LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

180 Ashish Gehani, Thomas LaBean, and John Reif
0 < ¯ ∆ ¯ 1
∆ (c− 1+ɛ
p (= )L
p ), is
1
n ′ ,wheren ′ 2 n
=(1−p) L .
Lemma 5. The probability that a word w in D ′ is a parsed word of the “plaintext”
DNA sequence is > 1 − 1
e .
Proof. Let X be a “plaintext” DNA sequence of length n. ConsiderD ′′ ,the
subset of D containing words of length between c ′ L and L
p . D′′ contains at least
(1 − p) 2 of the parsed words of X by Lemma 4. D ′ is the subset of D ′′ which
consists of only words that have frequency > 1
n ′ .Considerawordvparsed from
X. The probability that a word w from D ′ is not v is < 1 − 1
n ′ by construction.
X has an expected number n
L parsed words. By Lemma 1, there are an expected
2 n
number (1 − p) L words with length in the range between c′ L and L
p .The
probability that w is none of these words is therefore < (1 − 1
n ′ n (1−p)2 ) L ≈ 1
e .
Thus, a word w in D ′ is some parsed word of X with probability > 1 − 1
e .