Affiliation:
1. National Research University Higher School of Economics, Moscow, Russia
2. Towson University, Towson, MD, USA
Abstract
In a lossless compression system with target lengths, a compressor 𝒞 maps an integer
m
and a binary string
x
to an
m
-bit code
p
, and if
m
is sufficiently large, a decompressor 𝒟 reconstructs
x
from
p
. We call a pair (
m,x
)
achievable
for (𝒞,𝒟) if this reconstruction is successful. We introduce the notion of an optimal compressor 𝒞
opt
by the following universality property: For any compressor-decompressor pair (𝒞,𝒟), there exists a decompressor 𝒟
′
such that if
(m,x)
is achievable for (𝒞,𝒟), then (
m
+ Δ ,
x
) is achievable for (𝒞
opt
, 𝒟
′
), where Δ is some small value called the overhead. We show that there exists an optimal compressor that has only polylogarithmic overhead and works in probabilistic polynomial time. Differently said, for any pair (𝒞,𝒟), no matter how slow 𝒞 is, or even if 𝒞 is non-computable, 𝒞
opt
is a fixed compressor that in polynomial time produces codes almost as short as those of 𝒞. The cost is that the corresponding decompressor is slower.
We also show that each such optimal compressor can be used for distributed compression, in which case it can achieve optimal compression rates as given in the Slepian–Wolf theorem and even for the Kolmogorov complexity variant of this theorem.
Funder
Russian Science Foundation
National Science Foundation
Publisher
Association for Computing Machinery (ACM)
Subject
Artificial Intelligence,Hardware and Architecture,Information Systems,Control and Systems Engineering,Software
Reference39 articles.
1. B. Bauwens A. Makhlin N. Vereshchagin and M. Zimand. 2018. Short lists with short programs in short time. Comput. Complex. 27 1 (2018) 31–61.
2. Bruno Bauwens and Marius Zimand. 2014. Linear list-approximation for short programs (or the power of a few random bits). In Proceedings of the IEEE 29th Conference on Computational Complexity. IEEE, 241–247. DOI:DOI:10.1109/CCC.2014.32
3. Entropy and the complexity of the trajectories of a dynamic system;Brudno A. A.;Trudy Moskovskogo Matematicheskogo Obshchestva,1982
4. Individual communication complexity
5. M. R. Capalbo, O. Reingold, S. P. Vadhan, and A. Wigderson. 2002. Randomness conductors and constant-degree lossless expanders. In Proceedings of the STOC.John H. Reif (Ed.), ACM, 659–668.