Compression of Samplable Sources

Abstract

Abstract. We study the compression of polynomially samplable sources. In particular, we give efficient prefix-free compression and decompression algorithms for three classes of such sources (whose support is a subset of {0, 1} n ).1. We show how to compress sources X samplable by logspace machines to expected length H(X) + O(1).Our next results concern flat sources whose support is in P. If H(X)≤ k = n−O(log n), we show how to compress to expected length k + polylog(n − k).3. If the support of X is the witness set for a self-reducible NP relation, then we show how to compress to expected length H(X) + 5.

On Pseudorandom Encodings

“…Therefore, E X is an optimal and efficient compression algorithm for X, with decompression algorithm D X ; this is true even for the relaxation in the CRS model. The existence of efficient compression algorithms for various categories of samplers was thoroughly studied [TVZ05]. In particular, the existence of compression algorithms for all efficiently samplable sources implies the inexistence of one-way functions (this is an observation attributed to Levin in [GS85]) since compressing the output of a pseudorandom generator to its entropy would distinguish it from a random string, and the existence of one-way functions implies the existence of pseudorandom generators [HILL99]).…”

“…One can circumvent this impossibility by studying whether compression can be achieved for more restricted classes of distributions, as was done e.g. in [TVZ05]. Our work can be seen as pursuing an orthogonal direction.…”

“…The output of a compression algorithm is not required to look pseudorandom and, in some cases, admits a specific structure which makes it easily distinguishable from uniform randomness, e.g. instances using Levin search, [TVZ05].…”

“…It is essential that the decoding algorithm D S depends on the sampler S, since due to pseudorandomness, D S on input of a random string needs to produce a sample that is in some sense close to the distribution produced by S. This argument does not hold necessarily for the encode algorithm. In [TVZ05], universal compression was studied. This translates to the following definition of PREH with universal encoding.…”

Untitled

“…The most common model for logarithmic-space samplers is one with streaming/one-way access to the pure random input bits. Topics that have been studied concerning such logspace samplers include compression [34], extraction [24], and min-entropy estimation [40]. One more paper worth mentioning is [10], which considers Markov random fields as succinct descriptions of distributions (though these descriptions would not be considered "samplers").…”