analysis the algorithm complexity of the solution below:

Second solution:

The second solution we will implement to solve the shopping cart problem is data structure called a doubly linked list to represent the shopping cart. And one of the advantages of using this type of data structure is flexibility because it allows adding, removing, or updating items in any order within the cart efficiently.

Here is a step$-$by$-$step description of the algorithm:

We define a node structure CartItem that stores information about an item in the cart. And this includes the following:

a$.$ product: $($string$)$ name of the product.

b$.$ price: $($double$)$ price of the product.

c$.$ quantity: $($integer$)$ number of the product.

And we will create a doubly linked list to hold these CartItem nodes. And this allows for efficient insertion, deletion, and modification of items in any order within the cart.

$1-$ Initialization:

$-$ Create an empty doubly linked list to represent the shopping cart.

$-$ Define functions for adding, removing, updating, and displaying items in the cart.

$2-$ Adding items: When a user adds an item to the cart

a$.$ Check if the item already exists in the cart.

$-$if yes, update the quantity in the existing node.

$-$if no$,$ create a new CartItem node with the product details and quantity.

b$.$ add new or update node to the end $($or beginning$)$ of the linked list depending on the desired order $($FIFO or LIFO$).$

c$.$ update pointers of surrounding nodes in the doubly linked list.

$3-$ Removing items: when the user removes an item from the cart:

a$.$ Find the node containing the item in the linked list.

b$.$ Update pointers of surrounding nodes to bypass the removed node.

c$.$ Remove the node from the list.

$4-$ Updating items: when a user updates the quantity of an item:

a$.$ Find the node containing the item.

b$.$ Update the quantity field in the node with the new value.

$5-$ Displaying cart:

a$.$ Traverse the linked list and display the product name, price, and quantity for each item.

b$.$ Calculate and display the total cost of items in the cart $($price$*$quantity for each item$).$