II. Notes on Data Structuring * - Cornell University

150 ¢. A. R. HOARE
and has infinite cardinality, The required type is therefore declared as
sparse:
type carfile = sparse array owner of carset.
In a data processing application, a variable of carfile type would be known
as a random access file, and the owner would be known as the key element of
the file.
The next two examples are drawn from numerical applications. A vector
is a mapping from integers onto floating point numbers. A sparse vector is
one in which most of the elements are zero; consequently its initial value will
be the zero constant function, and all elements will remain zero unless an
explicit assignment is made of a different value:
type sparsevector = sparse array integer of real.
A sparse complex matrix may be defined in a similar way"
type irregular matrix = sparse array (row, column" integer)
of complex.
The next example is taken from the field of the translation of programming
languages to machine code. During the process of translation, the translator
needs to know certain information about each identifier declared in the
program, such as machine address allocated to the variable, its length and
type, etc. This information is assumed to belong to a type decode. The type of
an array which associates a decode with each identifier is given the name
dictionary and is declared:
type dictionary = sparse array ident of decode
Of course, the translator is interested in the decode only of those identifiers
actually declared in the source program. For the vast majority of possible
identifiers, the value given by any dictionary of this type will be that value of
the decode type which indicates that the identifier was undeclared.
The final example is of a type that causes familiar problems in a com-
mercial filing system and in real life--that of multidimensional cross-
classification. The customers of a firm are split up into a number of
geographical areas; they are also classified in a number of classes, in accord-
ance with the kind of product they purchase. On occasions it is required to
access all customers in an area, sequencing through all classes; on other
occasions to access all customers in a class, sequencing through the areas;
and finally, it is sometimes required to process all customers of a given class
in a given area. The abstract structure required to deal with this situation
is a two-dimensional sparse array of sparse sets"
sparse array (c:class; a:area) of sparse powerset customer.