Theoretical and Experimental DNA Computation (Natural ...

Unrestricted model
3.2 Filtering Models 47
Adleman [3] provided the impetus for recent work through his experimental
solution to the Hamiltonian Path Problem. This solution, however, was not
expressed within a formal model of computation, and is therefore described
later in Chap. 5. In [98], Lipton considered Adleman’s specific model and
showed how it can encompass solutions to one other NP-complete problem.
Here we summarize the operations within Adleman’s subsequent unrestricted
model [4]. All operations are performed on sets of strings over some alphabet
α.
• separate(T,S). Given a set T and a substring S, createtwonewsets
+(T,S)and−(T,S), where +(T,S)isallstringsinT containing S, and
−(T,S)isallstringsinTnotcontaining S.
• merge(T1,T2,...,Tn). Given set T1,T2,...,Tn, create∪(T1,T2,...,Tn) =
T1 ∪ T2 ∪ ...Tn.
• detect(T ). Given a set T ,returntrue if T is nonempty, otherwise return
false.
For example, given α = {A, B, C}, the following algorithm returns true only
if the initial multi-set contains a string composed entirely of “A”s:
Input(T)
T ←−(T,B)
T ←−(T,C)
Output(detect(T ))
In [4] Adleman describes an algorithm for the 3-vertex-colorability problem.
Recall, from Chap. 2, that in order to obtain a proper coloring of a graph
G =(V,E) colors are assigned to the vertices in such a way that no two
adjacent vertices are similarly colored.
We now describe Adleman’s algorithm in detail. The initial set, T ,consists
of strings of the form c1,c2,...,cn, whereci ∈{ri,gi,bi} and n is |V |,
the number of vertices in G. Thus each string represents one possible (not
necessarily proper) coloring of the given graph. With reference to Fig. 2.11,
the coloring represented in (c) would be encoded by the string b0,g1,b2,r3,
and the coloring in (d) would be encoded by b0,g1,r2,r3.
We assume that all possible colorings are represented in T . The algorithm
proceeds as follows:
(1) read initial set T
(2) for each vertex do
(3) From T ,createred tube containing strings encoding this
vertex red, and create blue/green tube containing
all other strings
(4) Create blue tube from blue/green tube,
and create red tube from remaining strings