Online Paging with Heterogeneous Cache Slots

It is natural to generalize the $k$-Server problem by allowing each request to specify not only a point $p$, but also a subset $S$ of servers that may serve it. To attack this generalization, we focus on uniform and star metrics. 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-Heterogeneous Paging. We parameterize the problem by specifying an arbitrary family ${\cal S} \subseteq 2^{[k]}$, and restricting the sets $S$ to ${\cal S}$. If all request sets are allowed (${\cal S}=2^{[k]}$), the optimal deterministic and randomized competitive ratios are exponentially worse than for standard Paging (${\cal S}=\{[k]\}$). As a function of $|{\cal S}|$ and the cache size $k$, the optimal deterministic ratio is polynomial: at most $O(k^2|{\cal S}|)$ and at least $\Omega(\sqrt{|{\cal S}|})$. For any laminar family ${\cal S}$ of height $h$, the optimal ratios are $O(hk)$ (deterministic) and $O(h^2\log k)$ (randomized). The special case 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. For All-or-One Paging the optimal competitive ratios are $\Theta(k)$ (deterministic) and $\Theta(\log k)$ (randomized), while the offline problem is NP-hard. We extend the deterministic upper bound to the weighted variant of All-Or-One Paging (a generalization of standard Weighted Paging), showing that it is also $\Theta(k)$.