Theoretical and Experimental DNA Computation (Natural ...

4.12 A Worked Example: The List Ranking Problem 103
for all k ∈ L in parallel do
if P (k) �= P (P (k)) then
begin
distance(k) :=distance(k)+distance(P (k));
P (k) :=P (P (k));
end;
for all k ∈ L in parallel do
rank(k) :=distance(k);
Obviously this solves the list ranking problem in O(log n) parallel steps, using
n processors and O(n) memory locations. We consider the realization of this
for n =2 m . In the translation, a total of 5n memory locations are used, which
are laid out as follows:
M[1],...,M[n] hold the list pointers next(k)(1 ≤ k ≤ n)
M[n +1],...,M[2n] holdthevaluesP (k)
M[2n +1],...,M[3n] holddistance(k)
M[3n +1],...,M[4n] are used for temporary results
M[4n +1],...,M[5n] hold the output values rank(k)
The unwound control program contains precisely m +2 in parallel do instructions,
the middle m of which are identical. For the first instruction we
have
for all k ∈ L in parallel do
begin
P (k) :=next(k);
if P (k) �= k then
distance(k) :=1;
else
distance(k) :=0;
end
In terms of the basic instruction set this translates to
1. LDA k
2. STA n + k /* P (k) :=next(k)
3. LDC k
4. SUB n + k
5. STA 3n + k
6. JEQ 3n + k, 10 /* P (k) �= k
7. LDC 1
8. STA 2n + k /* distance(k) :=1
9. JUMP 12