4 research outputs found

    Optimal Time-Backlog Tradeoffs for the Variable-Processor Cup Game

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    The \emph{p p-processor cup game} is a classic and widely studied scheduling problem that captures the setting in which a pp-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 pp 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 pp-processor game is Θ(logn)\Theta(\log n), independent of pp, the optimal backlog in the variable-processor game is Θ(n)\Theta(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 tt rounds, the optimal backlog is Θ(n)\Theta(n) if and only if tΩ(n3)t \ge \Omega(n^3). 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.Comment: 40 pages, published in International Conference on Automata, Languages, and Programming (ICALP), 2022. Abstract abridged for arXiv submission: see paper for full abstract. Updated to acknowledge additional fundin

    The Variable-Processor Cup Game

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    The problem of scheduling tasks on pp processors so that no task ever gets too far behind is often described as a game with cups and water. In the pp-processor cup game on nn cups, there are two players, a filler and an emptier, that take turns adding and removing water from a set of nn cups. In each turn, the filler adds pp units of water to the cups, placing at most 11 unit of water in each cup, and then the emptier selects pp cups to remove up to 11 unit of water from. The emptier's goal is to minimize the backlog, which is the height of the fullest cup. The pp-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 pp is fixed. This paper initiates the study of what happens when the number of available processors pp 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 pp-processor cup has optimal backlog Θ(logn)\Theta(\log n), the variable-processor game has optimal backlog Θ(n)\Theta(n). Moreover, there is an efficient filling strategy that yields backlog Ω(n1ϵ)\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 Δ\Delta, and any Δ\Delta-greedy-like randomized emptying algorithm A\mathcal{A}, there is a filling strategy that achieves backlog Ω(n1ϵ)\Omega(n^{1 - \epsilon}) against A\mathcal{A} in quasi-polynomial time

    Approximating fluid schedules in packet-switched networks

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    Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2004.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Includes bibliographical references (p. 145-151).We consider a problem motivated by the desire to provide exible, rate-based, quality of service guarantees for packets sent over switches and switch networks. Our focus is solving a type of on-line, traffic scheduling problem, whose input at each time step is a set of desired traffic rates through the switch network. These traffic rates in general cannot be exactly achieved since they treat the incoming data as fluid, that is, they assume arbitrarily small fractions of packets can be transmitted at each time step. The goal of the traffic scheduling problem is to closely approximate the given sequence of traffic rates by a sequence of switch uses throughout the network in which only whole packets are sent. We prove worst-case bounds on the additional delay and buffer use that result from using such an approximation. These bounds depend on the network topology, the resources available to the scheduler, and the types of fluid policy allowed.by Michael Aaron Rosenblum.Ph.D
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