It is natural to generalize the online $k$-Server problem by allowing each request to specify not only a point $p$, but also a subset $S$ of servers that may serve it. For uniform metrics, the problem is equivalent to a generalization of Paging in which each request specifies not only a page $p$, but also a subset $S$ of cache slots, and is satisfied by having a copy of $p$ in some slot in $S$. We call this problem Slot-Heterogenous Paging. We parameterize the problem by specifying a family $\mathcal S \subseteq 2^{[k]}$ of requestable slot sets, and we establish bounds on the competitive ratio as a function of the cache size $k$ and family $\mathcal S$: - If all request sets are allowed ($\mathcal S=2^{[k]}\setminus\{\emptyset\}$), the optimal deterministic and randomized competitive ratios are exponentially worse than for standard \Paging ($\mathcal S=\{[k]\}$). - As a function of $|\mathcal S|$ and $k$, the optimal deterministic ratio is polynomial: at most $O(k^2|\mathcal S|)$ and at least $\Omega(\sqrt{|\mathcal S|})$. - For any laminar family $\mathcal S$ of height $h$, the optimal ratios are $O(hk)$ (deterministic) and $O(h^2\log k)$ (randomized). - The special case of laminar $\mathcal S$ that we call All-or-One Paging extends standard Paging by allowing each request to specify a specific slot to put the requested page in. The optimal deterministic ratio for weighted All-or-One Paging is $\Theta(k)$. Offline All-or-One Paging is NP-hard. Some results for the laminar case are shown via a reduction to the generalization of Paging in which each request specifies a set $\mathcal P of pages, and is satisfied by fetching any page from $\mathcal P into the cache. The optimal ratios for the latter problem (with laminar family of height $h$) are at most $hk$ (deterministic) and $h\,H_k$ (randomized).

翻译：通过允许每个请求指定不仅是一个点$p$，而且还可以指定可服务该请求的服务器子集$S$，可以自然地将在线$k$服务器问题推广。对于均匀度量空间，该问题等价于一种分页问题的推广：每个请求不仅指定一个页面$p$，还指定一个缓存槽子集$S$，并通过在$S$的某个槽中拥有$p$的副本得到满足。我们称此问题为槽异构分页问题。我们通过指定可请求的槽集族$\mathcal S \subseteq 2^{[k]}$对该问题进行参数化，并建立关于缓存大小$k$和族$\mathcal S$的竞争比界限：- 若允许所有请求集（$\mathcal S=2^{[k]}\setminus\{\emptyset\}$），则最优确定性和随机化竞争比较标准分页问题（$\mathcal S=\{[k]\}$）呈指数级恶化。- 作为$|\mathcal S|$和$k$的函数，最优确定性竞争比呈多项式级：上界为$O(k^2|\mathcal S|)$，下界为$\Omega(\sqrt{|\mathcal S|})$。- 对任意高度为$h$的层状族$\mathcal S$，最优竞争比分别为$O(hk)$（确定性）和$O(h^2\log k)$（随机化）。- 层状族$\mathcal S$的特殊情况——我们称之为全或无分页问题——通过允许每个请求指定将请求页面放入某个特定槽中，扩展了标准分页问题。加权全或无分页问题的最优确定性竞争比为$\Theta(k)$。离线全或无分页问题是NP难问题。层状情况的某些结果通过归约到推广的分页问题得到证明：在该推广问题中，每个请求指定一个页面集$\mathcal P$，并通过将$\mathcal P$中的任意页面取入缓存得到满足。后者问题（对于高度为$h$的层状族）的最优竞争比不超过$hk$（确定性）和$h\,H_k$（随机化）。