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+tau) where tau is a small positive parameter. We demonstrate this model's attractiveness by showing that it enables efficient algorithms to problems such as Maximum and GeneralSumming that require Omega(W) bits even for constant factor approximations in the exact sliding window model. Additionally, for problems that admit sublinear approximation algorithms such as BasicSumming and CountDistinct, the slack model enables a further asymptotic improvement.
The main focus of our paper [{Ben Basat} et al., 2017] is on the widely studied BasicSumming problem of computing the sum of the last W integers from {0,1 ...,R} in a stream. While it is known that Omega(W log{R}) bits are needed in the exact window model, we show that approximate windows allow an exponential space reduction for constant tau.
Specifically, for tau=Theta(1), we present a space lower bound of Omega(log(RW)) bits. Additionally, we show an Omega(log ({W/epsilon})) lower bound for RW epsilon additive approximations and a Omega(log ({W/epsilon})+log log{R}) bits lower bound for (1+epsilon) 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 [Mayur Datar et al., 2002] for (1+epsilon)multiplicative approximation that requires O(epsilon^{1} log ({RW})log log ({RW})) space and performs updates in O(log ({RW})) worst case time. Finally, we show asymptotic improvements for the CountDistinct, GeneralSumming and Maximum problems.
BibTeX  Entry
@InProceedings{benbasat_et_al:LIPIcs:2018:9167,
author = {Ran Ben Basat and Gil Einziger and Roy Friedman},
title = {{Brief Announcement: Give Me Some Slack: Efficient Network Measurements}},
booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)},
pages = {163:1163:5},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770767},
ISSN = {18688969},
year = {2018},
volume = {107},
editor = {Ioannis Chatzigiannakis and Christos Kaklamanis and D{\'a}niel Marx and Donald Sannella},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2018/9167},
URN = {urn:nbn:de:0030drops91672},
doi = {10.4230/LIPIcs.ICALP.2018.163},
annote = {Keywords: Streaming, Algorithms, Sliding window, Lower bounds}
}
Keywords: 

Streaming, Algorithms, Sliding window, Lower bounds 
Collection: 

45th International Colloquium on Automata, Languages, and Programming (ICALP 2018) 
Issue Date: 

2018 
Date of publication: 

04.07.2018 