  Slepian-Wolf theorem is a well-known framework that targets almost lossless
compression of (two) data streams with symbol-by-symbol correlation between the
outputs of (two) distributed sources. However, this paper considers a different
scenario which does not fit in the Slepian-Wolf framework. We consider two
identical but spatially separated sources. We wish to study the universal
compression of a sequence of length $n$ from one of the sources provided that
the decoder has access to (i.e., memorized) a sequence of length $m$ from the
other source. Such a scenario occurs, for example, in the universal compression
of data from multiple mirrors of the same server. In this setup, the
correlation does not arise from symbol-by-symbol dependency of two outputs from
the two sources. Instead, the sequences are correlated through the information
that they contain about the unknown source parameter. We show that the
finite-length nature of the compression problem at hand requires considering a
notion of almost lossless source coding, where coding incurs an error
probability $p_e(n)$ that vanishes with sequence length $n$. We obtain a lower
bound on the average minimax redundancy of almost lossless codes as a function
of the sequence length $n$ and the permissible error probability $p_e$ when the
decoder has a memory of length $m$ and the encoders do not communicate. Our
results demonstrate that a strict performance loss is incurred when the two
encoders do not communicate even when the decoder knows the unknown parameter
vector (i.e., $m \to \infty$).
