Theoretical and Experimental DNA Computation (Natural ...

88 4 Complexity Issues
xANDy)) and depth 1, giving the size of the network as n(log n)(log n −
1) + 4n − 4 and the depth of the network as 0.5(log n)(log n +1).
Within the context of our strong model [10], the volumes of DNA and the
time required to simulate an n-input Batcher network within each model are
depicted in Table 4.2. K1 andK2 are constants, representing the number
of copies of a single strand required to give reasonable guarantees of correct
operation. The coefficient of 7 in the A&D time figure represents the number
of separate stages in a single level simulation. If we concentrate on the time
measure for n =2 k we arrive at the figures shown in Table 4.3.
Table 4.2. Model comparison for Batcher network simulation
Model Volume Time
O&R (K1)(n(log n)(log n − 1) + 4n − 4) n(log n)(log n − 1) + 4n +4
A&D (K2)(2.5(log n)(logn − 1) + 10n − 10) 7(log n)(log n +1)
Table 4.3. Time comparisons for different values of k
n k O&R A&D
1024 10 92196 770
2 20
20 4 ∗ 10 8
2940
2 40
40 1.7 ∗ 10 15
11480
Roweis et al. claim that their sticker model [133] is feasible using 2 56 distinct
strands. We therefore conclude that our implementation is technically feasible
for input sizes that could not be physically realized in silico using existing
fabrication techniques.
4.8 Example Application: Transitive Closure
We now demonstrate how a particular computation, transitive closure, may
be translated into DNA via Boolean circuits. In this way we demonstrate the
feasibility of converting a general algorithm into a sequence of molecular steps.
The transitive closure problem
The computation of the transitive closure of a directed graph is fundamental to
the solution of several other problems, including shortest path and connected
components problems. Several variants of the transitive closure problem exist;