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 $d$.
翻译:我们研究滑动窗口模型的算法,这是数据流模型的一个重要变体,在这种模型中,我们的目标是计算流中固定长度后缀的某些函数。我们将布拉弗曼和奥斯特罗夫斯基(FOCS 2007)的平滑直方图框架扩展至几乎平滑的函数,其中包括所有子相加功能。具体地说,我们显示,如果一个子追加函数在插入式平流流模型中接近于$(1 ⁇ epsilon)美元,那么它也可以是(2 ⁇ epsilon)美元在滑动窗口模型中直接接近于美元,在滑动-平流模型中也接近于(2 ⁇ epreslon)美元最高值后端端端。我们演示我们的框架是如何产生新的近似方算算算法,对于不认可平滑动直方图技术的各种各样的问题,例如,在频率-正流模式中,一个对调值为美元(2 ⁇ -lickralalal)的滚动 $(2 ⁇ -lickral dalation) 几乎是(Oxxxxxlation)。