Theoretical and Experimental DNA Computation (Natural ...

90 4 Complexity Issues
Input matrix:A+I
Matrix Squaring
Level 1
(A+I)
Matrix Squaring
Level 2
(A+I)
2
4
i
2
(A+I)
Matrix Squaring
Level i
(A+I) 2
i +1
n /2
(A+I)
Matrix Squaring
Level p(n=2 p )
n
(A+I)
Fig. 4.8. Matrix squaring
Each ∨ can be computed in two steps using three NAND gates, since x ∨ y =
NAND(NAND(x, x),NAND(y, y)).
In total we use 2p 2 parallel steps and a network of size 5p2 3p −p2 2p NAND
gates, i.e., of 2(log 2 n) 2 depth and of 5n 3 log 2 n − 2n 2 log 2 n size.
We now consider the biological resources required to execute such a simulation.
Each gate is encoded as a single strand; in addition, at level 1 ≤ k ≤ D(S)
we require an additional strand per gate for removal of output sections. The total
number of distinct strands required is therefore n(−1+2logn 2 (−1+5kn)),
where k is a constant, representing the number of copies of a single strand
required to give a reasonable guarantee of correct operation.
4.9 P-RAM Simulation
We now describe, after [11], our DNA-based P-RAM simulation. At the coarsest
level of description, the method simply describes how any so-called P-RAM