Theoretical and Experimental DNA Computation (Natural ...

96 4 Complexity Issues
The final stage is to reduce HP to a branch-free graph. Since cycles have
already been eliminated this involves merging conditional branches into a sequence
of instructions. This is accomplished by the transformation illustrated
in Fig. 4.9. In this process both branches of the conditional are executed; however,
only the branch for which the condition is true will affect the contents
of memory locations.
Q
C1
s
R
C1
C1^Q
~C1^R
Fig. 4.9. Merging of conditional branches
If GP is the initial control graph of P and LP is the graph resulting after all
of the transformations above have been effected, then LP is called the linear
control graph of the program P .
The total effect of the transformation described above is to replace the
initial program by a sequence of instructions , wherem
is at most a constant factor larger than the runtime of P .Furthermore,each
instruction Qi has the form
< condition >< active − instruction >
where < condition > is a composition of a constant number of tests arising
from JEQ instructions, and is any non-branching
instruction. The interpretation of this form is that
changes the contents of a memory location if and only if < condition > is
true.
We then convert the linear control graph into a series of combinational
circuits. For each of the R(n) parallel processing (i.e., par do) instructions in
SA we construct a combinational circuit Ck with the following specification:
Ck takes as input r × S(n) Boolean values arranged in S(n) blocksofr bits
(we assume, without loss of generality, that r ≤ S(n)); the values in the i th
s