Randomized k-server on hierarchical binary trees

Coté, Aaron; Meyerson, Adam; Poplawski, Laura J.

doi:10.1145/1374376.1374411

Cited by 21 publications

(35 citation statements)

References 27 publications

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“…I am also thankful to one of them for calling my attention to the very recent result [8]. This research is partially supported by OTKA Grant K76099.…”

Section: Acknowledgementmentioning

confidence: 86%

“…If additionally M < k holds, we maybe get a better competitive ratio. It may be an another genuine advance to combine this approach with other [8] and [17] to obtain improved randomized algorithms for the k-server problem.…”

Section: Discussionmentioning

confidence: 99%

“…Seiden [17] proved the existence of an O (polylog k)-competitive algorithm for Ω(k log k)-decomposable spaces, where the space can be partitioned into O (log k) uniform blocks, each having diameter 1, and where the distance of any two blocks is at least c · k · log k. In his work he also showed that for binary HST's (where each non-leaf node has exactly two children) there exists an O (log 3 k)-competitive algorithm, provided the parameter μ of the HST is sufficiently large. Very recently Coté et al [8] designed a randomized algorithm on binary trees with competitive ratio logarithmic in the diameter of the metric (but independent of k).…”

Section: Definitionmentioning

confidence: 99%

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Randomized algorithm for the k-server problem on decomposable spaces

Nagy-György

2009

Journal of Discrete Algorithms

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We study the randomized k-server problem on metric spaces consisting of widely separated subspaces. We give a method which extends existing algorithms to larger spaces with the growth rate of the competitive quotients being at most O (log k). This method yields o(k)-competitive algorithms solving the randomized k-server problem for some special underlying metric spaces, e.g. HSTs of "small" height (but unbounded degree). HSTs are important tools for probabilistic approximation of metric spaces.

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“…I am also thankful to one of them for calling my attention to the very recent result [8]. This research is partially supported by OTKA Grant K76099.…”

Section: Acknowledgementmentioning

confidence: 86%

Section: Discussionmentioning

confidence: 99%

Section: Definitionmentioning

confidence: 99%

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Randomized algorithm for the k-server problem on decomposable spaces

Nagy-György

2009

Journal of Discrete Algorithms

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“…The authors of [CMP08] give an O(log D)-competitive randomized algorithm on binary 2-HSTs with combinatorial depth at most D, and in [BBMN15], a major breakthrough was achieved when the authors exhibited an (log n) O(1) -competitive algorithm for general n-vertex HSTs. In the present work, we obtain a competitive ratio independent of the size of the underlying HST, thereby verifying a long-held belief.…”

Section: Introductionmentioning

confidence: 99%

k-server via multiscale entropic regularization

Bubeck

Cohen

Lee

et al. 2018

Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing

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We present an O((log k) 2 )-competitive randomized algorithm for the k-server problem on hierarchically separated trees (HSTs). This is the first o(k)-competitive randomized algorithm for which the competitive ratio is independent of the size of the underlying HST. Our algorithm is designed in the framework of online mirror descent where the mirror map is a multiscale entropy. When combined with Bartal's static HST embedding reduction, this leads to an O((log k) 2 log n)-competitive algorithm on any n-point metric space. We give a new dynamic HST embedding that yields an O((log k) 3 log ∆)-competitive algorithm on any metric space where the ratio of the largest to smallest non-zero distance is at most ∆.Perhaps the most widely-studied problem in the field of online algorithms and competitive analysis is the k-server problem, introduced in [MMS90] to generalize and abstract a number of related problems arising in the study of paging and caching. The problem has been the object of intensive study since its inception, motivated largely by two long-standing conjectures about the competitive ratios that can be achieved by deterministic and randomized algorithms, respectively.We recall the problem briefly; see Section 1.2 for a formal definition of the model. Fix a metric space (X, d) and k 1, as well as an initial placement ρ 0 ∈ X k of k servers in X. An online k-server algorithm operates as follows. At each time step, a request r t ∈ X comes online, and the algorithm must respond to this request by moving one of the servers to r t (unless there is already a server there). The cost of the algorithm is the total distance moved by all the servers over the course of the request sequence. An offline algorithm operates in the same manner, but is allowed access to the entire request sequence in advance. An online algorithm has competitive ratio α if, for every request sequence, its movement cost per unit time step is within an α factor of that achieved by the optimal offline algorithm.Randomization. The authors of [MMS90] stated the k-server conjecture: On any metric space with at least k +1 points, the best competitive ratio achieved by deterministic online algorithms is precisely k. They showed that the ratio is always at least k. While the conjecture is still open in general, Koutsoupias and Papadimitriou resolved it within a factor of two: The work function algorithm obtains a competitive ratio of 2k − 1 on any metric space [KP95]. We refer the reader to the book [BE98] for further background on online algorithms and the k-server problem.In the context of the k-paging problem, which is the special case of k-server on a metric space with all distances in the set {0, 1}, it was observed that randomness can help an online algorithm dramatically: It is known that the competitive ratio for k-paging is precisely the kth harmonic number H k for every k 1 [FKL + 91, MS91].Hierarchically separated trees. There is another class of metric spaces on which one can prove lower bounds on the competitive ratio even for randomized algo...

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“…This was improved substantially by Fiat and Mendel to O((log k log log k) 2 ) [40]. A more recent result by Cote, Meyerson, and Poplawski [41] gave an algorithm for metric spaces that can be hierarchically approximated by binary trees; the algorithm has competitive ratio O(log ∆), where ∆ is the diameter of the metric space.…”

mentioning

confidence: 98%

The -server problem

Koutsoupias

2009

Computer Science Review

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Randomized k-server on hierarchical binary trees

Cited by 21 publications

References 27 publications

Randomized algorithm for the k-server problem on decomposable spaces

Randomized algorithm for the k-server problem on decomposable spaces

k-server via multiscale entropic regularization

The -server problem

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