Theoretical and Experimental DNA Computation (Natural ...

3.2 Filtering Models 55
The maximum clique problem
A clique Ki isthecompletegraphoni vertices [67]. The problem of finding
a maximum independent set is closely related to the maximum clique problem.
Problem: Maximum clique
Given a graph G =(V,E) determine the largest i such that Ki is a subgraph
of G. HereKi isthecompletegraphoni vertices.
Solution:
• In parallel run the subgraph isomorphism algorithm for pairs of graphs
(G, Ki) for2≤i≤ n. The largest value of i for which a nonempty result
is obtained solves the problem.
• Complexity: O(|V |) parallel time.
The above examples fully illustrate the way in which the NP-complete problems
have a natural mode of expression within the model. The mode of solution
fully emulates the definition of membership of NP: that instances of problems
have candidate solutions that are polynomial-time verifiable and that there
are generally an exponential number of candidates.
Sticker model
We now introduce an alternative filtering-style model due to Roweis et al.
[133], named the sticker model. Within this model, operations are performed
on multisets of strings over the binary alphabet {0, 1}. Memory strands are
n characters in length, and contain k nonoverlapping substrings of length m.
Substrands are numbered contiguously, and there are no gaps between them
(Fig. 5.10a). Each substrand corresponds to a Boolean variable (or bit), so
within the model each substrand is either on or off. Weusethetermsbit and
substrand interchangeably
We now describe the operations available within the sticker model. They
are very similar in nature to those operations already described previously,
and we retain the same general notation. A tube is a multiset, its members
being memory strings.
• merge. Create the multiset union of two tubes.
• separate(N,i). Given a tube N andanintegeri, create two new tubes
+(N,i)and−(N,i), where +(N,i) contains all strings in N with substrand
i set to on, and−(N,i) contains all strings in N with substrand i set to
off.
• set(N,i). Given a tube N and an integer i, produce a new tube set(N,i)
in which the ith substrand of every memory strand is turned on.
• clear(N,i). Given a tube N and an integer i, produce a new tube set(N,i)
in which the ith substrand of every memory strand is turned off.