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 \(\left( 1+{{\varepsilon }}\right) \)-approximated in the insertion-only streaming model, then it can be \(\left( 2+{{\varepsilon }}\right) \)-approximated also in the sliding-window model with space complexity larger by factor \(O{\negmedspace }\left( {{\varepsilon }}^{-1}\log w\right) \), 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 \(\left( 2+{{\varepsilon }}\right) \)-approximation algorithm for it. Another example is for streaming matrices, where we derive a new sliding-window \(\left( \sqrt{2}+{{\varepsilon }}\right) \)-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 \(O{\negmedspace }\left( 1\right) \)-approximation algorithms for them almost for free (using known insertion-only algorithms). Finally, we design for every \(d\in \left( 1,2\right] \) an artificial function, based on the maximum-matching size, whose almost-smoothness parameter is exactly d.