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The \emph{p-processor cup game} is a classic and widely studied scheduling problem that captures the setting in which a p-processor machine must assign tasks to processors over time in order to ensure that no individual task ever falls too far behind. The problem is formalized as a multi-round game in which two players, a filler (who assigns work to tasks) and an emptier (who schedules tasks) compete. The emptier's goal is to minimize backlog, which is the maximum amount of outstanding work for any task. Recently, Kuszmaul and Westover (ITCS, 2021) proposed the \emph{variable-processor cup game}, which considers the same problem, except that the amount of resources available to the players (i.e., the number p of processors) fluctuates between rounds of the game. They showed that this seemingly small modification fundamentally changes the dynamics of the game: whereas the optimal backlog in the fixed p-processor game is Θ(logn), independent of p, the optimal backlog in the variable-processor game is Θ(n). The latter result was only known to apply to games with \emph{exponentially many} rounds, however, and it has remained an open question what the optimal tradeoff between time and backlog is for shorter games. This paper establishes a tight trade-off curve between time and backlog in the variable-processor cup game. Importantly, we prove that for a game consisting of t rounds, the optimal backlog is Θ(n) if and only if t≥Ω(n3). Our techniques also allow for us to resolve several other open questions concerning how the variable-processor cup game behaves in beyond-worst-case-analysis settings.
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Randomized Cup Game Algorithms Against Strong Adversaries
In each step of the cup game on n cups, a filler distributes up to 1" water among the cups, and then an emptier removes 1 unit of water from a single cup. The emptier’s goal is to minimize the height of the fullest cup, also known as the backlog. The cup emptying game has found extensive applications to processor scheduling, network- switch buffer management, quality of service guarantees, and data- structure deamortization. The greedy emptying algorithm (i.e., always remove from the fullest cup) is known to achieve backlog O(log n) and to be the optimal deterministic algorithm. Randomized algorithms can do significantly better, achieving backlog O(log log n) with high probability, as long as " is not too small. In order to achieve these improvements, the known randomized algorithms require that the filler is an oblivious adversary, unaware of which cups the emptier chooses to empty out of at each step. Such randomized guarantees are known to be impossible against fully adaptive fillers. We show that, even when the filler is just slightly non- adaptive, randomized emptying algorithms can still guarantee a backlog of O(log log n). In particular, we give randomized ran- domized algorithms against an elevated adaptive filler, which is an adaptive filler that can see the precise fills of every cup containing more than 3 units of water, but not of the cups containing less than 3 units.
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- PAR ID:
- 10298569
- Date Published:
- Journal Name:
- Proceedings of the annual ACMSIAM symposium on discrete algorithms
- ISSN:
- 2160-1445
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/https://doi.org/10.1145/3313276.3316342
