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# 1.2. Abstract Data Types¶

## 1.2.1. Abstract Data Types¶

This module presents terminology and definitions related to techniques for managing the tremendous complexity of computer programs. It also presents working definitions for the fundamental but somewhat slippery terms “data item” and “data structure”. We begin with the basic elements on which data structures are built.

A type is a collection of values. For example, the Boolean type consists of the values true and false. The integers also form a type. An integer is a simple type because its values contain no subparts. A bank account record will typically contain several pieces of information such as name, address, account number, and account balance. Such a record is an example of an aggregate type or composite type. A data item is a piece of information or a record whose value is drawn from a type. A data item is said to be a member of a type.

A data type is a type together with a collection of operations to manipulate the type. For example, an integer variable is a member of the integer data type. Addition is an example of an operation on the integer data type.

A distinction should be made between the logical concept of a data type and its physical implementation in a computer program. For example, there are two traditional implementations for the list data type: the linked list and the array-based list. The list data type can therefore be implemented using a linked list or an array. But we don’t need to know how the list is implemented when we wish to use a list to help in a more complex design. For example, a list might be used to help implement a graph data structure.

As another example, the term “array” could refer either to a data type or an implementation. “Array” is commonly used in computer programming to mean a contiguous block of memory locations, where each memory location stores one fixed-length data item. By this meaning, an array is a physical data structure. However, array can also mean a logical data type composed of a (typically homogeneous) collection of data items, with each data item identified by an index number. It is possible to implement arrays in many different ways besides as a block of contiguous memory locations. The sparse matrix refers to a large, two-dimensional array that stores only a relatively few non-zero values. This is often implemented with a linked structure, or possibly using a hash table. But it could be implemented with an interface that uses traditional row and column indices, thus appearing to the user in the same way that it would if it had been implemented as a block of contiguous memory locations.

An abstract data type (ADT) is the specification of a data type within some language, independent of an implementation. The interface for the ADT is defined in terms of a type and a set of operations on that type. The behavior of each operation is determined by its inputs and outputs. An ADT does not specify how the data type is implemented. These implementation details are hidden from the user of the ADT and protected from outside access, a concept referred to as encapsulation.

A data structure is the implementation for an ADT. In an object-oriented language, an ADT and its implementation together make up a class. Each operation associated with the ADT is implemented by a member function or method. The variables that define the space required by a data item are referred to as data members. An object is an instance of a class, that is, something that is created and takes up storage during the execution of a computer program.

The term data structure often refers to data stored in a computer’s main memory. The related term file structure often refers to the organization of data on peripheral storage, such as a disk drive or CD.

Example 1.2.1

The mathematical concept of an integer, along with operations that manipulate integers, form a data type. The int variable type is a physical representation of the abstract integer. The int variable type, along with the operations that act on an int variable, form an ADT. Unfortunately, the int implementation is not completely true to the abstract integer, as there are limitations on the range of values an int variable can store. If these limitations prove unacceptable, then some other representation for the ADT “integer” must be devised, and a new implementation must be used for the associated operations.

Example 1.2.2

An ADT for a list of integers might specify the following operations:

1. Insert a new integer at a particular position in the list.

2. Return True if the list is empty.

3. Reinitialize the list.

4. Return the number of integers currently in the list.

5. Retrieve the integer at a particular position in the list.

6. Delete the integer at a particular position in the list.

From this description, the input and output of each operation should be clear, but the implementation for lists has not been specified.

Given two applications that make use of an ADT, one might use particular operations more frequently than the other, or they might have different time constraints for the various operations. Fortunately, an ADT can accomodate these difference in requirements by providing differing implementations.

Example 1.2.3

Two popular implementations for large disk-based database applications are hashing and the B-tree. Both support efficient insertion and deletion of records, and both support exact-match queries. However, hashing is more efficient than the B-tree for exact-match queries. On the other hand, the B-tree can perform range queries efficiently, while hashing is hopelessly inefficient for range queries. Thus, if the database application limits searches to exact-match queries, hashing is preferred. On the other hand, if the application requires support for range queries, the B-tree is preferred. Despite these performance issues, both implementations solve versions of the same problem: updating and searching a large collection of records.

The concept of an ADT can help us to focus on key issues even in non-computing applications.

Example 1.2.4

When operating a car, the primary activities are steering, accelerating, and braking. On nearly all passenger cars, you steer by turning the steering wheel, accelerate by pushing the gas pedal, and brake by pushing the brake pedal. This design for cars can be viewed as an ADT with operations “steer”, “accelerate”, and “brake”. Two cars might implement these operations in radically different ways, say with different types of engine, or front- versus rear-wheel drive. Yet, most drivers can operate many different cars because the ADT presents a uniform method of operation that does not require the driver to understand the specifics of any particular engine or drive design. These differences are deliberately hidden.

The concept of an ADT is one instance of an important principle that must be understood by any successful computer scientist: managing complexity through abstraction. A central theme of computer science is complexity and techniques for handling it. Humans deal with complexity by assigning a label to an assembly of objects or concepts and then manipulating the label in place of the assembly. Cognitive psychologists call such a label a metaphor. A particular label might be related to other pieces of information or other labels. This collection can in turn be given a label, forming a hierarchy of concepts and labels. This hierarchy of labels allows us to focus on important issues while ignoring unnecessary details.

Example 1.2.5

We apply the label “hard drive” to a collection of hardware that manipulates data on a particular type of storage device, and we apply the label “CPU” to the hardware that controls execution of computer instructions. These and other labels are gathered together under the label “computer”. Because even the smallest home computers today have millions of components, some form of abstraction is necessary to comprehend how a computer operates.

Consider how you might go about the process of designing a complex computer program that implements and manipulates an ADT. The ADT is implemented in one part of the program by a particular data structure. While designing those parts of the program that use the ADT, you can think in terms of operations on the data type without concern for the data structure’s implementation. Without this ability to simplify your thinking about a complex program, you would have no hope of understanding or implementing it.

Example 1.2.6

Consider the design for a relatively simple database system stored on disk. Typically, records on disk in such a program are accessed through a buffer pool rather than directly. Variable length records might use a memory manager to find an appropriate location within the disk file to place the record. Multiple index structures will typically be used to support access to a collection of records using multiple search keys. Thus, we have a chain of classes, each with its own responsibilities and access privileges. A database query from a user is implemented by searching an index structure. This index requests access to the record by means of a request to the buffer pool. If a record is being inserted or deleted, such a request goes through the memory manager, which in turn interacts with the buffer pool to gain access to the disk file. A program such as this is far too complex for nearly any human programmer to keep all of the details in their head at once. The only way to design and implement such a program is through proper use of abstraction and metaphors. In object-oriented programming, such abstraction is handled using classes.

Data types have both a logical form and a physical form. The definition of the data type in terms of an ADT is its logical form. The implementation of the data type as a data structure is its physical form. Sometimes you might see the term concrete implementation, but the word concrete is redundant. The figure below illustrates this relationship between logical and physical forms for data types. When you implement an ADT, you are dealing with the physical form of the associated data type. When you use an ADT elsewhere in your program, you are concerned with the associated data type’s logical form. Some sections of this book focus on physical implementations for a given data structure. Other sections use the logical ADT for the data structure in the context of a higher-level task.

Figure 1.2.1: The relationship between data items, abstract data types, and data structures.

The ADT defines the logical form of the data type. The data structure implements the physical form of the data type. Users of an ADT are typically programmers working in the same language as the implementer of the ADT. Typically, these programmers want to use the ADT as a component in another application. The interface to an ADT is also commonly referred to as the Application Programmer Interface, or API, for the ADT. The interface becomes a form of communication between the two programmers.

Example 1.2.7

A particular programming environment might provide a library that includes a list class. The logical form of the list is defined by the public functions, their inputs, and their outputs that define the class. This might be all that you know about the list class implementation, and this should be all you need to know. Within the class, a variety of physical implementations for lists is possible.