Theoretical and Experimental DNA Computation (Natural ...

4.4 Ogihara and Ray’s Boolean Circuit Model 77
operation. We justify a new time complexity of O(n 2 ) as follows: at each iteration
of the for loop we perform one copy operation, nremoveoperations and
one union operation. The remove operation is itself a compound operation,
consisting of 2n pour operations. The copy and union operations consist of n
pour operations.
Similar considerations cause us to reassess the complexities of the algorithms
described in [65], according to Table 4.1.
Table 4.1. Time comparison of algorithms within the weak and strong models
Algorithm Weak Strong
Three coloring O(n) O(n 2 )
Hamiltonian path O(1) O(n)
Subgraph isomorphism O(n) O(n 2 )
Maximum clique O(n) O(n 2 )
Maximum independent set O(n) O(n 2 )
Although we have concentrated here on adjusting time complexities of algorithms
described in [65], similar adjustments can be made to other work. An
example is given in the following section.
4.4 Ogihara and Ray’s Boolean Circuit Model
Several authors [50, 113, 128] have described models of DNA computation
that are Turing-complete. In other words, they have shown that any process
that could naturally be described as an algorithm can be realized by a DNA
computation. [50] and [128] show how any Turing Machine computation may
be simulated by the addition of a splice operation to the models already
described in this book. In [113], Ogihara and Ray describe the simulation of
Boolean circuits within a model of DNA computation. The complexity of these
simulations is therefore of general interest. We first describe that of Ogihara
and Ray [113].
Boolean circuits are an important Turing-equivalent model of parallel computation
(see [54, 75]). An n-input bounded fan-in Boolean circuit may be
viewed as a directed, acyclic graph, S, withtwotypesofnode:ninputnodes
with in-degree (i.e., input lines) zero, and gate nodes with maximum in-degree
two. Each input node is associated with a unique Boolean variable xi from the
input set Xn =(x1,x2,...,xn). Each gate node, gi is associated with some
Boolean function fi ∈ Ω. We refer to Ω as the circuit basis. Acomplete basis
is a set of functions that are able to express all possible Boolean functions.
It is well known [143] that the NAND function provides a complete basis by
itself, but for the moment we consider the common basis, according to which