6,941 research outputs found

    Multi-Embedding of Metric Spaces

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    Metric embedding has become a common technique in the design of algorithms. Its applicability is often dependent on how high the embedding's distortion is. For example, embedding finite metric space into trees may require linear distortion as a function of its size. Using probabilistic metric embeddings, the bound on the distortion reduces to logarithmic in the size. We make a step in the direction of bypassing the lower bound on the distortion in terms of the size of the metric. We define "multi-embeddings" of metric spaces in which a point is mapped onto a set of points, while keeping the target metric of polynomial size and preserving the distortion of paths. The distortion obtained with such multi-embeddings into ultrametrics is at most O(log Delta loglog Delta) where Delta is the aspect ratio of the metric. In particular, for expander graphs, we are able to obtain constant distortion embeddings into trees in contrast with the Omega(log n) lower bound for all previous notions of embeddings. We demonstrate the algorithmic application of the new embeddings for two optimization problems: group Steiner tree and metrical task systems

    Ramsey-type theorems for metric spaces with applications to online problems

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    A nearly logarithmic lower bound on the randomized competitive ratio for the metrical task systems problem is presented. This implies a similar lower bound for the extensively studied k-server problem. The proof is based on Ramsey-type theorems for metric spaces, that state that every metric space contains a large subspace which is approximately a hierarchically well-separated tree (and in particular an ultrametric). These Ramsey-type theorems may be of independent interest.Comment: Fix an error in the metadata. 31 pages, 0 figures. Preliminary version in FOCS '01. To be published in J. Comput. System Sc

    Using genetic algorithms to create meaningful poetic text

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    Work carried out when all authors were at the University of Edinburgh.Peer reviewedPostprin

    Metrical Service Systems with Multiple Servers

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    We study the problem of metrical service systems with multiple servers (MSSMS), which generalizes two well-known problems -- the kk-server problem, and metrical service systems. The MSSMS problem is to service requests, each of which is an ll-point subset of a metric space, using kk servers, with the objective of minimizing the total distance traveled by the servers. Feuerstein initiated a study of this problem by proving upper and lower bounds on the deterministic competitive ratio for uniform metric spaces. We improve Feuerstein's analysis of the upper bound and prove that his algorithm achieves a competitive ratio of k((k+ll)āˆ’1)k({{k+l}\choose{l}}-1). In the randomized online setting, for uniform metric spaces, we give an algorithm which achieves a competitive ratio O(k3logā”l)\mathcal{O}(k^3\log l), beating the deterministic lower bound of (k+ll)āˆ’1{{k+l}\choose{l}}-1. We prove that any randomized algorithm for MSSMS on uniform metric spaces must be Ī©(logā”kl)\Omega(\log kl)-competitive. We then prove an improved lower bound of (k+2lāˆ’1k)āˆ’(k+lāˆ’1k){{k+2l-1}\choose{k}}-{{k+l-1}\choose{k}} on the competitive ratio of any deterministic algorithm for (k,l)(k,l)-MSSMS, on general metric spaces. In the offline setting, we give a pseudo-approximation algorithm for (k,l)(k,l)-MSSMS on general metric spaces, which achieves an approximation ratio of ll using klkl servers. We also prove a matching hardness result, that a pseudo-approximation with less than klkl servers is unlikely, even for uniform metric spaces. For general metric spaces, we highlight the limitations of a few popular techniques, that have been used in algorithm design for the kk-server problem and metrical service systems.Comment: 18 pages; accepted for publication at COCOON 201

    Randomized online computation with high probability guarantees

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    We study the relationship between the competitive ratio and the tail distribution of randomized online minimization problems. To this end, we define a broad class of online problems that includes some of the well-studied problems like paging, k-server and metrical task systems on finite metrics, and show that for these problems it is possible to obtain, given an algorithm with constant expected competitive ratio, another algorithm that achieves the same solution quality up to an arbitrarily small constant error a with high probability; the "high probability" statement is in terms of the optimal cost. Furthermore, we show that our assumptions are tight in the sense that removing any of them allows for a counterexample to the theorem. In addition, there are examples of other problems not covered by our definition, where similar high probability results can be obtained.Comment: 20 pages, 2 figure
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