II. Notes on Data Structuring * - Cornell University

NOTES ON DATA STRUCTURING 151
A similar example may arise in the description of family relationship
among persons:
type children = sparse array (mother, father: person) of
sparse powerset person:
This array caters for multiple marriages better than the more tree-like
representations of a family, which can be defined as a recursive structure.
In the case of sparse arrays, it is sometimes useful to regard them as
partial rather than total mappings. A partial mapping is one which does not
necessarily give a value for each member of its domain type. In other words,
the actual domain over which it is defined is a subset of the domain type.
For such an array type it is necessary to introduce an additional constant
omega, denoting a mapping which is everywhere undefined. It is also useful
to introduce a function
domain (x)
which delivers as result the set of subscripts for elements of x which are
actually defined. Thus the programmer can sequence through all the defined
elements, or test whether a particular element is defined or not. Many of the
examples quoted above might well have been declared as partial instead of
sparse. In the case of a partial mapping, the default value does not have to be
recorded.
10.1 REPRESENTATION
Sparse sets and arrays are usually represented by simply keeping a record
of the default value and those members or elements which are significant;
thus the representation borrows techniques which are used in the case of the
sequence type to deal with structures of changeable size. A sparse set may be
regarded as a special case of a sparse mapping, which maps all its members
onto the Boolean value true, and all its non-members onto the default value
false. Thus their representations are closely similar to those of sparse arrays,
and do not require separate treatment.
A sparse mapping consists of a number of elements. Each element
of the mapping is represented as the Cartesian product of its subscript and
its value; in this case the subscript is known as the key, and the value is
known as the information associated with the element, and the juxtaposition
of the two will be known as an entry. In the case of a set which is sparse,
there is no need to record any information, since the presence of the key
itself is sufficient to indicate that this value is a member of the set. Thus an
entry for a sparse set consists only of a key.