### Abstract

Many problems in processor scheduling, deamortization, and buffer management can be modeled as single- and multi-processor cup games. At the beginning of the single-processor n-cup game, all cups are empty. In each step of the game, a filler distributes 1 − ε units of water among the cups, and then an emptier selects a cup and removes up to 1 unit of water from it. The goal of the emptier is to minimize the amount of water in the fullest cup, also known as the backlog. The greedy algorithm (i.e., empty from the fullest cup) is known to achieve backlog O(log n), and no deterministic algorithm can do better. We show that the performance of the greedy algorithm can be exponentially improved with a small amount of randomization: After each step and for any k ≥ Ω(log ε^{−1}), the emptier achieves backlog at most O(k) with probability at least 1 − O(2^{−2k} ). We call our algorithm the smoothed greedy algorithm because if follows from a smoothed analysis of the (standard) greedy algorithm. In each step of the p-processor n-cup game, the filler distributes p(1 − ε) unit of water among the cups, with no cup receiving more than 1 − δ units of water, and then the emptier selects p cups and removes 1 unit of water from each. Proving nontrivial bounds on the backlog for the multi-processor cup game has remained open for decades. We present a simple analysis of the greedy algorithm for the multi-processor cup game, establishing a backlog of O(ε^{−1} log n), as long as δ > 1/poly(n). Turning to randomized algorithms, we find that the backlog drops to constant. Specifically, we show that if ε and δ satisfy reasonable constraints, then there exists an algorithm that bounds the backlog after a given step by 3 with probability at least 1 − O(exp(−Ω(ε^{2}p)). We prove that our results are asymptotically optimal for constant ε, in the sense that no algorithms can achieve better bounds, up to constant factors in the backlog and in p. Moreover, we prove robustness results, demonstrating that our randomized algorithms continue to behave well even when placed in bad starting states.

Original language | English (US) |
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Title of host publication | STOC 2019 - Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing |

Editors | Moses Charikar, Edith Cohen |

Publisher | Association for Computing Machinery |

Pages | 1148-1157 |

Number of pages | 10 |

ISBN (Electronic) | 9781450367059 |

DOIs | |

State | Published - Jun 23 2019 |

Event | 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019 - Phoenix, United States Duration: Jun 23 2019 → Jun 26 2019 |

### Publication series

Name | Proceedings of the Annual ACM Symposium on Theory of Computing |
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ISSN (Print) | 0737-8017 |

### Conference

Conference | 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019 |
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Country | United States |

City | Phoenix |

Period | 6/23/19 → 6/26/19 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Software

### Keywords

- Cup emptying
- Deamortization
- Discretized scheduling
- Parallelism
- Processor sharing
- Smoothed analysis

### Cite this

*STOC 2019 - Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing*(pp. 1148-1157). (Proceedings of the Annual ACM Symposium on Theory of Computing). Association for Computing Machinery. https://doi.org/10.1145/3313276.3316342

}

*STOC 2019 - Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing.*Proceedings of the Annual ACM Symposium on Theory of Computing, Association for Computing Machinery, pp. 1148-1157, 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019, Phoenix, United States, 6/23/19. https://doi.org/10.1145/3313276.3316342

**Achieving optimal backlog in multi-processor cup games.** / Bender, Michael A.; Farach-Colton, Martín; Kuszmaul, William.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

TY - GEN

T1 - Achieving optimal backlog in multi-processor cup games

AU - Bender, Michael A.

AU - Farach-Colton, Martín

AU - Kuszmaul, William

PY - 2019/6/23

Y1 - 2019/6/23

N2 - Many problems in processor scheduling, deamortization, and buffer management can be modeled as single- and multi-processor cup games. At the beginning of the single-processor n-cup game, all cups are empty. In each step of the game, a filler distributes 1 − ε units of water among the cups, and then an emptier selects a cup and removes up to 1 unit of water from it. The goal of the emptier is to minimize the amount of water in the fullest cup, also known as the backlog. The greedy algorithm (i.e., empty from the fullest cup) is known to achieve backlog O(log n), and no deterministic algorithm can do better. We show that the performance of the greedy algorithm can be exponentially improved with a small amount of randomization: After each step and for any k ≥ Ω(log ε−1), the emptier achieves backlog at most O(k) with probability at least 1 − O(2−2k ). We call our algorithm the smoothed greedy algorithm because if follows from a smoothed analysis of the (standard) greedy algorithm. In each step of the p-processor n-cup game, the filler distributes p(1 − ε) unit of water among the cups, with no cup receiving more than 1 − δ units of water, and then the emptier selects p cups and removes 1 unit of water from each. Proving nontrivial bounds on the backlog for the multi-processor cup game has remained open for decades. We present a simple analysis of the greedy algorithm for the multi-processor cup game, establishing a backlog of O(ε−1 log n), as long as δ > 1/poly(n). Turning to randomized algorithms, we find that the backlog drops to constant. Specifically, we show that if ε and δ satisfy reasonable constraints, then there exists an algorithm that bounds the backlog after a given step by 3 with probability at least 1 − O(exp(−Ω(ε2p)). We prove that our results are asymptotically optimal for constant ε, in the sense that no algorithms can achieve better bounds, up to constant factors in the backlog and in p. Moreover, we prove robustness results, demonstrating that our randomized algorithms continue to behave well even when placed in bad starting states.

AB - Many problems in processor scheduling, deamortization, and buffer management can be modeled as single- and multi-processor cup games. At the beginning of the single-processor n-cup game, all cups are empty. In each step of the game, a filler distributes 1 − ε units of water among the cups, and then an emptier selects a cup and removes up to 1 unit of water from it. The goal of the emptier is to minimize the amount of water in the fullest cup, also known as the backlog. The greedy algorithm (i.e., empty from the fullest cup) is known to achieve backlog O(log n), and no deterministic algorithm can do better. We show that the performance of the greedy algorithm can be exponentially improved with a small amount of randomization: After each step and for any k ≥ Ω(log ε−1), the emptier achieves backlog at most O(k) with probability at least 1 − O(2−2k ). We call our algorithm the smoothed greedy algorithm because if follows from a smoothed analysis of the (standard) greedy algorithm. In each step of the p-processor n-cup game, the filler distributes p(1 − ε) unit of water among the cups, with no cup receiving more than 1 − δ units of water, and then the emptier selects p cups and removes 1 unit of water from each. Proving nontrivial bounds on the backlog for the multi-processor cup game has remained open for decades. We present a simple analysis of the greedy algorithm for the multi-processor cup game, establishing a backlog of O(ε−1 log n), as long as δ > 1/poly(n). Turning to randomized algorithms, we find that the backlog drops to constant. Specifically, we show that if ε and δ satisfy reasonable constraints, then there exists an algorithm that bounds the backlog after a given step by 3 with probability at least 1 − O(exp(−Ω(ε2p)). We prove that our results are asymptotically optimal for constant ε, in the sense that no algorithms can achieve better bounds, up to constant factors in the backlog and in p. Moreover, we prove robustness results, demonstrating that our randomized algorithms continue to behave well even when placed in bad starting states.

KW - Cup emptying

KW - Deamortization

KW - Discretized scheduling

KW - Parallelism

KW - Processor sharing

KW - Smoothed analysis

UR - http://www.scopus.com/inward/record.url?scp=85068662651&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85068662651&partnerID=8YFLogxK

U2 - 10.1145/3313276.3316342

DO - 10.1145/3313276.3316342

M3 - Conference contribution

AN - SCOPUS:85068662651

T3 - Proceedings of the Annual ACM Symposium on Theory of Computing

SP - 1148

EP - 1157

BT - STOC 2019 - Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

A2 - Charikar, Moses

A2 - Cohen, Edith

PB - Association for Computing Machinery

ER -