Online Paging with Heterogeneous Cache Slots

TL;DR

This paper parameterizes the Slot-Heterogenous Paging problem, and establishes bounds on the competitive ratio as a function of the cache size k and family S, and specifies a family S.

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. 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 $Ω(\sqrt{|\mathcal S|})$. - 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 $Θ(k)$. 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).

Figures (11)

• Figure 1: Online algorithm ExhSearch for Slot-Heterogenous Paging.
• Figure 2: Illustration of the proof of Theorem \ref{['thm: slot hetero lower bound']} Part (ii) for $k = 35$, $m = 6$, and $\ell = 7$. The figure shows the partition of all slots into $m-1 = 5$ sets $B^1, \ldots, B^5$, each represented by a cycle. To avoid clutter, each slot $b^e_c$ is represented by its index $c$ within $B^e$. The picture shows set ${S}=\{b^1_2, b^1_3, b^2_6, b^2_7, b^4_4, b^4_5\}\in\mathcal{G}\xspace$, marked by dashed ovals. It also shows $Z_{{S}'}\in\mathcal{Z}\xspace$, represented by orange/shaded circles, for ${S}'=\{b^1_2, b^1_3, b^3_4, b^3_5, b^4_7, b^4_1\}$.
• Figure 3: An example of a laminar family ${\mathcal{P}}$ of height $4$.
• Figure 4: The algorithm that transforms $(\pi, C)$ into $(\sigma, C')$, by repairing each phase.
• Figure 5: At the top, an instance of Slot-Laminar Paging and its solution $C$. In the middle row, the corresponding solution $C'$ of the relaxed instance. The bottom row shows a cheaper solution of the relaxed instance.

Theorems & Definitions (44)

• Theorem 3.1
• proof : Proof of Theorem \ref{['thm: subset server upper bound']}.
• Theorem 3.2
• Lemma 3.1
• proof
• proof : Proof of Theorem \ref{['thm: slot hetero lower bound']}
• Theorem 3.3
• Lemma 3.2
• proof
• Claim 3.4