The problem of scheduling tasks on $p$ processors so that no task ever gets too far behind is often described as a game with cups and water. In the $p$-processor cup game on $n$ cups, there are two players, a filler and an emptier, that take turns adding and removing water from a set of $n$ cups. In each turn, the filler adds $p$ units of water to the cups, placing at most $1$ unit of water in each cup, and then the emptier selects $p$ cups to remove up to $1$ unit of water from. The emptier's goal is to minimize the backlog, which is the height of the fullest cup. The $p$-processor cup game has been studied in many different settings, dating back to the late 1960's. All of the past work shares one common assumption: that $p$ is fixed. This paper initiates the study of what happens when the number of available processors $p$ varies over time, resulting in what we call the \emph{variable-processor cup game}. Remarkably, the optimal bounds for the variable-processor cup game differ dramatically from its classical counterpart. Whereas the $p$-processor cup has optimal backlog $\Theta(\log n)$, the variable-processor game has optimal backlog $\Theta(n)$. Moreover, there is an efficient filling strategy that yields backlog $\Omega(n^{1 - \epsilon})$ in quasi-polynomial time against any deterministic emptying strategy. We additionally show that straightforward uses of randomization cannot be used to help the emptier. In particular, for any positive constant $\Delta$, and any $\Delta$-greedy-like randomized emptying algorithm $\mathcal{A}$, there is a filling strategy that achieves backlog $\Omega(n^{1 - \epsilon})$ against $\mathcal{A}$ in quasi-polynomial time.

中文翻译：

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可变处理器杯赛

在$ p $处理器上安排任务以使任何任务都不会落后得太远的问题通常被描述为带有杯子和水的游戏。在$ n $杯子的$ p $处理器杯子游戏中，有两个玩家，一个填充器和一个空容器，轮流从一组$ n $杯子中添加和去除水。在每个回合中，填充器会向杯子中添加$ p $单位水，每个杯子中​​最多放置$ 1 $单位水，然后倒空者选择$ p $杯子中的水分最多可删除$ 1 $单位水。容器的目标是最大程度地减少积压，即积满的杯子的高度。可以在许多不同的环境中研究$ p $处理器杯游戏，其历史可以追溯到1960年代后期。过去的所有工作都有一个共同的假设：$ p $是固定的。本文着手研究当可用处理器的数量$ p $随时间变化时会发生什么，从而导致我们称之为\ emph {variable-processor cup game}。值得注意的是，变量处理器杯赛的最佳界限与经典比赛有很大的不同。$ p $处理器杯具有最佳积压$ \ Theta（\ log n）$，而变量处理器游戏具有最佳积压$ \ Theta（n）$。而且，存在一种有效的填充策略，它相对于任何确定性的清空策略在准多项式时间内产生积压的\\ Omega（n ^ {1-\ epsilon}）$。我们还表明，不能将随机化的直接用法用于帮助倒空者。特别是，对于任何正常数$ \ Delta $和任何$ \ Delta $ -greedy-like随机清空算法$ \ mathcal {A} $，