  We study algorithms for the sliding-window model, an important variant of the
data-stream model, in which the goal is to compute some function of a
fixed-length suffix of the stream. We extend the smooth-histogram framework of
Braverman and Ostrovsky (FOCS 2007) to almost-smooth functions, which includes
all subadditive functions. Specifically, we show that if a subadditive function
can be $(1+\epsilon)$-approximated in the insertion-only streaming model, then
it can be $(2+\epsilon)$-approximated also in the sliding-window model with
space complexity larger by factor $O(\epsilon^{-1}\log w)$, where $w$ is the
window size.
  We demonstrate how our framework yields new approximation algorithms with
relatively little effort for a variety of problems that do not admit the
smooth-histogram technique. For example, in the frequency-vector model, a
symmetric norm is subadditive and thus we obtain a sliding-window
$(2+\epsilon)$-approximation algorithm for it. Another example is for streaming
matrices, where we derive a new sliding-window
$(\sqrt{2}+\epsilon)$-approximation algorithm for Schatten $4$-norm. We then
consider graph streams and show that many graph problems are subadditive,
including maximum submodular matching, minimum vertex-cover, and maximum
$k$-cover, thereby deriving sliding-window $O(1)$-approximation algorithms for
them almost for free (using known insertion-only algorithms). Finally, we
design for every $d\in (1,2]$ an artificial function, based on the
maximum-matching size, whose almost-smoothness parameter is exactly $d$.
