What Is An Abstract Data Type

In computer science, an abstract data type is a mathematical model for data types. An abstract data type is defined by its behavior from the point of view of a user, of the data, specifically in terms of possible values, possible operations on data of this type, and the behavior of these operations.

Abstract Data type (ADT) is a type (or class) for objects whose behaviour is defined by a set of value and a set of operations.

The definition of ADT only mentions what operations are to be performed but not how these operations will be implemented. It does not specify how data will be organized in memory and what algorithms will be used for implementing the operations. It is called “abstract” because it gives an implementation-independent view. The process of providing only the essentials and hiding the details is known as abstraction.

ADT

The user of data type does not need to know how that data type is implemented, for example, we have been using Primitive values like int, float, char data types only with the knowledge that these data type can operate and be performed on without any idea of how they are implemented. So a user only needs to know what a data type can do, but not how it will be implemented. Think of ADT as a black box which hides the inner structure and design of the data type. Now we’ll define three ADTs namely List ADT, stack ADT, Queue ADT.

  1. List ADT
    • The data is generally stored in key sequence in a list which has a head structure consisting of countpointers and address of compare function needed to compare the data in the list.ListADTStructure
    • The data node contains the pointer to a data structure and a self-referential pointer which points to the next node in the list.//List ADT Type Definitionstypedef struct node{ void *DataPtr; struct node *link;} Node;typedef struct{ int count; Node *pos; Node *head; Node *rear; int (*compare) (void *argument1, void *argument2)} LIST; 
    • The List ADT Functions is given below:ListADTFunctions
    A list contains elements of the same type arranged in sequential order and following operations can be performed on the list.
    • get() – Return an element from the list at any given position.
    • insert() – Insert an element at any position of the list.
    • remove() – Remove the first occurrence of any element from a non-empty list.
    • removeAt() – Remove the element at a specified location from a non-empty list.
    • replace() – Replace an element at any position by another element.
    • size() – Return the number of elements in the list.
    • isEmpty() – Return true if the list is empty, otherwise return false.
    • isFull() – Return true if the list is full, otherwise return false.
  2. Stack ADT
    • In Stack ADT Implementation instead of data being stored in each node, the pointer to data is stored.
    • The program allocates memory for the data and address is passed to the stack ADT.StackADT
    • The head node and the data nodes are encapsulated in the ADT. The calling function can only see the pointer to the stack.
    • The stack head structure also contains a pointer to top and count of number of entries currently in stack.//Stack ADT Type Definitionstypedef struct node{ void *DataPtr; struct node *link;} StackNode;typedef struct{ int count; StackNode *top;} STACK;
    A Stack contains elements of the same type arranged in sequential order. All operations take place at a single end that is top of the stack and following operations can be performed:
    • push() – Insert an element at one end of the stack called top.
    • pop() – Remove and return the element at the top of the stack, if it is not empty.
    • peek() – Return the element at the top of the stack without removing it, if the stack is not empty.
    • size() – Return the number of elements in the stack.
    • isEmpty() – Return true if the stack is empty, otherwise return false.
    • isFull() – Return true if the stack is full, otherwise return false.
  3. Queue ADT
    • The queue abstract data type (ADT) follows the basic design of the stack abstract data type.QueueADT
    • Each node contains a void pointer to the data and the link pointer to the next element in the queue. The program’s responsibility is to allocate memory for storing the data.//Queue ADT Type Definitionstypedef struct node{ void *DataPtr; struct node *next;} QueueNode;typedef struct { QueueNode *front; QueueNode *rear; int count;} QUEUE;
    A Queue contains elements of the same type arranged in sequential order. Operations take place at both ends, insertion is done at the end and deletion is done at the front. Following operations can be performed:
    • enqueue() – Insert an element at the end of the queue.
    • dequeue() – Remove and return the first element of the queue, if the queue is not empty.
    • peek() – Return the element of the queue without removing it, if the queue is not empty.
    • size() – Return the number of elements in the queue.
    • isEmpty() – Return true if the queue is empty, otherwise return false.
    • isFull() – Return true if the queue is full, otherwise return false.

From these definitions, we can clearly see that the definitions do not specify how these ADTs will be represented and how the operations will be carried out. There can be different ways to implement an ADT, for example, the List ADT can be implemented using arrays, or singly linked list or doubly linked list. Similarly, stack ADT and Queue ADT can be implemented using arrays or linked lists

Some authors describe object-oriented programming as programming abstract data types and their relationships. Within this section we introduce abstract data types as a basic concept for object-orientation and we explore concepts used in the list example of the last section in more detail.

3.1 Handling Problems

The first thing with which one is confronted when writing programs is the problem. Typically you are confronted with “real-life” problems and you want to make life easier by providing a program for the problem. However, real-life problems are nebulous and the first thing you have to do is to try to understand the problem to separate necessary from unnecessary details: You try to obtain your own abstract view, or model, of the problem. This process of modeling is called abstraction and is illustrated in Figure 3.1. 

The model defines an abstract view to the problem. This implies that the model focusses only on problem related stuff and that you try to define properties of the problem. These properties include

  • the data which are affected and
  • the operations which are identified

by the problem.

As an example consider the administration of employees in an institution. The head of the administration comes to you and ask you to create a program which allows to administer the employees. Well, this is not very specific. For example, what employee information is needed by the administration? What tasks should be allowed? Employees are real persons who can be characterized with many properties; very few are:

  • name,
  • size,
  • date of birth,
  • shape,
  • social number,
  • room number,
  • hair colour,
  • hobbies.

Certainly not all of these properties are necessary to solve the administration problem. Only some of them are problem specific. Consequently you create a model of an employee for the problem. This model only implies properties which are needed to fulfill the requirements of the administration, for instance name, date of birth and social number. These properties are called the data of the (employee) model. Now you have described real persons with help of an abstract employee.

Of course, the pure description is not enough. There must be some operations defined with which the administration is able to handle the abstract employees. For example, there must be an operation which allows you to create a new employee once a new person enters the institution. Consequently, you have to identify the operations which should be able to be performed on an abstract employee. You also decide to allow access to the employees’ data only with associated operations. This allows you to ensure that data elements are always in a proper state. For example you are able to check if a provided date is valid.

To sum up, abstraction is the structuring of a nebulous problem into well-defined entities by defining their data and operations. Consequently, these entities combine data and operations. They are not decoupled from each other.

3.2 Properties of Abstract Data Types

The example of the previous section shows, that with abstraction you create a well-defined entity which can be properly handled. These entities define the data structure of a set of items. For example, each administered employee has a name, date of birth and social number.

The data structure can only be accessed with defined operations. This set of operations is called interface and is exported by the entity. An entity with the properties just described is called an abstract data type (ADT).

Figure 3.2 shows an ADT which consists of an abstract data structure and operations. Only the operations are viewable from the outside and define the interface. 

Once a new employee is “created” the data structure is filled with actual values: You now have an instance of an abstract employee. You can create as many instances of an abstract employee as needed to describe every real employed person.

Let’s try to put the characteristics of an ADT in a more formal way:

Definition (Abstract Data Type) An abstract data type (ADT) is characterized by the following properties:1.It exports a type.2.It exports a set of operations. This set is called interface.3.Operations of the interface are the one and only access mechanism to the type’s data structure.4.Axioms and preconditions define the application domain of the type.

With the first property it is possible to create more than one instance of an ADT as exemplified with the employee example. You might also remember the list example of chapter 2. In the first version we have implemented a list as a module and were only able to use one list at a time. The second version introduces the “handle” as a reference to a “list object”. From what we have learned now, the handle in conjunction with the operations defined in the list module defines an ADT List:1.When we use the handle we define the corresponding variable to be of type List.2.The interface to instances of type List is defined by the interface definition file.3.Since the interface definition file does not include the actual representation of the handle, it cannot be modified directly.4.The application domain is defined by the semantical meaning of provided operations. Axioms and preconditions include statements such as

  • “An empty list is a list.”
  • “Let l=(d1, d2, d3, …, dN) be a list. Then l.append(dM) results in l=(d1, d2, d3, …, dN, dM).”
  • “The first element of a list can only be deleted if the list is not empty.”

However, all of these properties are only valid due to our understanding of and our discipline in using the list module. It is in our responsibility to use instances of List according to these rules.

Importance of Data Structure Encapsulation

The principle of hiding the used data structure and to only provide a well-defined interface is known as encapsulation. Why is it so important to encapsulate the data structure?

To answer this question consider the following mathematical example where we want to define an ADT for complex numbers. For the following it is enough to know that complex numbers consists of two parts: real part and imaginary part. Both parts are represented by real numbers. Complex numbers define several operations: addition, substraction, multiplication or division to name a few. Axioms and preconditions are valid as defined by the mathematical definition of complex numbers. For example, it exists a neutral element for addition.

To represent a complex number it is necessary to define the data structure to be used by its ADT. One can think of at least two possibilities to do this:

  • Both parts are stored in a two-valued array where the first value indicates the real part and the second value the imaginary part of the complex number. If x denotes the real part and y the imaginary part, you could think of accessing them via array subscription: x=c[0] and y=c[1].
  • Both parts are stored in a two-valued record. If the element name of the real part is r and that of the imaginary part is ix and y can be obtained with: x=c.r and y=c.i.

Point 3 of the ADT definition says that for each access to the data structure there must be an operation defined. The above access examples seem to contradict this requirement. Is this really true?

Let’s look again at the two possibilities for representing imaginary numbers. Let’s stick to the real part. In the first version, x equals c[0]. In the second version, x equals c.r. In both cases x equals “something”. It is this “something” which differs from the actual data structure used. But in both cases the performed operation “equal” has the same meaning to declare x to be equal to the real part of the complex number c: both cases archieve the same semantics.

If you think of more complex operations the impact of decoupling data structures from operations becomes even more clear. For example the addition of two complex numbers requires you to perform an addition for each part. Consequently, you must access the value of each part which is different for each version. By providing an operation “add” you can encapsulate these details from its actual use. In an application context you simply “add two complex numbers” regardless of how this functionality is actually archieved.

Once you have created an ADT for complex numbers, say Complex, you can use it in the same way like well-known data types such as integers.

Let’s summarize this: The separation of data structures and operations and the constraint to only access the data structure via a well-defined interface allows you to choose data structures appropriate for the application environment.

Generic Abstract Data Types

ADTs are used to define a new type from which instances can be created. As shown in the list example, sometimes these instances should operate on other data types as well. For instance, one can think of lists of apples, cars or even lists. The semantical definition of a list is always the same. Only the type of the data elements change according to what type the list should operate on.

This additional information could be specified by a generic parameter which is specified at instance creation time. Thus an instance of a generic ADT is actually an instance of a particular variant of the ADT. A list of apples can therefore be declared as follows:

    List<Apple> listOfApples;

The angle brackets now enclose the data type for which a variant of the generic ADT List should be created. listOfApples offers the same interface as any other list, but operates on instances of type Apple.

Notation

As ADTs provide an abstract view to describe properties of sets of entities, their use is independent from a particular programming language. We therefore introduce a notation here which is adopted from [3]. Each ADT description consists of two parts:

  • Data: This part describes the structure of the data used in the ADT in an informal way.
  • Operations: This part describes valid operations for this ADT, hence, it describes its interface. We use the special operation constructor to describe the actions which are to be performed once an entity of this ADT is created and destructor to describe the actions which are to be performed once an entity is destroyed. For each operation the provided arguments as well as preconditions and postconditions are given.

As an example the description of the ADT Integer is presented. Let k be an integer expression:ADT Integer isData
A sequence of digits optionally prefixed by a plus or minus sign. We refer to this signed whole number as N.OperationsconstructorCreates a new integer.add(k)Creates a new integer which is the sum of N and k.

Consequently, the postcondition of this operation is sum = N+k. Don’t confuse this with assign statements as used in programming languages! It is rather a mathematical equation which yields “true” for each value sumN and k after add has been performed.sub(k)Similar to add, this operation creates a new integer of the difference of both integer values. Therefore the postcondition for this operation is sum = Nk.set(k)Set N to k. The postcondition for this operation is N = k.end

The description above is a specification for the ADT Integer. Please notice, that we use words for names of operations such as “add”. We could use the more intuitive “+” sign instead, but this may lead to some confusion: You must distinguish the operation “+” from the mathematical use of “+” in the postcondition. The name of the operation is just syntax whereas the semantics is described by the associated pre- and postconditions. However, it is always a good idea to combine both to make reading of ADT specifications easier.

Real programming languages are free to choose an arbitrary implementation for an ADT. For example, they might implement the operation add with the infix operator “+” leading to a more intuitive look for addition of integers.

Abstract Data Types and Object-Orientation

ADTs allows the creation of instances with well-defined properties and behaviour. In object-orientation ADTs are referred to as classes. Therefore a class defines properties of objects which are the instances in an object-oriented environment.

ADTs define functionality by putting main emphasis on the involved data, their structure, operations as well as axioms and preconditions. Consequently, object-oriented programming is “programming with ADTs”: combining functionality of different ADTs to solve a problem. Therefore instances (objects) of ADTs (classes) are dynamically created, destroyed and used.

Excercises

 1.ADT Integer.(a)Why are there no preconditions for operations add and sub?(b)Obviously, the ADT description of Integer is incomplete. Add methods muldiv and any other one. Describe their impacts by specifying pre- and postconditions.2.Design an ADT Fraction which describes properties of fractions.(a)What data structures can be used? What are its elements?(b)What does the interface look like?(c)Name a few axioms and preconditions.3.Describe in your own words properties of abstract data types.4.Why is it necessary to include axioms and preconditions to the definition of an abstract data type?5.Describe in your own words the relationship between

  • instance and abstract data type,
  • generic abstract data type and corresponding abstract data type,
  • instances of a generic abstract data type.

Benefits of using Abstract Data Types

  • Code is easier to understand (e.g., it is easier to see “high-level” steps being performed, not obscured by low-level code).
  • Implementations of ADTs can be changed (e.g., for efficiency) without requiring changes to the program that uses the ADTs.
  • ADTs can be reused in future programs.

Fortunately for us, object-oriented programming languages (like Java) make it easy for programmers to use ADTs: each ADT corresponds to a class (or Java interface – more on this later) and the operations on the ADT are the class/interface’s public methods. The user, or client, of the ADT only needs to know about the method interfaces (the names of the methods, the types of the parameters, what the methods do, and what, if any, values they return), not the actual implementation (how the methods are implemented, the private data members, private methods, etc.).

Abstract Data Types

There are two parts to each ADT:

  1. The public or external part, which consists of:
    • the conceptual picture (the user’s view of what the object looks like, how the structure is organized)
    • the conceptual operations (what the user can do to the ADT)
  2. The private or internal part, which consists of:
    • the representation (how the structure is actually stored)
    • the implementation of the operations (the actual code)

In general, there are many possible operations that could be defined for each ADT; however, they often fall into these categories:

  1. initialize
  2. add data
  3. access data
  4. remove data

In this class, we will study a number of different abstract data types, different ways to implement them, and different ways to use them. Our first ADT (coming up in the next set of notes) is the List.

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