Theoretical and Experimental DNA Computation (Natural ...

94 4 Complexity Issues
denoted by P (t1,...,tn), is the maximum size of | L | in an instruction for
x ∈ L par do ...that is executed by A. The processor requirements in inputs
of size n, denoted P (n), is
P (n) =max {t1,...,tn} P (t1,...,tn).
Finally, the space required by A on , denoted by S(t1,...,tn), is
the maximum value, s, such that M[s] is referenced during the computation
of A. The space required by A on inputs of size n, denoted by S(n), is S(n) =
max {t1,...,tn} S(t1,...,tn). It is assumed that S(n) ≥ P (n), i.e., that there is
a memory location that is associated with each processor activated.
Let f :N→N be some function over N. We say that a function Q :N→N
has parallel-time (space, processor) complexity f(n)isthereexistsaCREWP-
RAM algorithm computing Q andsuchthatT (n) ≤ f(n)(S(n) ≤ f(n),P(n) ≤
f(n)).
We are particularly concerned with the complexity class of problems that
can be solved with parallel algorithms with a polynomial number of processors
and having polylogarithmic parallel runtime, i.e., the class of functions M for
which there is an algorithm A achieving T (n) ≤ (log n) k and P (n) ≤ n r for
some constants k and r; these form the complexity class NC of problems
regarded as having efficient parallel algorithms.
4.10 The Translation Process
We now provide a description of the translation from CREW P-RAM algorithm
to DNA. The input to the process is a CREW P-RAM program A,
which takes n input items, uses S(n) memory locations, employs P (n) different
processors, and has a total running time of T (n). The final output takes
the form of a DNA algorithm Q with the following characteristics: Q takes as
input an encoding of the initial input to A, returns as its output an encoding
of the final state of the S(n) memory locations after the execution of A, hasa
total running time of O(T (n)logS(n)), and requires a total volume of DNA
O(T (n)P (n)S(n)logS(n)). The translation involves the following stages:
1. Translate the control program into a straight-line program
2. For each parallel processing instruction, construct a combinational circuit
to simulate it
3. Cascade the circuits generated in Stage 2
4. Translate the resulting circuit into a DNA algorithm
We first translate the control program to a straight-line program SA. SA consists
of a sequence of R(n) ≤ T (n) instructions, Ik(1 ≤ k ≤ R(n)), each of
which is of the form for x ∈ Lk par do instk(x). We “unwind” the control
program by replacing all loops involving m repetitions of an instruction sequence
I by m copies I1,...,Im of I in sequence. Since the loops in the control