GG给定由一个循环链表表示的两个多项式数，任务是通过将同一变量的幂的系数相加来将这两个多项式相加。
注意：在给定的多项式中，包含x的较高幂的项将首先出现。
例子：
Input:
1st Number = 5x^2 * y^1 + 4x^1 * y^2 + 3x^1 * y^1 + 2x^1
2nd Number = 3x^1 * y^2 + 4x^1
Output:
5x^2 * y^1 + 7x^1 * y^2 + 3x^1 * y^1 + 6x^1
Explanation:
The coefficient of x^2 * y^1 in 1st numbers is 5 and 0 in the 2nd number. Therefore, sum of the coffiecient of x^2 * Y^1 is 5.
The coefficient of x^1 * y^2 in 1st numbers is 4 and 3 in the 2nd number. Therefore, sum of the coefficient of x^1 * Y^2 is 7.
The coefficient of x^1 * y^1 in 1st numbers is 3 and 0 in the 2nd number. Therefore, sum of the coefficient of x^1 * Y^1 is 2.
The coefficient of x^1 * Y^0 in 1st numbers is 2 and 4 in the 2nd number. Therefore, sum of the coefficient of x^1 * Y^0 is 6.
Input:
1st Number = 3x^3 * y^2 + 2x^2 + 5x^1 * y^1 + 9y^1 + 2
2nd Number = 4x^3 * y^3 + 2x^3 * y^2 + 1y^2 + 3
Output:
4x^3 * y^3 + 5x^3 * y^2 + 2x^2 + 5x^1 * y^1 + 1y^2 + 9y^1 + 5
处理方法：按照以下步骤解决问题：
- 创建两个循环链表，其中每个节点将由系数、 x 的幂、 y 的幂和指向下一个节点的指针组成。
- 遍历两个多项式并检查以下条件：
- 如果第一个多项式的 x的幂大于第二个多项式的 x的幂，则将第一个多项式的节点存储在结果多项式中并增加多项式1 的计数器。
- 如果第一个多项式的 x的幂小于第二个多项式的 x的幂，则将第二个多项式的节点存储在结果多项式中并增加多项式2 的计数器。
- 如果第一个多项式的 x的幂等于第二个多项式的 x的幂，并且第一个多项式的 y的幂大于第二个多项式的 y的幂，则将第一个多项式的节点存储在结果多项式中并增加多项式1 的计数器。
- 如果第一个多项式的 x的幂等于第二个多项式的 x的幂，并且第一个多项式的 y的幂等于第二个多项式的 y的幂，则将两个多项式的系数之和存储在结果多项式中，并增加两个多项式的计数器1和多项式2 。
- 如果在第一个多项式或第二个多项式中还有节点要遍历，则将它们附加到结果多项式中。
- 最后，打印结果多项式。
下面是上述方法的实现：
C++
// C++ program to implement
// the above approach
#include
using namespace std;
// Structure of a node
// in a circular linked list
struct Node {
// Stores coefficient
// of a node
int coeff;
// Stores power of'
// variable x of a node
int powx;
// Stores power of
// variable y of a node
int powy;
// Stores pointer
// to next node
struct Node* next;
};
// Function to dynamically create a node
void create_node(int c, int p1, int p2,
struct Node** temp)
{
// Stores new node
struct Node *r;
// Stores temp node
struct Node *z
= *temp;
// Dyanamically create a new node
r = (struct Node*)malloc(
sizeof(struct Node));
// Update coefficient
// of r
r->coeff = c;
// Update power of
// variable x in r
r->powx = p1;
// Update power of
// variable y in r
r->powy = p2;
// If z is null
if (z == NULL) {
// Update temp node
(*temp) = r;
// Update next pointer
// of temp node
(*temp)->next = (*temp);
}
else {
// Update next pointer
// of z
r->next = z->next;
// Update next pointer
// of z
z->next = r;
// Update temp Node
(*temp) = r;
}
}
// Function to add polynomial of two list
void add_poly(struct Node* poly1,
struct Node* poly2, struct Node** temp)
{
// Stores head node of polynomial1
struct Node *start1 = poly1;
// Stores head node of polynomial1
struct Node *start2 = poly2;
// Update poly1
poly1 = poly1->next;
// Update poly2
poly2 = poly2->next;
// Traverse both circular linked list
while ((poly1 != start1 &&
poly2 != start2)) {
// Stores new node
struct Node* r;
// Stores temp node
struct Node* z
= *temp;
// Dynamically create a new node
r = (struct Node*)malloc(
sizeof(struct Node));
// Update coefficient of r
r->coeff = 0;
// If power of x of poly1 is
// greater than power of x of poly2
if (poly1->powx > poly2->powx) {
// Update coefficient of r
r->coeff = poly1->coeff;
// Update of power of x in r
r->powx = poly1->powx;
// Update of power of y in r
r->powy = poly1->powy;
// Update poly1
poly1 = poly1->next;
}
// If power of x of 1st polynomial is
// less than power of x of 2nd poly
else if (poly1->powx < poly2->powx) {
// Update coefficient OF r
r->coeff = poly2->coeff;
// Update power of x in r
r->powx = poly2->powx;
// Update power of y in r
r->powy = poly2->powy;
// Update ploy2
poly2 = poly2->next;
}
// If power of x of 1st polynomial is
// equal to power of x of 2nd poly
else {
// Power of y of 1st polynomial is
// greater than power of y of poly2
if (poly1->powy > poly2->powy) {
// Update coefficient of r
r->coeff = poly1->coeff;
// Update power of x in r
r->powx = poly1->powx;
// Update power of y in r
r->powy = poly1->powy;
// Update poly1
poly1 = poly1->next;
}
// If power of y of poly1 is
// less than power of y of ploy2
else if (poly1->powy <
poly2->powy) {
// Update coefficient of r
r->coeff = poly2->coeff;
// Update power of x in r
r->powx = poly2->powx;
// Update power of y in r
r->powy = poly2->powy;
// Update poly2
poly2 = poly2->next;
}
// If power of y of 1st poly is
// equal to power of y of ploy2
else {
// Update coefficient of r
r->coeff = poly2->coeff
+ poly1->coeff;
// Update power of x in r
r->powx = poly1->powx;
// Update power of y in r
r->powy = poly1->powy;
// Update poly1
poly1 = poly1->next;
// Update poly2
poly2 = poly2->next;
}
}
// If z is null
if (z == NULL) {
// Update temp
(*temp) = r;
// Update next pointer
// of temp
(*temp)->next = (*temp);
}
else {
// Update next pointer
// of r
r->next = z->next;
// Update next pointer
// of z
z->next = r;
// Update temp
(*temp) = r;
}
}
// If there are nodes left to be
// traversed in poly1 or poly2 then
// append them in resultant polynomial .
while (poly1 != start1 ||
poly2 != start2) {
// If poly1 is not empty
if (poly1 != start1) {
// Stores new node
struct Node *r;
// Stores temp node
struct Node *z = *temp;
// Create new node
r = (struct Node*)malloc(
sizeof(struct Node));
// Update coefficient or r
r->coeff = poly1->coeff;
// Update power of x in r
r->powx = poly1->powx;
// Update power of y in r
r->powy = poly1->powy;
// Update poly1
poly1 = poly1->next;
// If z is null
if (z == NULL) {
// Update temp
(*temp) = r;
// Update pointer
// to next node
(*temp)->next = (*temp);
}
else {
// Update next pointer
// of r
r->next = z->next;
// Update next pointer of z
z->next = r;
// Update temp
(*temp) = r;
}
}
// If poly2 is not empty
if (poly2 != start2) {
// Stores new node
struct Node *r;
// Stores temp node
struct Node *z = *temp;
// Create new node
r = (struct Node*)malloc(
sizeof(struct Node));
// Update coefficient of z
z->coeff = poly2->coeff;
// Update power of x in z
z->powx = poly2->powx;
// Update power of y in z
z->powy = poly2->powy;
// Update poly2
poly2 = poly2->next;
// If z is null
if (z == NULL) {
// Update temp
(*temp) = r;
// Update next pointer
// of temp
(*temp)->next = (*temp);
}
else {
// Update next pointer
// of r
r->next = z->next;
// Update next pointer
// of z
z->next = r;
// Update temp
(*temp) = r;
}
}
}
// Stores new node
struct Node *r;
// Stores temp node
struct Node *z = *temp;
// Create new node
r = (struct Node*)malloc(
sizeof(struct Node));
// Update coefficient of r
r->coeff = 0;
// If power of x of start1 greater than
// power of x of start2
if (start1->powx > start2->powx) {
// Update coefficient of r
r->coeff = start1->coeff;
// Update power of x in r
r->powx = start1->powx;
// Update power of y in r
r->powy = start1->powy;
}
// If power of x of start1 less than
// power of x of start2
else if (start1->powx < start2->powx) {
// Update coefficient of r
r->coeff = start2->coeff;
// Update power of x in r
r->powx = start2->powx;
// Update power of y in r
r->powy = start2->powy;
}
// If power of x of start1 equal to
// power of x of start2
else {
// If power of y of start1 greater than
// power of y of start2
if (start1->powy > start2->powy) {
// Update coefficient of r
r->coeff = start1->coeff;
// Update power of x in r
r->powx = start1->powx;
// Update power of y in r
r->powy = start1->powy;
}
// If power of y of start1 less than
// power of y of start2
else if (start1->powy <
start2->powy) {
// Update coefficient of r
r->coeff = start2->coeff;
// Update power of x in r
r->powx = start2->powx;
// Update power of y in r
r->powy = poly2->powy;
}
// If power of y of start1 equal to
// power of y of start2
else {
// Update coefficient of r
r->coeff = start2->coeff
+ start1->coeff;
// Update power of x in r
r->powx = start1->powx;
// Update power of y in r
r->powy = start1->powy;
}
}
// If z is null
if (z == NULL) {
// Update temp
(*temp) = r;
// Update next pointer
// of temp
(*temp)->next = (*temp);
}
else {
// Update next pointer of r
r->next = z->next;
// Update next pointer of z
z->next = r;
// Update temp
(*temp) = r;
}
}
// Display the circular linked list
void display(struct Node* node)
{
// Stores head node of list
struct Node* start = node;
// Update node
node = node->next;
// Traverse the list
while (node != start &&
node->coeff != 0) {
// Print coefficient of
// current node
cout << node->coeff;
// If power of variable x
// is not zero
if (node->powx != 0)
cout << "x^" << node->powx;
// If power of variable x
// and y is not zero
if(node->powx != 0 &&
node->powy != 0)
cout<<" * ";
// If power of variable y
// is not zero
if (node->powy != 0)
cout << "y^" << node->powy;
// Add next term of
// the polynomial
if (node != start &&
node->next->coeff != 0) {
cout << " + ";
}
// Update node
node = node->next;
}
// Print coefficient of
// current node
cout << node->coeff;
// If power of variable x
// is not zero
if (node->powx != 0)
cout << "x^" << node->powx;
// If power of variable y
// is not zero
if (node->powy != 0)
cout << "y^" << node->powy;
cout << "\n\n";
}
// Driver Code
int main()
{
// Stores node of
// first polynomial
struct Node *poly1 = NULL;
// Stores node of
// second polynomial
struct Node *poly2 = NULL;
// Stores node of resultant
// polynomial
struct Node *store = NULL;
// Create first polynomial
create_node(5, 2, 1, &poly1);
create_node(4, 1, 2, &poly1);
create_node(3, 1, 1, &poly1);
create_node(2, 1, 0, &poly1);
create_node(3, 0, 1, &poly1);
create_node(2, 0, 0, &poly1);
// Create second polynomial
create_node(3, 1, 2, &poly2);
create_node(4, 1, 0, &poly2);
create_node(2, 0, 1, &poly2);
create_node(6, 0, 0, &poly2);
// Function call to add
// two polynomial
add_poly(poly1, poly2, &store);
// Display polynomial 1
cout << "Polynomial 1"
<< "\n";
display(poly1);
// Display polynomial 2
cout << "Polynomail 2"
<< "\n";
display(poly2);
// Display final addition of 2-variable polynomial
cout << "Polynomial after addition"
<< "\n";
display(store);
return 0;
} 输出：
Polynomial 1
5x^2 * y^1 + 4x^1 * y^2 + 3x^1 * y^1 + 2x^1 + 3y^1 + 2
Polynomail 2
3x^1 * y^2 + 4x^1 + 2y^1 + 6
Polynomial after addition
5x^2 * y^1 + 7x^1 * y^2 + 3x^1 * y^1 + 6x^1 + 5y^1 + 8
时间复杂度： O(M + N)，其中 M 和 N 分别是第一个和第二个列表中的节点数。
辅助空间： O(M + N)