terms until one of the input polynomials is exhausted.
- Return the resulting linked list as the sum of the two input polynomials.
The time complexity of this algorithm is O(n), where n is the total number of terms in both input polynomials, since we're iterating through each term once. The space complexity is also O(n) because we're storing the result polynomial in a new linked list.
Optimized Solutions
Here are optimized solutions for the Polynomial Addition problem in various programming languages:
Java
class Node {
int value;
int exponent;
public Node(int value, int exponent) {
this.value = value;
this.exponent = exponent;
}
}
public class PolynomialAddition {
public static Node[] addPolynomials(Node[] poly1, Node[] poly2) {
Node[] result = new Node[0];
for (int i = 0; i < Math.max(poly1.length, poly2.length); i++) {
int value = 0;
if (i < poly1.length && poly1[i].exponent == i) {
value += poly1[i].value;
}
if (i < poly2.length && poly2[i].exponent == i) {
value += poly2[i].value;
}
if (value != 0 || result.length == 0) {
Node newNode = new Node(value, i);
Node[] temp = new Node[result.length + 1];
System.arraycopy(result, 0, temp, 0, result.length);
temp[temp.length - 1] = newNode;
result = temp;
}
}
return result;
}
}