TY - JOUR

T1 - Give me some slack

T2 - Efficient network measurements

AU - Ben Basat, Ran

AU - Einziger, Gil

AU - Friedman, Roy

N1 - Funding Information:
We would like to thank Eden Saig for presenting our work as a brief announcement in ICALP 2018. This work was partially funded by the Israel Science Foundation grant # 1505/16 and the Technion-HPI research school. Ran Ben Basat was also supported by the Zuckerman Institute and the Technion Hiroshi Fujiwara cyber security research center.
Publisher Copyright:
© 2019 Elsevier B.V.

PY - 2019/10/29

Y1 - 2019/10/29

N2 - 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−1log(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.

AB - 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−1log(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.

KW - Algorithms

KW - Approximation algorithms

KW - Lower bounds

KW - Sliding windows

KW - Streaming

UR - http://www.scopus.com/inward/record.url?scp=85066114995&partnerID=8YFLogxK

U2 - 10.1016/j.tcs.2019.05.012

DO - 10.1016/j.tcs.2019.05.012

M3 - Article

AN - SCOPUS:85066114995

SN - 0304-3975

VL - 791

SP - 87

EP - 108

JO - Theoretical Computer Science

JF - Theoretical Computer Science

ER -