511 research outputs found

    On Randomized Memoryless Algorithms for the Weighted kk-server Problem

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    The weighted kk-server problem is a generalization of the kk-server problem in which the cost of moving a server of weight βi\beta_i through a distance dd is βid\beta_i\cdot d. The weighted server problem on uniform spaces models caching where caches have different write costs. We prove tight bounds on the performance of randomized memoryless algorithms for this problem on uniform metric spaces. We prove that there is an αk\alpha_k-competitive memoryless algorithm for this problem, where αk=αk12+3αk1+1\alpha_k=\alpha_{k-1}^2+3\alpha_{k-1}+1; α1=1\alpha_1=1. On the other hand we also prove that no randomized memoryless algorithm can have competitive ratio better than αk\alpha_k. To prove the upper bound of αk\alpha_k we develop a framework to bound from above the competitive ratio of any randomized memoryless algorithm for this problem. The key technical contribution is a method for working with potential functions defined implicitly as the solution of a linear system. The result is robust in the sense that a small change in the probabilities used by the algorithm results in a small change in the upper bound on the competitive ratio. The above result has two important implications. Firstly this yields an αk\alpha_k-competitive memoryless algorithm for the weighted kk-server problem on uniform spaces. This is the first competitive algorithm for k>2k>2 which is memoryless. Secondly, this helps us prove that the Harmonic algorithm, which chooses probabilities in inverse proportion to weights, has a competitive ratio of kαkk\alpha_k.Comment: Published at the 54th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2013

    Memoryless Algorithms for the Generalized kk-server Problem on Uniform Metrics

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    We consider the generalized kk-server problem on uniform metrics. We study the power of memoryless algorithms and show tight bounds of Θ(k!)\Theta(k!) on their competitive ratio. In particular we show that the \textit{Harmonic Algorithm} achieves this competitive ratio and provide matching lower bounds. This improves the 22k\approx 2^{2^k} doubly-exponential bound of Chiplunkar and Vishwanathan for the more general setting of uniform metrics with different weights

    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)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(k3logl)\mathcal{O}(k^3\log l), beating the deterministic lower bound of (k+ll)1{{k+l}\choose{l}}-1. We prove that any randomized algorithm for MSSMS on uniform metric spaces must be Ω(logkl)\Omega(\log kl)-competitive. We then prove an improved lower bound of (k+2l1k)(k+l1k){{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

    Non-Zero Sum Games for Reactive Synthesis

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    In this invited contribution, we summarize new solution concepts useful for the synthesis of reactive systems that we have introduced in several recent publications. These solution concepts are developed in the context of non-zero sum games played on graphs. They are part of the contributions obtained in the inVEST project funded by the European Research Council.Comment: LATA'16 invited pape

    Weighted k-Server Bounds via Combinatorial Dichotomies

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    The weighted kk-server problem is a natural generalization of the kk-server problem where each server has a different weight. We consider the problem on uniform metrics, which corresponds to a natural generalization of paging. Our main result is a doubly exponential lower bound on the competitive ratio of any deterministic online algorithm, that essentially matches the known upper bounds for the problem and closes a large and long-standing gap. The lower bound is based on relating the weighted kk-server problem to a certain combinatorial problem and proving a Ramsey-theoretic lower bound for it. This combinatorial connection also reveals several structural properties of low cost feasible solutions to serve a sequence of requests. We use this to show that the generalized Work Function Algorithm achieves an almost optimum competitive ratio, and to obtain new refined upper bounds on the competitive ratio for the case of dd different weight classes.Comment: accepted to FOCS'1

    Power Aware Wireless File Downloading: A Constrained Restless Bandit Approach

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    This paper treats power-aware throughput maximization in a multi-user file downloading system. Each user can receive a new file only after its previous file is finished. The file state processes for each user act as coupled Markov chains that form a generalized restless bandit system. First, an optimal algorithm is derived for the case of one user. The algorithm maximizes throughput subject to an average power constraint. Next, the one-user algorithm is extended to a low complexity heuristic for the multi-user problem. The heuristic uses a simple online index policy and its effectiveness is shown via simulation. For simple 3-user cases where the optimal solution can be computed offline, the heuristic is shown to be near-optimal for a wide range of parameters

    Value Iteration for Long-run Average Reward in Markov Decision Processes

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    Markov decision processes (MDPs) are standard models for probabilistic systems with non-deterministic behaviours. Long-run average rewards provide a mathematically elegant formalism for expressing long term performance. Value iteration (VI) is one of the simplest and most efficient algorithmic approaches to MDPs with other properties, such as reachability objectives. Unfortunately, a naive extension of VI does not work for MDPs with long-run average rewards, as there is no known stopping criterion. In this work our contributions are threefold. (1) We refute a conjecture related to stopping criteria for MDPs with long-run average rewards. (2) We present two practical algorithms for MDPs with long-run average rewards based on VI. First, we show that a combination of applying VI locally for each maximal end-component (MEC) and VI for reachability objectives can provide approximation guarantees. Second, extending the above approach with a simulation-guided on-demand variant of VI, we present an anytime algorithm that is able to deal with very large models. (3) Finally, we present experimental results showing that our methods significantly outperform the standard approaches on several benchmarks

    Any-Order Online Interval Selection

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    We consider the problem of online interval scheduling on a single machine, where intervals arrive online in an order chosen by an adversary, and the algorithm must output a set of non-conflicting intervals. Traditionally in scheduling theory, it is assumed that intervals arrive in order of increasing start times. We drop that assumption and allow for intervals to arrive in any possible order. We call this variant any-order interval selection (AOIS). We assume that some online acceptances can be revoked, but a feasible solution must always be maintained. For unweighted intervals and deterministic algorithms, this problem is unbounded. Under the assumption that there are at most kk different interval lengths, we give a simple algorithm that achieves a competitive ratio of 2k2k and show that it is optimal amongst deterministic algorithms, and a restricted class of randomized algorithms we call memoryless, contributing to an open question by Adler and Azar 2003; namely whether a randomized algorithm without access to history can achieve a constant competitive ratio. We connect our model to the problem of call control on the line, and show how the algorithms of Garay et al. 1997 can be applied to our setting, resulting in an optimal algorithm for the case of proportional weights. We also discuss the case of intervals with arbitrary weights, and show how to convert the single-length algorithm of Fung et al. 2014 into a classify and randomly select algorithm that achieves a competitive ratio of 2k. Finally, we consider the case of intervals arriving in a random order, and show that for single-lengthed instances, a one-directional algorithm (i.e. replacing intervals in one direction), is the only deterministic memoryless algorithm that can possibly benefit from random arrivals. Finally, we briefly discuss the case of intervals with arbitrary weights.Comment: 19 pages, 11 figure
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