II. Notes on Data Structuring * - Cornell University

NOTES ON DATA STRUCTURING 127
each occasion. A third method is to split the bitpattern into small parts, and
use table lookup on each part, adding together the results.
(5) The up and down operations can obviously be accomplished by right
or left shifts.
(6) The min of a set can be efficiently discovered by a standardise instruc-
tion, which automatically counts the number of shifts required to move the
first one-bit into the position next to the sign.
(7) The for statement may also be emciently constructed using standardi-
sation, masking off each one-bit as it is reached.
(8) The range operation can be accomplished by two shifts, the first of
which regenerates the sign bit.
Thus when the cardinality of the domain type is not greater than the
number of bits in the largest computer word to which logical and shift
operations can be applied, all these operations can be carried out with great
efficiency. If significantly more than one such word is involved, it will usually
pay to use selective updating operations rather than the normal result-
producing operators. Furthermore, operations such as size and min can
become rather inefficient, and it will often pay to store these values re-
dundantly together with the set, and keep them up to date whenever the value
of the set is updated, rather than recomputing them whenever they are
required.
When it is known that the cardinality of the base type is very large (perhaps
even infinite) compared with the size of the typical set, the bitpattern repre-
sentation altogether loses its attraction, since it no longer pays to store and
operate upon large areas of zeroes. The treatment of such sparse sets is
postponed to Section 10.
7.3. EXAMPLE
Problem: Write a program to construct a set
primes:powerset 2.. N;
containing all prime numbers in its base type.
Use the method of Eratosthenes' sieve to avoid all multiplications and
divisions.
The method of Eratosthenes is first to put all numbers in the "sieve" and
repeat the following until the sieve is empty:
Select and remove the smallest number remaining in the sieve (necessarily a
prime), and then step through the sieve, removing all multiples of that
number.