For a caching system with multiple users, we aim to characterize the memory-rate tradeoff for caching with uncoded cache placement, under nonuniform file popularity. Focusing on the modified coded caching scheme (MCCS) recently proposed by Yu, etal., we formulate the cache placement optimization problem for the MCCS to minimize the average delivery rate under nonuniform file popularity, restricting to a class of popularity-first placements. We then present two information-theoretic lower bounds on the average rate for caching with uncoded placement, one for general cache placements and the other restricted to the popularity-first placements. By comparing the average rate of the optimized MCCS with the lower bounds, we prove that the optimized MCCS attains the general lower bound for the two-user case, providing the exact memory-rate tradeoff. Furthermore, it attains the popularity-first-based lower bound for the case of general K users with distinct file requests. In these two cases, our results also reveal that the popularity-first placement is optimal for the MCCS, and zero-padding used in coded delivery incurs no loss of optimality. For the case of K users with redundant file requests, our analysis shows that there may exist a gap between the optimized MCCS and the lower bounds due to zero-padding. We next fully characterize the optimal popularity-first cache placement for the MCCS, which is shown to possess a simple file-grouping structure and can be computed via an efficient algorithm using closed-form expressions. Finally, we extend our study to accommodate both nonuniform file popularity and sizes, where we show that the optimized MCCS attains the lower bound for the two-user case, providing the exact memory-rate tradeoff. Numerical results show that, for general settings, the gap between the optimized MCCS and the lower bound only exists in limited cases and is very small.
翻译:对于使用多个用户的缓冲系统,我们的目标是在非统一文件的受欢迎度下,将存储率转换成未编码的缓存位置。 聚焦于Yu, Contal最近提出的修改的编码缓存机制(MCCS), 我们为 MCCS 制定了缓存放置优化问题, 以在非统一文件受欢迎度下将平均交付率最小化, 限制在受欢迎程度第一的等级。 然后我们展示了两个信息- 精度较低范围, 即未编码的缓存位置的平均比率, 一个用于一般缓存位置, 而另一个限制在首级位置。 通过比较优化的 优化的 MCCS 的平均比率, 我们证明最优化的缓存系统的平均比率(MCCS ) 和较低的缩放范围。 此外, 最优化的 CMCS 的缓存范围( MCCS) 达到一般的下限范围, 并且最优化的中央端端端端端点显示我们最短的直径端点 。