Abstract
We study the Double Coverage (DC) algorithm for the k-server problem in the (h, k)-setting, i.e. when DC with k servers is compared against an offline optimum algorithm with \(h \le k\) servers. It is well-known that DC is k-competitive for \(h=k\). We prove that even if \(k>h\) the competitive ratio of DC does not improve; in fact, it increases up to \(h+1\) as k grows. In particular, we show matching upper and lower bounds of \(\frac{k(h+1)}{k+1}\) on the competitive ratio of DC on any tree metric.
Supported by NWO grant 639.022.211, ERC consolidator grant 617951, NCN grant DEC-2013/09/B/ST6/01538,NSF grants CCF-1115575, CNS-1253218, CCF-1421508, and an IBM Faculty Award.
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Notes
- 1.
Actually [5] shows a slightly stronger upper bound WFA\(_k \le 2h\)OPT\(_h - \)OPT\(_k + \text{ const }\) where OPT\(_k\) and OPT\(_h\) are the optimal cost using k and h servers respectively.
- 2.
If the online algorithm knows h, it can simply disable its \(k-h\) extra servers and be \(2h-1\) competitive (which is slightly better than 2h). However, Koutsoupias (and also us) consider the setting where the online algorithm does not know h.
- 3.
Here, we view the uniform metric as a star graph where requests appear to the leaves. A proof of this result will be given in the full version of the paper.
- 4.
- 5.
Consider the instance where all servers are at \(x=0\) initially. A request arrives at \(x=2\), upon which both DC and offline move a server there and pay 2. Then a request arrives at \(x=1\). DC moves both servers there and pays 2 while offline pays 1. All servers are now at \(x=1\) and the instance repeats.
- 6.
We remark that this property does not hold (simultaneously) for every offline server, but only for a single fixed offline server y.
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Bansal, N., Eliáš, M., Jeż, Ł., Koumoutsos, G., Pruhs, K. (2015). Tight Bounds for Double Coverage Against Weak Adversaries. In: Sanità, L., Skutella, M. (eds) Approximation and Online Algorithms. WAOA 2015. Lecture Notes in Computer Science(), vol 9499. Springer, Cham. https://doi.org/10.1007/978-3-319-28684-6_5
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