Almost-Smooth Histograms and Sliding-Window Graph Algorithms

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