B. FILTERING MODELS
231
B
Filtering models
¶
1.
Filtering models:
Operate by fltering out oF the solution molecules
that are not part oF the solution.
¶
2. A solution can be treated mathematically as a fnite
bag
or
multiset
oF
molecules,
and fltering operations can be treated as operations to produce multi
sets From multisets.
¶
3.
Initial multiset:
Typically, For a problem oF size
n
,s
t
r
ing
so
Fs
ize
O
(
n
)arerequ
ired
.
Should contain enough strings to include many copies all possible so
lutions.
ThereFore, For an exponential problem, we will have
O
(
k
n
)str
ings
.
¶
4. This is essentially a bruteForce method.
B.1
Adleman: HPP
B.1.a
Review of HPP
¶
1.
Hamiltonian Path Problem (HPP):
The
Hamiltonian Path Prob
lem
is to determine, For a given directed graph
G
=(
V,E
)andtwoo
F
its vertices
v
in
,v
out
2
V
, whether there is a HP From
v
in
to
v
out
,tha
t
is, a path that goes through each vertex exactly once.
¶
2.
NPcomplete:
HPP is an NPcomplete problem.
¶
3. We will see that For Adleman’s algorithm the
number of algorithm steps
is linear in problem size.
¶
4.
Laboratory demonstration:
Leonard Adleman gave a laboratory
demonstration oF the procedure in 1994 (For
n
=7)
.
(By the way, he is the “A” oF “RSA.”)
¶
5. “In 2002, he and his research group managed to solve a ‘nontriv
ial’ problem using DNA computation. Specifcally, they solved a 20
variable SAT problem having more than 1 million potential solutions.
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CHAPTER IV. MOLECULAR COMPUTATION
7
1
2
3
4
5
6
Figure IV.6: HPP solved by Adleman. The HP is indicated by the dotted
edges. [source: Amos, Fig. 5.2]
They did it in a manner similar to the one Adleman used in his seminal
1994 paper.”
4
B.1.b
Problem Representation
¶
1. The heart of Adleman’s algorithm is a clever way to encode candidate
paths in DNA.
¶
2.
Vertices:
Vertices were represented by singlestranded 20mers, that
is, sequences of 20nt (nucleotides).
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 Fall '13
 BruceMacLennan
 DNA, Hamiltonian path problem, Adleman, Leonard Adleman

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