analysis the algorithm complexity of the solution below:

The cart problem is a classic problem in data structures and algorithms known as the "Shopping Cart Problem" or "Supermarket Shopping Cart Problem." It involves efficiently implementing operations for a shopping cart, such as adding items, removing items, updating quantities, and calculating the total cost.

One common algorithm used to solve the cart problem efficiently is to use a data structure called a "hash table" or "dictionary" to represent the shopping cart. In this approach, each item in the cart is stored as a key$-$value pair in the hash table, where the item name or identifier is the key, and the quantity is the value.

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

Initialize an empty hash table to represent the shopping cart.

To add an item to the cart:

a$.$ Check if the item is already present in the hash table.

b$.$ If it is present, update the quantity of the item by incrementing it$.$

c$.$ If it is not present, add a new key$-$value pair to the hash table, with the item name as the key and the quantity as the value.

To remove an item from the cart:

a$.$ Check if the item is present in the hash table.

b$.$ If it is present, decrement the quantity of the item.

c$.$ If the quantity becomes zero, remove the item from the hash table.

d$.$ If the item is not present or the quantity is already zero, display an appropriate error message.

To update the quantity of an item in the cart:

a$.$ Check if the item is present in the hash table.

b$.$ If it is present, update the quantity to the new value.

c$.$ If the item is not present, display an appropriate error message.

To calculate the total cost of the items in the cart:

a$.$ Initialize a variable "totalCost" to zero.

b$.$ Iterate over each key$-$value pair in the hash table.

c$.$ For each item, multiply the quantity by its unit price and add it to the totalCost variable.

d$.$ Return the totalCost as the final result.

This algorithm provides an efficient way to manage a shopping cart by utilizing a hash table to store and update the items and their quantities. The operations can be performed in constant or near$-$constant time, resulting in a fast and scalable solution for the cart problem in data structures.