II. Notes on Data Structuring * - Cornell University

126 c.A.R. HOARE
7.2 REPRESENTATION
In choosing a computer representation for powersets, it is desirable to
ensure that all the basic operations can be executed simply by single machine
code instructions; and further, that the amount of store occupied is
minimised. For most data structure storage methods, there is a fundamental
conflict between these two objectives, and consequently a choice between
representation methods must be made by the programmer; but in the case
of" powersets the two objectives can be fully reconciled, provided that the
base type is not too large.
The recommended method of representation is to allocate as many bits
in the store as there are potential members in the set. Thus to each value
of the base type there is a single bit which takes the value one if it is in fact a
member, or zero if it is not. For example, each value of type colour can be
represented in three bits; the most significant corresponding to the primary
colour red, and the least significant corresponding to blue. Thus the orange
colour is represented as 110 and red as 100. Each set of size n is represented
as a bitpattern with exactly n ones in the appropriate positions. The null set
is accordingly represented as an all-zero bitpattern.
Another example is afforded by the "hand" type, which requires fifty-two
bits for its representation, one corresponding to each value of type cardface.
In this case, it is advisable to use the minimal representation of the base
type, to avoid unused gaps in the bitpattern representation.
Since the number of values of a powerset type is always an exact power of
two, for powersets of small base there can be no more economical method
of utilising storage on a binary computer than that of the bitpattern repre-
sentation. It remains to show that the operations defined over the powerset
type can be executed with high efficiency.
(1) The unitset of x may be obtained by loading a single 1 into the signbit
position, and shifting it right x places. On computers on which shifting is
slow, the same effect may be obtained by table lookup. The construction of a
set out of components may be achieved by taking the logical union of all the
corresponding unit sets.
(2) A membership test x in s may be made by shifting s up x places and
looking at the most significant bit: 1 stands for true and 0 for false.
(3) Logical intersection, union, and complementation are often available
as single instructions on binary computers.
(4) The size of a set can sometimes be discovered by a builtin machine
code instruction for counting the bits in a word. Otherwise the size can be
determined by repeated standardisation, masking off the next-to-sign bit on