The problem of online checkpointing is a classical problem with numerous applications that has been studied in various forms for almost 50 years. In the simplest version of this problem, a user has to maintain k memorized checkpoints during a long computation, where the only allowed operation is to move one of the checkpoints from its old time to the current time, and his goal is to keep the checkpoints as evenly spread out as possible at all times. Bringmann, Doerr, Neumann, and Sliacan studied this problem as a special case of an online/offline optimization problem in which the deviation from uniformity is measured by the natural discrepancy metric of the worst case ratio between real and ideal segment lengths. They showed this discrepancy is smaller than 1.59-o(1) for all k and smaller than ln 4-o(1)≈ 1.39 for the sparse subset of k ’s, which are powers of 2. In addition, they obtained upper bounds on the achievable discrepancy for some small values of k . In this article, we solve the main problems left open in the above-mentioned paper by proving that ln 4 is a tight upper and lower bound on the asymptotic discrepancy for all large k and by providing tight upper and lower bounds (in the form of provably optimal checkpointing algorithms, some of which are in fact better than those of Bringmann et al.) for all the small values of k ≤ 10. In the last part of the article, we describe some new applications of this online checkpointing problem.

中文翻译：

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在线检查点算法的严格界限

在线检查点问题是一个经典问题，有许多应用程序，近 50 年来以各种形式进行了研究。在这个问题的最简单版本中，用户必须维护ķ在长时间计算期间记忆检查点，其中唯一允许的操作是将其中一个检查点从其旧时间移动到当前时间，他的目标是始终保持检查点尽可能均匀分布。Bringmann、Doerr、Neumann 和 Sliacan 将此问题作为在线/离线优化问题的一个特例进行了研究，在该问题中，与均匀性的偏差是通过真实和理想段长度之间的最坏情况比率的自然差异度量来衡量的。他们表明这种差异对于所有人来说都小于 1.59-o(1)ķ并且对于 的稀疏子集小于 ln 4-o(1)≈ 1.39ķ的，它们是 2 的幂。此外，他们获得了一些小的值的可实现差异的上限ķ. 在本文中，我们通过证明 ln 4 是所有大的渐近差异的紧上下界来解决上述论文中未解决的主要问题ķ并且通过为ķ≤ 10。在文章的最后部分，我们描述了这个在线检查点问题的一些新应用。