II. Notes on Data Structuring * - Cornell University

NOTES ON DATA STRUCTURING 115
6. TIlE ARRAY
The array is for many programmers the most familiar data structure, and in
some programming languages it is the only structure explicitly available.
From the abstract point of view, an array may be regarded as a mapping
between a domain of one type (the subscript range) and a range of some
possibly different type (the type of the array, or more accurately, the type of
its elements).
The type of a mapping is normally specified by a mathematician using an
arrow:
M: D ~ R;
where D is the domain type and R is the range type. An alternative notation
which will be more familiar to programmers is"
M:array D of R.
This notation is more expressive of the manner in which the data is repre-
sented, whereas the mathematical notation emphasises the abstract character
of the structure, independent of its representation.
When a particular value M of a mapping type is applied to a value x of the
domain type, it specifies some unique element of the range type, which is
known as M of x, and is written using either round or square brackets
M(x) or Mix].
Another name for a mapping is a function" the term "mapping" is used to
differentiate the data structure from a piece of program which actually
computes a value in its range from an argument in its domain. The essence of
the difference is that a mapping M is specified not by giving a computation
method but by explicitly listing the value of M(x) for each possible value x
in its domain. Thus an array can be used only for functions defined at a
finite set of points, whereas the domain of a computed function may be
infinite.
An example of a finite mapping is a monthtable, which specifies for each
month of the year the number of days it has:
type monthtable = array month of 28.. 31.
The domain is the month type and the range type consists of the integers
between 28 and 31 inclusive. A typical value of this tyiae may be simply
specified by listing the values of M(x) as x ranges over its domain. Thus
if M: monthtable is specified as
monthtable (Jan: 31, Feb: 28, March: 31, April: 30,
May: 31, June: 30, July: 31, Aug: 31,
Sept: 30, Oct: 31, Nov: 30, Dec: 31)
then M[Jan] = 31, M[Feb] = 28, and so on.