Online Paging with Heterogeneous Cache Slots

Abstract

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. To date, only a few special cases of this problem have been studied. The objective of the work presented in this paper has been to more systematically explore this generalization in the case of 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-Heterogenous Paging. In realistic settings only certain subsets of cache slots or servers would appear in requests. Therefore 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 \(\mathbb{N}\mathbb{P}\)-hard.

Some results for the laminar case are shown via a reduction to the generalization of Paging in which each request specifies a set \(P\) of pages, and is satisfied by fetching any page from \(P\) into the cache. The optimal ratios for the latter problem (with laminar family of height h) are at most hk (deterministic) and \(hH_k\) (randomized).