II. Notes on Data Structuring * - Cornell University

128 c.A.R. HOARE
The program can be written easily
begin n, next:2..N; sieve:powerset 2..N;
end primefinder.
sieve: = range (2, N);
primes: = { };
while sieve ~ empty do
begin next: = min (sieve);
end
primes: v (next };
for n'= next step next until N do
sieve" - {n}
But if N is significantly large, say of the order of 10 000, this program
cannot be directly executed with any acceptable degree of efficiency. The
solution is to use this program as an abstract model of the algorithm, and
rewrite it in a more efficient fashion, using only operations on sets not
exceeding the word-length of the computer. We therefore need to declare
an array of words to represent the two sets, assuming that "wordlength"
is an environment enquiry giving the number of bits in a word"
primes, sieve:array O.. W of powerset O.. wordlength - 1
where W = (N + 1) + wordlength + 1.
This means that the two sets may be slightly larger than N, but for con-
venience we shall accept that harmless extension.
In order to access an individual bit of these sets, it is necessary to know
both the wordnumber and the bitnumber. Since we do not wish to use
division to find these, we will represent the counting variables n and next as
Cartesian products
n, next: (w, b :integer);
where w indicates the wordnumber and b indicates the bitnumber.
It is now as well to check the efficiency of this representation by recoding
the innermost loop first.
is recoded as"
for n: = next step next until N flo sieve:- {n };
n: = next;
while n. w ~< W do
begin sieve [n. w]: - {n. b };
n.b: = n.b + next.b;