Add Two Polynomials Represented as Linked Lists

class Solution {
public:
    PolyNode* mergePolynomials(PolyNode* poly1, PolyNode* poly2) {
        PolyNode* head1 = poly1;
        PolyNode* head2 = poly2;

        PolyNode* dummy = new PolyNode();
        PolyNode* tail = dummy;

        while (head1 != NULL && head2 != NULL) {
            if (head1->power == head2->power) {
                if (head1->coefficient + head2->coefficient != 0) {
                    tail->next = new PolyNode(head1->coefficient + head2->coefficient, head1->power);
                    tail = tail->next;
                }
                head1 = head1->next;
                head2 = head2->next;
            } else if (head1->power > head2->power) {
                tail->next = head1;
                head1 = head1->next;
                tail = tail->next;
            } else {
                tail->next = head2;
                head2 = head2->next;
                tail = tail->next;
            }
        }
        
        if (head1 == NULL) {
            tail->next = head2;
        } else {
            tail->next = head1;
        }
        return dummy->next;
    }
};

Merge two polynomials represented as linked lists using a C++ function in an efficient manner. The provided solution initiates merging by comparing the powers of each node in the linked lists. Proceed with the following steps to perform the merge:

1. Initialize pointers for both polynomials (head1 and head2), and create a dummy node to ease the merge process.
2. Use a tail pointer linked to this dummy node to build the resulting linked list.
3. Traverse both linked lists simultaneously:
   • If nodes have the same power, combine their coefficients (only if nonzero) and move both pointers forward.
   • If the power in poly1 (head1) is greater, attach head1 to the merged list and advance the head1 pointer.
   • Conversely, if the power in poly2 (head2) is greater, attach head2 to the merged list and advance the head2 pointer.
4. Once the end of one list is reached, link the remaining part of the non-empty list directly to the merged list.
5. Return the starting node of the merged polynomial by omitting the dummy head.

This approach effectively merges two polynomials by traversing each only once, ensuring the solution is as efficient as possible in both time and space complexity, suitable for handling large polynomials in real-world applications.

class Solution {

    public PolyNode addPolynomials(PolyNode poly1, PolyNode poly2) {
        PolyNode ptr1 = poly1;
        PolyNode ptr2 = poly2;
        PolyNode dummyHead = new PolyNode();
        PolyNode tail = dummyHead;

        while (ptr1 != null && ptr2 != null) {
            if (ptr1.power == ptr2.power) {
                int newCoefficient = ptr1.coefficient + ptr2.coefficient;
                if (newCoefficient != 0) {
                    tail.next = new PolyNode(newCoefficient, ptr1.power);
                    tail = tail.next;
                }
                ptr1 = ptr1.next;
                ptr2 = ptr2.next;
            } else if (ptr1.power > ptr2.power) {
                tail.next = ptr1;
                ptr1 = ptr1.next;
                tail = tail.next;
            } else {
                tail.next = ptr2;
                ptr2 = ptr2.next;
                tail = tail.next;
            }
        }
        tail.next = (ptr1 != null) ? ptr1 : ptr2;

        return dummyHead.next;
    }
}

In the given Java solution, the task of adding two polynomials that are represented as linked lists is efficiently handled. This method involves defining a helper class PolyNode (not fully detailed here but assumed to have coefficient, power, and next properties).

To implement the addition:

1. Initialize two pointers, ptr1 and ptr2, to the heads of the two linked lists poly1 and poly2.
2. Create a dummy head node to simplify list manipulations and a tail pointer to keep track of the end of the resultant list.
3. Traverse both lists simultaneously. Compare the powers of the current nodes pointed to by ptr1 and ptr2:
   • If the powers are equal, add the coefficients. If the resultant coefficient is not zero, create a new node with the summed coefficient and current power, then attach this node to the list formed by tail.
   • Update ptr1 and ptr2 to their next nodes.
   • If ptr1's power is greater, attach ptr1 to tail and shift ptr1 to the next node.
   • If ptr2's power is higher, follow a similar step but with ptr2.
4. After exiting the loop, attach any remaining nodes of the longer list to the resultant list.
5. The head of the resulting polynomial is provided by dummyHead.next.

This solution effectively manages the addition of two polynomials in a way that is optimal for sparse representations, preserving space and time by only creating new nodes when necessary and directly attaching unchanged nodes from the input polynomials. This strategy ensures that the addition operation is both efficient and straightforward.

class Solution:

    def mergePoly(self, first, second):
        listA = first
        listB = second
        # Create initial placeholder node
        result_head = PolyNode()
        # Pointer to the last node added
        current_node = result_head

        # Process each polynomial term
        while listA is not None and listB is not None:
            if listA.power == listB.power:
                # Add coefficients if they won't cancel each other out
                if listA.coefficient + listB.coefficient != 0:
                    current_node.next = PolyNode(
                        listA.coefficient + listB.coefficient, listA.power
                    )
                    current_node = current_node.next
                listA = listA.next
                listB = listB.next
            elif listA.power > listB.power:
                current_node.next = listA
                listA = listA.next
                current_node = current_node.next
            else:
                current_node.next = listB
                listB = listB.next
                current_node = current_node.next

        # Attach any remaining terms
        if listA is None:
            current_node.next = listB
        else:
            current_node.next = listA
        return result_head.next

In this solution, you'll learn how to merge two polynomials represented as linked lists using Python. Each term of the polynomial is represented by a node in the respective linked list. Each node (here called PolyNode) consists of two properties: coefficient and power.

The mergePoly function takes two linked lists (first and second) as input representing the two polynomials you need to add together. The function systematically processes each polynomial:

1. Begin by initializing listA to reference first and listB to reference second.
2. Establish a placeholder starting node result_head of PolyNode().
3. Utilize a pointer current_node initialized to result_head which keeps track of the last node added to the resulting linked list.
4. As long as there are terms in both polynomial lists:
   • Compare the powers of the current terms from listA and listB.
   • If they have the same power, add up their coefficients and create a new node unless their sum is zero (which would effectively cancel out the term).
   • Append the new node to the current_node.
   • If listA has a term with a greater power than the corresponding term in listB, move listA's node to the result list without modification, and vice versa.
   • Advance current_node to reference the node just moved or created.
5. Once one of the lists is exhausted, connect the current_node to the non-empty list.
6. Return the next of result_head, which effectively skips the initial placeholder to return the head of the new merged polynomial list.

Using this approach, terms are added in a manner that conserves the order inherent to polynomial expressions, typically descending powers, and directly combines like terms. Ensure that terms with zero coefficients are not included in the resulting polynomial, keeping the output concise and true to mathematical representation.