Theoretical and Experimental DNA Computation (Natural ...

76 4 Complexity Issues
4.3 A Strong Model of DNA Computation
In what follows we refine the weak model just described. We assume that the
initial tube (which takes at most linear time to construct) is already set up.
The pour operation is fundamental to all compound operations within
our weak model. We must therefore reassess the time complexity of these
operations. The remove operation requires adding to U
1. i tubes containing short primers to mark selected strands
2. restriction enzymes to destroy marked strands
This operation is inherently sequential, since there must be a pause between
steps 1 and 2 in order to allow the primers to mark selected strands. Therefore,
the remove operation takes O(i) time. Creating the union of i tubes is an
inherently sequential operation, since the assistant must first pour U1 into U,
then U2, andsoon,uptoUi. Rather than taking constant time, the union
operation actually takes O(i) time. It is clear that the copy operation may be
thought of as a reverse-union operation, since the contents of a single tube U
are split into many tubes, {Ui}. Therefore, copy takes O(i) time.
We may strengthen the strong model even further by taking kinetic issues
into consideration. All filtering models rely upon the fact that every possible
solution to the given problem is present in the initial tube with equal
probability. These solutions are created by the action of component molecules
“sticking” together to form long chains. Given a particular problem of size n,
we may naively assume that the creation of the initial library takes O(log n)
time [93]. Given a particular solution, we may imagine a balanced binary tree,
where each leaf node represents a component molecule, an internal node represents
a “sticking-together” reaction, and the root node represents the string
encoding the solution. Since the depth of the tree is log n, the time complexity
of the creation phase immediately follows.
However, Kurtz et al.[93] present a more detailed analysis of the kinetic
issues involved, suggesting that the creation phase actually takes Ω(n 2 )time.
Given the space available, and the fact that in what follows we assume that
the initial library is set up, we omit a detailed description of the arguments
presented in [93]. However, we acknowledge the importance of taking kinetic
issues into consideration in the future.
Complexity comparisons
In this section we compare time complexities for algorithms previously described
[65] within both the weak and strong models. In particular, we examine
in detail the problem of generating a set of permutations. This will
characterize the general form of comparisons that can be made, so that in the
space available we merely tabulate comparisons for other algorithms.
In [65] the authors claimed a time complexity for the Permutations algorithm
of O(n). This is based on the assumption that remove is a constant-time