Theoretical and Experimental DNA Computation (Natural ...

74 4 Complexity Issues
The weak parallel filtering model
Here we recall the basic legal operations on sets (or tubes) [65] within what we
now refer to as the weak model. The operation set described here is constrained
by biological feasibility, but all operations are currently realisable with current
technology. The biological implementation of this operation set is described
in detail in Chap. 5.
• remove(U, {Si}). This operation removes from the tube U, in parallel, any
string which contains at least one occurrence of any of the substrings Si.
• union({Ui},U). This operation, in parallel, creates the tube U, whichis
the set union of the tubes Ui.
• copy(U, {Ui}). In parallel, this operation produces a number of copies, Ui,
of the tube U.
• select(U). This operation selects an element of U uniformly at random, if
U is the empty set then empty is returned.
We now describe an example algorithm within the model. The problem solved
is that of generating the set of all permutations of the integers 1 to n. The
initial set and the filtering out of strings which are not permutations were
described earlier. The only non-self-evident notation employed below is ¬i to
mean (in this context) any integer in the range which is not equal to i.
Problem: Permutations
Generate the set Pn of all permutations of the integers {1, 2,...,n}.
Solution:
• Input: The input set U consists of all strings of the form p1i1p2i2 ...pnin
where, for all j, pj uniquely encodes “position j” and each ij is in
{1, 2,...,n}. Thus each string consists of n integers with (possibly) many
occurrences of the same integer.
• Algorithm:
for j =1ton − 1 do
begin
copy(U, {U1,U2,...,Un})
for i =1,2,... , n and all k>j
in parallel do remove(Ui, {pj¬i, pki})
union({U1,U2,...,Un},U)
end
Pn ← U
• Complexity: O(n) parallel-time.