LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

DNA-based Cryptography 179
can be compressed without loss of information. Random sequences can not be
compressed and therefore have an entropy rate of 1.
Lossless data compression with an algorithm such as Lempel-Ziv [51], is theoretically
asymptotically optimal. For sequences whose length n is large, the
compression ratio approaches the entropy rate of the source. In particular, it
is of the form (1 + ɛ(n))HS, whereɛ(n) → 0forn → ∞. Algorithms such
as Lempel-Ziv build an indexed dictionary of all subsequences parsed that can
not be constructed as a catenation of current dictionary entries. Compression is
performed by sequentially parsing the input text, finding maximal length subsequences
already present in the dictionary, and outputting their index number
instead. When a subsequence is not found in the dictionary, it is added to it (including
the case of single symbols). Algorithms can achieve better compression
by making assumptions about the distribution of the data [4]. It is possible to
use a dictionary of bounded size, consisting of the most frequent subsequences.
Experimentation on a wide variety of text sources shows that this method can
be used to achieve compression within a small percentage of the ideal [43]. In
particular, the compression ratio is of the form (1 + ɛ)HS, forasmallconstant
ɛ>0 typically of at most 1/10 if the source length is large.
Lemma 1. The expected length of a parsed word is between L
1+ɛ
L = logb n
HS .
and L, where
Proof. Assume the source data has an alphabet of size b. An alphabet of the
same size can be used for the compressed data. The dictionary can be limited to
a constant size. We can choose an algorithm that achieves a compression ratio
within 1 + ɛ of the asymptotic optimal, for a small ɛ>0. Therefore, for large n,
we can expect the compression ratio to approach (1 + ɛ)HS.
Each parsed word is represented by an index into the dictionary, and so its size
would be log b n if the source had no redundancy. By the choice of compression
algorithm, the length of the compressed data is between HS and HS(1+ɛ) times
the length of the original data. From these two facts, it follows that the expected
length of a code word will be between log b n
(1+ɛ)HS and log b n
HS .
Lemma 2. A parsed word has length ≤ L
p
with probability ≥ 1 − p.
Proof. The probability of a parsed word having length > L
p is
1 − 1
c(1+ɛ) and c′ = c −
c− 1
1+ɛ
p
> 0.
Proof. The maximum length of a parsed word has an upper bound in practice.
We assume that this is cL for a small constant c>1. We use ∆ to denote
the difference between the maximum possible and the actual length of a parsed
word, and ¯ ∆ to denote the difference’s expected value. Applying Lemma 1,