Trevisan, Luca, Salil Vadhan, and David Zuckerman. “
Compression of samplable sources.”
Computational Complexity: Special Issue on CCC'04 14, no. 3 (2005): 186227.
Publisher's VersionAbstract
We study the compression of polynomially samplable sources. In particular, we give efficient prefixfree compression and decompression algorithms for three classes of such sources (whose support is a subset of \(\{0, 1\}^n\)).

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 \(\mathbf{P}\).

If \(H(X) ≤ k = n−O(\log n)\), we show how to compress to expected length \(k + \mathrm{polylog}(n − k)\).

If the support of \(X\) is the witness set for a selfreducible \(\mathbf{NP}\) relation, then we show how to compress to expected length \(H(X)+ 5\).
CC2005.pdf BenSasson, Eli, Oded Goldreich, Prahladh Harsha, Madhu Sudan, and Salil Vadhan. “
Short PCPs verifiable in polylogarithmic time.” In
Proceedings of the 20th Annual IEEE Conference on Computational Complexity (CCC '05), 120134, 2005, 120134.
Publisher's VersionAbstractWe show that every language in NP has a probabilistically checkable proof of proximity (i.e., proofs asserting that an instance is "close" to a member of the language), where the verifier's running time is polylogarithmic in the input size and the length of the probabilistically checkable proof is only polylogarithmically larger that the length of the classical proof. (Such a verifier can only query polylogarithmically many bits of the input instance and the proof. Thus it needs oracle access to the input as well as the proof, and cannot guarantee that the input is in the language  only that it is close to some string in the language.) If the verifier is restricted further in its query complexity and only allowed q queries, then the proof size blows up by a factor of 2/sup (log n)c/q/ where the constant c depends only on the language (and is independent of q). Our results thus give efficient (in the sense of running time) versions of the shortest known PCPs, due to BenSasson et al. (STOC '04) and BenSasson and Sudan (STOC '05), respectively. The time complexity of the verifier and the size of the proof were the original emphases in the definition of holographic proofs, due to Babai et al. (STOC '91), and our work is the first to return to these emphases since their work. Of technical interest in our proof is a new complete problem for NEXP based on constraint satisfaction problems with very low complexity constraints, and techniques to arithmetize such constraints over fields of small characteristic.