# Almost-Smooth Histograms and Sliding-Window Graph Algorithms

## TL;DR

This paper extends the smooth-histogram framework to almost-smooth functions, enabling efficient sliding-window algorithms for various subadditive problems in data streams, including graphs, matrices, and norms.

## Contribution

It introduces a generalized framework for almost-smooth functions, broadening the applicability of sliding-window algorithms to new problems.

## Key findings

- Developed a $(2+\epsilon)$-approximation for subadditive functions in sliding windows.
- Derived new algorithms for Schatten 4-norm and graph problems with near-optimal approximation ratios.
- Showed many graph problems are subadditive, enabling efficient sliding-window solutions.

## 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 $(1+\epsilon)$-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 $d$.