II. Notes on Data Structuring * - Cornell University

NOTES ON DATA STRUCTURING 149
The representation and manipulation of powersets and mappings with
infinite domains can be accomplished, provided that consideration is re-
stricted to sets with only a finite number of members, and mappings in which
only a finite number of elements take significant values; where "significant"
is defined as different from some specified null or default value. The powerset
of an infinite set is obviously also infinite; but since each value of the powerset
type contains only a finite number of elements, each value can be specified
simply by listing those elements in a finite period of time, and the list will
occupy only a finite amount of storage. Similarly, each value of a mapping
type with infinite domain can be finitely specified by listing all elements of
the domain which map onto significant values of the range type, together
with the value mapped in each case. A type which is restricted in this way is
known as sparse.
In fact the concept of sparsity is not confined to infinite bases and domains;
it may also be applied to very large but finite powersets, when the pro-
grammer knows that each actual set in which he is interested will contain
only a very small proportion of the potential members. For example, the
base type may contain hundreds of millions of values, but the programmer
may know that he only has to deal with sets of less than a hundred in size,
and perhaps most of them less than ten. It would be impossible to use the
bitpattern representation, since this requires hundreds of millions of bits;
but since each value actually used in a program contains only a few members,
these members can readily be listed in a comparatively small amount of
store. A powerset type of this sort is known as sparse. Similarly, arrays
with a very large domain, nearly all of which map onto the same default
value of the range, are said to belong to a sparse array type.
Sparse sets and arrays are frequently encountered in advanced data
processing applications, and their representation and manipulation present a
number of familiar problems. Our first example is the definition of a type
whose values are sets of car numbers. The cardinality of the carnumber type
is perhaps something like four thousand million; but the programmer
wishes only to deal with sets of cars owned by a single person; most of these
will have only one member, and very few will have more than ten. The
carset type may therefore be declared as sparse powerset:
type carset = sparse powerset carnumber;
As an example of a sparse array, we may take the type of mappings
between car owners and the set of cars they own. Each owner is represented
by name and address; since these are of arbitrary length, the owner type
may be defined:
type owner = sequence character;