Theory of ADTs
Students must be able to explain how a data structure stores information and the operations that can be performed on the structure. Students must be able to explain how nodes are added and removed, how the data in a node is accessed, and how the structure is traversed[edit]
There are many different types of data structures, each with its own set of characteristics and operations. Here are a few examples:
Arrays: An array is a contiguous block of memory that stores a fixed number of values of the same data type. Elements in an array are accessed using an index, and the index must be an integer. Adding or removing elements from an array can be costly, as the entire array may need to be resized and elements may need to be shifted.
Linked lists: A linked list is a linear data structure in which each element (called a "node") stores a value and a reference to the next node in the list. Linked lists can be used to store a variable number of elements, and elements can be added or removed from the list efficiently by updating the references of the surrounding nodes. Linked lists can be traversed by starting at the head node and following the references to each successive node.
Trees: A tree is a hierarchical data structure in which each element (called a "node") has zero or more child nodes. The top node in the tree is called the root, and the nodes below the root are called the children. Trees can be used to store and organize data in a hierarchical fashion, and they can be traversed by starting at the root and exploring the children of each node in a depth-first or breadth-first manner.
Hash tables: A hash table is a data structure that uses a hash function to map keys to indices in an array, allowing for fast insertion, deletion, and lookup of values. Hash tables can be used to implement efficient associative arrays, which allow you to store key-value pairs and access the value associated with a given key in constant time.
Graphs: A graph is a data structure that consists of a set of nodes and a set of edges connecting the nodes. Graphs can be used to represent relationships between different objects, and they can be traversed using algorithms such as depth-first search or breadth-first search.
Students must be able to explain how the following ADTs work: Singly linked lists; Doubly linked lists; Circularly linked lists; Binary trees; Stacks; and Queues. Students must be able to discuss the advantages of using a given structure, and sketch the structure of the ADTs based on the required operation[edit]
Sure! Here is a brief overview of these ADTs and their characteristics:
Singly linked list: A singly linked list is a linear data structure in which each element (called a "node") stores a value and a reference to the next node in the list. The nodes are connected in a one-way chain, and the last node has a reference to null to mark the end of the list. Singly linked lists are simple to implement and require only a small amount of memory, but they do not allow for efficient traversal in the reverse direction.
Doubly linked list: A doubly linked list is similar to a singly linked list, but each node also stores a reference to the previous node in the list. This allows for efficient traversal in both the forward and backward directions, but it requires twice as much memory to store the extra references.
Circularly linked list: A circularly linked list is a variant of the singly linked list in which the last node has a reference to the first node, forming a loop. This allows for efficient traversal in a circular fashion, but it can make it more difficult to determine the end of the list.
Binary tree: A binary tree is a hierarchical data structure in which each node has at most two child nodes. The top node in the tree is called the root, and the nodes below the root are called the children. Binary trees can be used to store and organize data in a hierarchical fashion, and they can be traversed using algorithms such as in-order, pre-order, and post-order traversal.
Stack: A stack is a linear data structure in which elements are added and removed from the same end (called the "top"). Stacks follow the "last in, first out" (LIFO) principle, meaning that the most recently added element is the first one to be removed. Stacks are often used to store temporary data, such as function arguments and return values.
Queue: A queue is a linear data structure in which elements are added to one end (called the "back") and removed from the other end (called the "front"). Queues follow the "first in, first out" (FIFO) principle, meaning that the first element added to the queue is the first one to be removed. Queues are often used to store data that needs to be processed in a first-come, first-served manner.
Each of these ADTs has its own set of advantages and disadvantages, and the best choice for a particular task will depend on the specific requirements of the problem.
Students must be able to explain how iterative and recursive solutions exist for the same problem, and how recursion can be used to solve problems which are sometimes more elegant[edit]
Iterative solutions are ones that use a looping construct (such as a for loop) to solve a problem by repeating a set of steps until a certain condition is met. Recursive solutions, on the other hand, are ones that solve a problem by breaking it down into smaller subproblems and calling themselves with the smaller subproblems.
There are often multiple ways to solve a given problem, and iterative and recursive approaches are two common options. In some cases, one approach may be more elegant or easier to understand than the other.
Recursion can be a powerful tool for solving problems, as it allows you to express solutions in a compact and intuitive way. Many problems that are difficult to solve iteratively can be solved elegantly using recursion. For example, the factorial function (n!) can be defined recursively as follows:
if n == 0:
return 1
else:
return n * factorial(n-1)
This recursive definition captures the idea that the factorial of 0 is 1, and the factorial of any other number is the product of that number and the factorial of the number one less than it.
However, it is important to note that recursive solutions can be less efficient than iterative ones, as they require the overhead of function calls and the maintenance of a call stack. In some cases, it may be necessary to use an iterative solution to achieve acceptable performance.