Question

Are problems or shortanswer questions. a. Show how the numbers 1066 and 1492 would be represented in a linked list with one digit per node. b. Use a linked list to represent the sum of these integers. c. Outline an algorithm to show how the calculation might be carried out in a computer.

Short answer

1066 and 1492 are represented with linked lists, sum is 2558; algorithm adds digits with carry.

Step-by-step solution

Represent Numbers as Linked Lists

To represent the numbers 1066 and 1492 each with one digit per node in a linked list, you begin by creating a linked list for each number. For 1066: - Start with a node containing '1', followed by a node containing '0', then a node containing '6', followed by another node containing '6'. For 1492: - Start with a node containing '1', followed by a node containing '4', then a node containing '9', and finally a node containing '2'. Each node points to the next node so that the entire number is sequentially linked.

Add Corresponding Digits

To find the sum of two numbers represented as linked lists, you need to add corresponding digits from each list, starting from the least significant digit (the end of the list). Align both lists by their end and add: 1. Add 6 (from 1066) and 2 (from 1492) to get 8. 2. Add 6 and 9, resulting in 15. Write down 5 and carry over 1. 3. Add 0 and 4, including the carry-over of 1, to get 5. 4. Add 1 and 1 to get 2. Thus, the number 2558 is represented.

Store the Result in a Linked List

Create a new linked list to hold the result from the addition process: - Start with a node containing '2'. - The next node will contain '5'. - The following node contains another '5'. - The last node of the list contains '8'. This list represents the digits of the sum 2558, with each node pointing to the next.

Outline the Algorithm

The algorithm to perform this addition is as follows: 1. Initialize two pointers, each starting at the head of each linked list (1066 and 1492). 2. Traverse each list from the start to end, adding respective digits along with any carry from the previous addition (starting with carry=0). 3. For each addition, create a new node in the result list with the last digit of the sum. 4. If the sum is greater than 9, set the carry for the next addition. 5. If any list becomes exhausted, continue adding with the carry. 6. If there is a remaining carry after the lists are exhausted, add a new node to handle it. 7. Continue until there are no more nodes in either list and no carry.

Key concepts

Linked Lists

A linked list is a popular data structure that consists of a sequence of nodes, each containing some data and a reference to the next node in the sequence. This design is crucial in scenarios where dynamic memory allocation, efficient insertion, and deletion operations are necessary. Linked lists come in various types, such as singly linked lists, doubly linked lists, and circular linked lists.
The fundamental building block of a linked list is the node. In our example with the numbers 1066 and 1492, each digit of the number is stored in a separate node, starting with the most significant digit. This structure enables flexibility, allowing the list to grow as needed by simply adding new nodes. This capability is particularly beneficial in algorithms that require frequent and dynamic data modification.
Linked lists are advantageous for representing numbers in computations because of their efficient sequential access properties. However, they do have some drawbacks, such as consuming more space due to the storage of references and having slower element access time than arrays since it requires traversal from the head to any given node.

Algorithm Design

Algorithm design involves creating a step-by-step method to solve a specific problem. It requires an understanding of the problem space and an efficient approach to reaching a solution. In our problem of adding numbers using linked lists, the algorithm must carefully manage pointers and potentially manage carryovers from one digit addition to the next.
The core idea is to emulate the process of adding numbers on paper, where digits are added sequentially starting from the least significant end. Each digit from the two lists is added together, just like traditional columnar addition:

• Carry over any excess value greater than 9 to the next higher digit.
• If one list is shorter, assume a digit of zero for its missing parts.
• Continue until all nodes and carry overs are processed.

This design principle allows the algorithm to dynamically handle numbers of varying lengths without extra pre-processing of the input data. By using a linked list, this algorithm handles overflow naturally and ensures that only the necessary computations are performed, maintaining efficiency.

Integer Addition

Adding integers, especially large ones, is a common operation in computer science. When working with linked lists to represent numbers, each digit of the integer is stored in a separate node. This allows the handling of very large numbers which aren't feasible with basic data types.
When adding integers using linked lists, we start at the least significant digit, which aligns with the end of the list, and add corresponding digits from each list, similar to how we do it manually on paper:

• Any sum that exceeds 9 results in a carry to the next nodes' addition.
• The sum of current digits and any carry is stored in a new linked list.
• If there's a remaining carry at the end of the addition, a new node is created to account for it.

This method of integer addition is beneficial in handling big numbers that exceed typical integer limits in programming languages, leveraging the linked list's dynamic growth capability.

Computer Representation of Numbers

In computer science, numbers are often represented in a binary format suitable for computation. However, some operations and applications require more complex representations. Representing numbers in linked lists is one such method, especially useful for operations that exceed standard data types' capacities.

• This representation mimics manual numeric operations, such as addition and subtraction, allowing flexibility.
• Each node effectively represents a digit of the number, facilitating direct manipulation.
• Digits are processed linearly, simplifying the algorithmic processing of these numbers.

Using linked lists for number representation is beneficial when dealing with large numbers or needing to perform complex manipulations that default data types cannot efficiently accommodate. This representation can simplify problems where number lengths vary significantly or where high precision is required.