II. Notes on Data Structuring * - Cornell University

116 c.A.R. HOARE
The array provides a method of representing a particular arrangement of
cards in a pack, since each arrangement may be regarded as a mapping which
indicates for each of the fifty-two possible positions in a pack the value of the
card which occupies that position. Thus each possible arrangement may be
regarded as a value of the mapping type:
type cardpack = array 1.. 52 of cardface.
Of course, not all values of this type represent actual cardpacks, since there is
nothing to prevent some value of the type from mapping two different
positions onto the same card; which in real life is impossible.
Arrays with elements that are of Cartesian product type are sometimes
known as tables.
A third example of an array is that which represents all possible con-
figurations of character punching on a conventional punched card. This
may be regarded as a mapping M which maps each column number into a
character, namely the character punched in that column.
type punchcard = array 1.. 80 of character.
Any possible text punched into a card may be regarded as a single value of
type punchcard.
A fourth example shows an array which represents a possible value of a
page on a cathode ray tube display device. There are assumed to be 40 rows
and 27 character positions in each row. The effect of two dimensions can be
achieved by specifying the domain of the mapping as a Cartesian product of
the possible rows and the possible character positions within each row. This
is written as follows"
type spot = (row:l.. 40; column :l.. 27);
type display page = array spot of character.
An alternative method of dealing with a multidimensional array is to
regard it as an array of rows, where each row is an array of characters"
type display page = array I..40 of row;
type row = array 1..27 of character.
This is a more suitable abstract structure if the rows are to be processed
separately and the columns are not.
The cardinality of an array type is computed by raising the cardinality
of the range type to the power of the cardinality of the domain type, i.e.
cardinality (D -~ R) = cardinality (R) cardinality (D)
This may be proved by considering the number of decisions which have to
be made to specify completely a value of an array type. For each value of
the domain we have to choose between cardinality (R) possible values of the
range type. We have to make such a choice independently for each element
of the array, that is cardinality (D) times.