Asymptotic rate-distortion functions for coding precedence relations (Corresp.)

An improved analysis for an information system is described in which the input and output alphabets consist of (equiprobable) ordered lists of m items, with two distortion criteria: 1) the fraction of item-pairs found out of order and 2) the fraction of items found in wrong positions. For the former it is shown that, as
m \rightarrow \infty
, the rate-distortion function
R_{1} (D)
is equivalent to
mG(D)
with, for small
D
,
G(D)=\log (2/D)-2+ O(D)
. For the latter,
R_{2} (D) = m \log m(1 - D) + m(l - D)[\log(1 - D) - 1] + o(m)
.