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Almost-Smooth Histograms and Sliding-Window Graph Algorithms

Abstract

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+)(1+\epsilon)-approximated in the insertion-only streaming model, then it can be (2+)(2+\epsilon)-approximated also in the sliding-window model with space complexity larger by factor O(1logw)O(\epsilon^{-1}\log w), where ww 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+)(2+\epsilon)-approximation algorithm for it. Another example is for streaming matrices, where we derive a new sliding-window (2+)(\sqrt{2}+\epsilon)-approximation algorithm for Schatten 44-norm. We then consider graph streams and show that many graph problems are subadditive, including maximum submodular matching, minimum vertex-cover, and maximum kk-cover, thereby deriving sliding-window O(1)O(1)-approximation algorithms for them almost for free (using known insertion-only algorithms). Finally, we design for every d(1,2]d\in (1,2] an artificial function, based on the maximum-matching size, whose almost-smoothness parameter is exactly dd

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