## Abstract

Many networking applications require timely access to recent network measurements, which can be captured using a sliding window model. Maintaining such measurements is a challenging task due to the fast line speed and scarcity of fast memory in routers. In this work, we study the impact of allowing slack in the window size on the asymptotic requirements of sliding window problems. That is, the algorithm can dynamically adjust the window size between W and W(1+τ) where τ is a small positive parameter. We demonstrate this model's attractiveness by showing that it enables efficient algorithms to problems such as MAXIMUM and GENERAL-SUMMING that require Ω(W) bits even for constant factor approximations in the exact sliding window model. Additionally, for problems that admit sub-linear approximation algorithms such as BASIC-SUMMING and COUNT-DISTINCT, the slack model enables a further asymptotic improvement. The main focus of the paper is on the widely studied BASIC-SUMMING problem of computing the sum of the last W integers from {0,1…,R} in a stream. While it is known that Ω(WlogR) bits are needed in the exact window model, we show that approximate windows allow an exponential space reduction for constant τ. Specifically, for τ=Θ(1), we present a space lower bound of Ω(log(RW)) bits. Additionally, we show an Ω(log(W/ϵ)) lower bound for RWϵ additive approximations and a Ω(log(W/ϵ)+loglogR) bits lower bound for (1+ϵ) multiplicative approximations. Our work is the first to study this problem in the exact and additive approximation settings. For all settings, we provide memory optimal algorithms that operate in worst case constant time. This strictly improves on the work of [17] for (1+ϵ)-multiplicative approximation that requires O(ϵ^{−1}log(RW)loglog(RW)) space and performs updates in O(log(RW)) worst case time. Finally, we show asymptotic improvements for the COUNT-DISTINCT, GENERAL-SUMMING, and MAXIMUM problems.

Original language | English |
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Pages (from-to) | 87-108 |

Number of pages | 22 |

Journal | Theoretical Computer Science |

Volume | 791 |

DOIs | |

State | Published - 29 Oct 2019 |

## Keywords

- Algorithms
- Approximation algorithms
- Lower bounds
- Sliding windows
- Streaming

## All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)