LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

An Algorithm for Testing Structure Freeness of Biomolecular Sequences 273
Proof. (1)Thesetofhairpins|x) such that x ∈ W (α : i, β : j) corresponds to
the set of all hairpins which could have the boundary configuration v1. However,
since there exist infinitely many such x’s, it is not clear whether the minimum
of E( |x) ) exists or not.
Recall that n is the length of each string in S and L is the constant introduced
in Section 2.2. Let x ∈ W (α : i, β : j) such that |x| >L+5n. Since
|x| > 5n, wecanwritex = α[i, n]α ′ γβ ′ β[1,j]forsomeα ′ ,β ′ ∈ S + and γ ∈ S.
Then,wehavethatx ′ = α[i, n]α ′ β ′ β[1,j]isalsoinW (α : i, β : j) andthat
|x| > |x ′ | >L.Sincethevaluesofh1 for |x) and|x ′ ) are equal to each other
and h2(|x|) ≥ h2(|x ′ |) holds (recall that h2 is weakly monotonically increasing),
we have E(|x ′ )) ≤ E(|x)). Therefore, we can conclude that E(|x)) takes the
minimum value for some x ∈ W (α:i, β:j) with|x| ≤L +5n. Thus,thefirstequation
holds. Note that the value of E(|x)) depends only on the loop length and
the four bases x[1],x[2],x[|x|−1],x[|x|]. Since, for each loop length, all possible
combinations of those four bases can be computed in O(|S|) time,minH(v1) can
also be computed in O(|S|) time.
(2)Thesetof2-cycles |−→ x|
|y|
such that x ∈ W (α : i, γ : k) andy∈ W (δ : l, β : j)
←−
corresponds to the set of all 2-cycles which could have boundary configuration
(v1,v2). However, since there exist infinitely many such x’s and y’s, it is not
clear whether the minimum of E( |−→ x|
|y|
)existsornot.
←−
Let x ∈ W (α : i, γ : k) andy∈ W (δ : l, β : j). Suppose that |x| > 6n. Since
|x| > 6n, wecanwritex = α[i, n]α ′ ηγ ′ γ[1,k]forsomeα ′ ,γ ′ ∈ S + and η ∈ S.
Then, we have that x ′ = α[i, n]α ′ γ ′ γ[1,j]isalsoinW (α:i, γ:k) andthat|x| >
|x ′ | > 5n. If|y| ≤5n, sety ′ = y. Otherwise, we can write y = δ[l, n]δ ′ ρβ ′ β[1,j]
for some δ ′ ,β ′ ∈ S + and ρ ∈ S, andsety ′ = δ[l, n]δ ′ β ′ β[1,j]. Then, we have
y ′ ∈ W (δ :l, β :j). Note that the loop length of |−→ x|
|y
←− | is greater than that of |−→ x ′ |
|y ′ |
.
←−
Further, note that the loop length mismatch of |−→ x|
|y
←− | is greater than or equal
to that of |−→ x ′ |
|y ′
←− | .Sincethevaluesofs1, b1 and e1 for |−→ x|
|y|
are equal to those for
←−
−→ ′ | x |
|y ′
←− | , respectively, we have E( |−→ x ′ |
|y ′
←− | ) ≤ E( |−→ x|
|y
←− | ) (recall that b2, e2 and e3 are weakly
monotonically increasing).
In case that |y| > 6n, in a similar manner, we can show the existence of
x ′ ∈ W (α:i, γ:k) andy ′ ∈ W (δ:l, β:j) such that E( |−→ x ′ |
|y ′
←− | ) ≤ E( |−→ x|
|y|
).
←−
Therefore, we can conclude that E( |−→ x|
|y|
) takes the minimum value for some
←−
x ∈ W (α : i, γ : k) andy∈ W (δ : l, β : j) such that |x| ≤ 6n and |y| ≤ 6n.