LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

274 Satoshi Kobayashi, Takashi Yokomori, and Yasubumi Sakakibara
Thus, the second equation holds. Note that the value of E( |−→ x|
|y|
) depends only on
←−
the loop length, the loop length mismatch, and the eight bases x[1],x[2],x[|x|−
1],x[|x|],y[1],y[2],y[|y| −1],y[|y|]. Since, for each loop length and loop length
mismatch, all possible combinations of those eight bases can be computed in
O(|S|) time,minI(v1,v2) can also be computed in O(|S|) time.
−−−−→
xα[1,i]|
(3) The set of free end structures β[j, n]y
←−−−− | such that x, y ∈ S∗ corresponds
to the set of all free end structures which could have boundary configuration
−−−−→
xα[1,i]|
v1. It is clear that E( β[j, n]y|
) takes the minimum value for some x, y ∈ S∗
←−−−−
such that |x|, |y| ≤ n. Thus, the third equation holds. Let s = xα[1,i]and
t = β[j, n]y. Then, note that the value of E( −→ s|
←−
t|
) depends only on the four
bases s[|s| −1],s[|s|],t[1],t[2]. Since for fixed lengths of s and t, all possible
combinations of those four bases can be computed in O(|S|) time,minD(v1)
can also be computed in O(|S|) time. ⊓⊔
5 Algorithm for Testing the Structure Freeness
For a set S of strings, we construct a weighted directed graph G(S) =(V,E,w),
called the Hydrogen Bond Network graph (HBN graph) of the set S, whereV
and E are defined as follows:
V = V ′ ∪{d, h},
V ′ ⎛
−→
α (i)
= { ⎝ ·
β
←− (j)
⎞
⎠ | α, β ∈ S, θ(α[i]) = β[j], 1 ≤ i, j ≤ n },
E =(V ′ × V ′ ) ∪ ({d}×V ′ ) ∪ (V ′ ×{h}).
Furthermore, the weight function w is defined as follows:
(1) for v1,v2 ∈ V ′ , we define:
(2) for v ∈ V ′ , we define:
(3) for v ∈ V ′ , we define:
w((v1,v2)) = minI(v1,v2),
w((d, v)) = minD(v),
w((v, h)) = minH(v).
For a path p in G,byw(p) we denote the sum of weights of the edges contained
in p, i.e., the weight of the path p.