LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

An Algorithm for Testing Structure Freeness of Biomolecular Sequences 275
Theorem 2. For a finite set S of strings of the same length and a real value F ,
the following two statements are equivalent.
(1) There is an E-minimal structured string α(T ) such that α ∈ S + , T �= ∅ and
E(α(T )) = F ,
(2) There is a path p from d to h of the graph G(S) such that w(p) =F .
Proof. (1)→(2) : Let α(T ) be an E-minimal structured string such that α ∈
S + , T �= ∅. Letcf(α(T )) = v0, ..., vk. Then, by ci (i =1, ..., k) we denote a 2cycle
in α(T ) with the boundary configuration (vi−1,vi), by c0 we denote a free
end structure in α(T ) with the boundary configuration v0,andbyck+1 we denote
a hairpin in α(T ) with the boundary configuration vk. By the definition of the
weight function w, foreachi =1, ..., k, wehavew((vi−1,vi)) = minI(vi−1,vi).
By the definition of minI(vi−1,vi), E(ci) ≥ minI(vi−1,vi) holds for each i =
1, ..., k. Suppose E(ci) >minI(vi−1,vi) forsomei, andletc ′ be a 2-cycle with
the boundary configuration (vi−1,vi) such that E(c ′ )=minI(vi−1,vi). Then,
by replacing ci by c ′ , we will obtain a new structured string α ′ (T ′ ) such that
α(T ) ≡ α ′ (T ′ )andE(α ′ (T ′ ))