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How asymmetry helps buffer management: achieving optimal tail size in cup games

Author:

William KuszmaulAuthors Info & Claims

STOC 2021: Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing

Pages 1248 - 1261

https://doi.org/10.1145/3406325.3451033

Published: 15 June 2021 Publication History

Abstract

The cup game on n cups is a multi-step game with two players, a filler and an emptier. At each step, the filler distributes 1 unit of water among the cups, and then the emptier selects a single cup to remove (up to) 1 unit of water from.

There are several objective functions that the emptier might wish to minimize. One of the strongest guarantees would be to minimize tail size, which is defined to be the number of cups with fill 2 or greater. A simple lower-bound construction shows that the optimal tail size for deterministic emptying algorithms is Θ(n), however.

We present a simple randomized emptying algorithm that achieves tail size Õ(logn) with high probability in n for poly n steps. Moreover, we show that this is tight up to doubly logarithmic factors. We also extend our results to the multi-processor cup game, achieving tail size Õ(logn + p) on p processors with high probability in n. We show that the dependence on p is near optimal for any emptying algorithm that achieves polynomial-bounded backlog.

A natural question is whether our results can be extended to give unending guarantees, which apply to arbitrarily long games. We give a lower bound construction showing that no monotone memoryless emptying algorithm can achieve an unending guarantee on either tail size or the related objective function of backlog. On the other hand, we show that even a very small (i.e., 1 / poly n) amount of resource augmentation is sufficient to overcome this barrier.

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Cited By

Hohne FSchmitt Svan Stee R(2022)SIGACT News Online Algorithms Column 38ACM SIGACT News10.1145/3510382.351039652:4(80-96)Online publication date: 3-Jan-2022
https://dl.acm.org/doi/10.1145/3510382.3510396
Bansal NKuszmaul W(2022)Balanced Allocations: The Heavily Loaded Case with Deletions2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS54457.2022.00081(801-812)Online publication date: Oct-2022
https://doi.org/10.1109/FOCS54457.2022.00081
Bilò DGualà LLeucci SProietti GScornavacca G(2022)Cutting bamboo down to sizeTheoretical Computer Science10.1016/j.tcs.2022.01.027909:C(54-67)Online publication date: 28-Mar-2022
https://dl.acm.org/doi/10.1016/j.tcs.2022.01.027

Index Terms

How asymmetry helps buffer management: achieving optimal tail size in cup games
1. Theory of computation
  1. Design and analysis of algorithms
    1. Online algorithms

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Published In

cover image ACM Conferences

STOC 2021: Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing

June 2021

1797 pages

ISBN:9781450380539

DOI:10.1145/3406325

General Chair:
Samir Khuller
Northwestern University, USA
,
Program Chair:
Virginia Vassilevska Williams
Massachusetts Institute of Technology, USA

Copyright © 2021 Owner/Author.

This work is licensed under a Creative Commons Attribution International 4.0 License.

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SIGACT: ACM Special Interest Group on Algorithms and Computation Theory

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Association for Computing Machinery

New York, NY, United States

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Published: 15 June 2021

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STOC '21: 53rd Annual ACM SIGACT Symposium on Theory of Computing

June 21 - 25, 2021

Virtual, Italy

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Cited By

Hohne FSchmitt Svan Stee R(2022)SIGACT News Online Algorithms Column 38ACM SIGACT News10.1145/3510382.351039652:4(80-96)Online publication date: 3-Jan-2022
https://dl.acm.org/doi/10.1145/3510382.3510396
Bansal NKuszmaul W(2022)Balanced Allocations: The Heavily Loaded Case with Deletions2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS54457.2022.00081(801-812)Online publication date: Oct-2022
https://doi.org/10.1109/FOCS54457.2022.00081
Bilò DGualà LLeucci SProietti GScornavacca G(2022)Cutting bamboo down to sizeTheoretical Computer Science10.1016/j.tcs.2022.01.027909:C(54-67)Online publication date: 28-Mar-2022
https://dl.acm.org/doi/10.1016/j.tcs.2022.01.027

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