Online Paging with Heterogeneous Cache Slots
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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 S⊆ 2^[k], and restricting the sets S to S. If all request sets are allowed (S=2^[k]), the optimal deterministic and randomized competitive ratios are exponentially worse than for standard Paging (S={[k]}). As a function of | S| and the cache size k, the optimal deterministic ratio is polynomial: at most O(k^2| S|) and at least Ω(√(| S|)). For any laminar family S of height h, the optimal ratios are O(hk) (deterministic) and O(h^2log 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 Θ(k) (deterministic) and Θ(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 Θ(k).
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