# Smooth Histograms for Sliding Windows

@article{Braverman2007SmoothHF, title={Smooth Histograms for Sliding Windows}, author={Vladimir Braverman and Rafail Ostrovsky}, journal={48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)}, year={2007}, pages={283-293} }

In the streaming model elements arrive sequentially and can be observed only once. Maintaining statistics and aggregates is an important and non-trivial task in the model. This becomes even more challenging in the sliding windows model, where statistics must be maintained only over the most recent n elements. In their pioneering paper, Datar, Gionis, Indyk and Motwani [15] presented exponential histograms, an effective method for estimating statistics on sliding windows. In this paper we…

## 56 Citations

Almost-Smooth Histograms and Sliding-Window Graph Algorithms

- Computer Science, MathematicsAlgorithmica
- 2022

The smooth-histogram framework of Braverman and Ostrovsky (FOCS 2007) is extended to almost-smooth functions, which includes all subadditive functions, and it is shown that if a sub additive function can be $(1+\epsilon)-approximated in the insertion-only streaming model, then it can be$.

A Unified Approach for Clustering Problems on Sliding Windows

- Computer ScienceArXiv
- 2015

A data structure that extends smooth histograms as introduced by Braverman and Ostrovsky to operate on a broader class of functions is introduced, and it is shown that using only polylogarithmic space the authors can maintain a summary of the current window from which they can construct an O(1)-approximate clustering solution.

Numerical Linear Algebra in the Sliding Window Model

- Computer Science
- 2018

This work gives a deterministic algorithm that achieves spectral approximation in the sliding window model that can be viewed as a generalization of smooth histograms, using the Loewner ordering of PSD matrices, and gives algorithms for both spectral approximation and low-rank approximation that are space-optimal up to polylogarithmic factors.

Improved Sliding Window Algorithms for Clustering and Coverage via Bucketing-Based Sketches

- Computer ScienceSODA
- 2022

This work proposes a new algorithmic framework for designing efficient sliding window algorithms via bucketing-based sketches and develops space-efficient slidingwindow algorithms for k-cover, k-clustering and diversity maximization problems.

Symmetric Norm Estimation and Regression on Sliding Windows

- Computer Science, MathematicsCOCOON
- 2021

This work observes that the symmetric norm streaming algorithm of Braverman et al. (STOC 2017) can be reduced to identifying and approximating the frequency of heavy-hitters in a number of substreams, and introduces a heavy-hitter algorithm that gives a (1 + )-approximation to each of the reported frequencies in the sliding window model.

k-Center Clustering with Outliers in Sliding Windows

- Computer ScienceAlgorithms
- 2022

This work provides efficient algorithms for metric k-center clustering in the streaming model under the sliding window setting and shows, as a by-product, how to estimate the effective diameter of the window W, which is a measure of the spread of thewindow points, disregarding a given fraction of noisy distances.

Sliding Window Algorithms for k-Clustering Problems

- Computer ScienceNeurIPS
- 2020

This work provides simple and practical algorithms that update the solution efficiently with each arrival rather than recomputing it from scratch, and finds solutions with costs only slightly higher than those returned by algorithms with access to the full dataset.

Nearly Optimal Distinct Elements and Heavy Hitters on Sliding Windows

- Computer ScienceAPPROX-RANDOM
- 2018

The composable histogram along with a careful combination of existing techniques to track either the identity or frequency of a few specific items suffices to obtain algorithms for both distinct elements and $\ell_p$-heavy hitters that are nearly optimal in both $n$ and $\epsilon$.

Dynamic Graphs in the Sliding-Window Model

- Computer ScienceESA
- 2013

An extensive set of positive results including algorithms for constructing basic graph synopses like combinatorial sparsifiers and spanners as well as approximating classic graph properties such as the size of a graph matching or minimum spanning tree are presented.

Submodular Maximization over Sliding Windows

- Computer ScienceArXiv
- 2016

The first algorithms in the sliding window model for maximizing a monotone/non-monotone submodular function under cardinality and matroid constraints are obtained.

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