This work studies a well-known shared-cache coded caching scenario where each cache can serve an arbitrary number of users, analyzing the case where there is some knowledge about such number of users (i.e., the topology) during the content placement phase. Under the assumption of regular placement and a cumulative cache size that can be optimized across the different caches, we derive the fundamental limits of performance by introducing a novel cache-size optimization and placement scheme and a novel information-theoretic converse. The converse employs new index coding techniques to bypass traditional uniformity requirements, thus finely capturing the heterogeneity of the problem, and it provides a new approach to handle asymmetric settings. The new fundamental limits reveal that heterogeneous topologies can in fact outperform their homogeneous counterparts where each cache is associated to an equal number of users. These results are extended to capture the scenario of topological uncertainty where the perceived/estimated topology does not match the true network topology. This scenario is further elevated to the stochastic setting where the user-to-cache association is random and unknown, and it is shown that the proposed scheme is robust to such noisy or inexact knowledge on the topology.

翻译：这项工作研究了一个众所周知的共享缓存编码编码缓存情景, 每个缓存可以为任意数量的用户服务, 分析在内容放置阶段对用户数量有一定了解的案例( 即表层学 ) 。 在假设定期放置和累积缓存大小可以优化到不同缓存时, 我们通过引入一个新的缓存大小优化和放置计划以及一个新的信息理论对口, 得出基本性能限制。 反向使用新的索引编码技术绕过传统的统一要求, 从而细微捕捉到问题的异质性, 它提供了处理不对称设置的新方法 。 新的基本限制显示, 在每一个缓存与同等数量用户相关的情况下, 混杂的表层学可能事实上超过其同质对应方。 这些结果将扩展到可以捕捉表层不确定性的假设, 即感知/ 估计的表层学与真实的网络表学不匹配。 这一假设进一步提升到用户对缓存联系是随机和未知的, 并且显示拟议的方案在顶部或顶部知识上是坚固的。