Online caching is among the most fundamental and well-studied problems in the area of online algorithms. Innovative algorithmic ideas and analysis -- including potential functions and primal-dual techniques -- give insight into this still-growing area. Here, we introduce a new analysis technique that first uses a potential function to upper bound the cost of an online algorithm and then pairs that with a new dual-fitting strategy to lower bound the cost of an offline optimal algorithm. We apply these techniques to the Caching with Reserves problem recently introduced by Ibrahimpur et al. [10] and give an O(log k)-competitive fractional online algorithm via a marking strategy, where k denotes the size of the cache. We also design a new online rounding algorithm that runs in polynomial time to obtain an O(log k)-competitive randomized integral algorithm. Additionally, we provide a new, simple proof for randomized marking for the classical unweighted paging problem.

翻译：在线缓存是在线算法领域中最基础且研究最深入的问题之一。创新的算法思想与分析（包括势函数和原始-对偶技术）为这一持续发展的领域提供了深刻洞见。本文提出一种新的分析技术，首先利用势函数上界约束在线算法的成本，再将该方法与新的对偶拟合策略相结合，用于下界约束离线最优算法的成本。我们将这些技术应用于Ibrahimpur等人[10]近期提出的带保留缓存问题，并通过标记策略给出了一个O(log k)-竞争的分数阶在线算法，其中k表示缓存大小。我们还设计了一种新的多项式时间在线舍入算法，从而获得一个O(log k)-竞争的随机整数算法。此外，我们针对经典的无权分页问题，给出了随机标记算法的一个新的简洁证明。