562 research outputs found
Memoryless Algorithms for the Generalized -server Problem on Uniform Metrics
We consider the generalized -server problem on uniform metrics. We study
the power of memoryless algorithms and show tight bounds of 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 doubly-exponential bound of Chiplunkar and
Vishwanathan for the more general setting of uniform metrics with different
weights
On Randomized Memoryless Algorithms for the Weighted -server Problem
The weighted -server problem is a generalization of the -server problem
in which the cost of moving a server of weight through a distance
is . 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 -competitive memoryless
algorithm for this problem, where ;
. On the other hand we also prove that no randomized memoryless
algorithm can have competitive ratio better than .
To prove the upper bound of 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
-competitive memoryless algorithm for the weighted -server problem
on uniform spaces. This is the first competitive algorithm for 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 .Comment: Published at the 54th Annual IEEE Symposium on Foundations of
Computer Science (FOCS 2013
Weighted k-Server Bounds via Combinatorial Dichotomies
The weighted -server problem is a natural generalization of the -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 -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 different weight classes.Comment: accepted to FOCS'1
The Randomized Competitive Ratio of Weighted k-Server Is at Least Exponential
The weighted k-server problem is a natural generalization of the k-server problem in which the cost incurred in moving a server is the distance traveled times the weight of the server. Even after almost three decades since the seminal work of Fiat and Ricklin (1994), the competitive ratio of this problem remains poorly understood even on the simplest class of metric spaces - the uniform metric spaces. In particular, in the case of randomized algorithms against the oblivious adversary, neither a better upper bound that the doubly exponential deterministic upper bound, nor a better lower bound than the logarithmic lower bound of unweighted k-server, is known. In this paper, we make significant progress towards understanding the randomized competitive ratio of weighted k-server on uniform metrics. We cut down the triply exponential gap between the upper and lower bound to a singly exponential gap by proving that the competitive ratio is at least exponential in k, substantially improving on the previously known lower bound of about ln k
Metrical Service Systems with Multiple Servers
We study the problem of metrical service systems with multiple servers
(MSSMS), which generalizes two well-known problems -- the -server problem,
and metrical service systems. The MSSMS problem is to service requests, each of
which is an -point subset of a metric space, using 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 . In the randomized
online setting, for uniform metric spaces, we give an algorithm which achieves
a competitive ratio , beating the deterministic lower
bound of . We prove that any randomized algorithm for
MSSMS on uniform metric spaces must be -competitive. We then
prove an improved lower bound of on
the competitive ratio of any deterministic algorithm for -MSSMS, on
general metric spaces. In the offline setting, we give a pseudo-approximation
algorithm for -MSSMS on general metric spaces, which achieves an
approximation ratio of using servers. We also prove a matching
hardness result, that a pseudo-approximation with less than 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 -server problem and metrical service systems.Comment: 18 pages; accepted for publication at COCOON 201
Competitive algorithms for generalized k-server in uniform metrics
The generalized k-server problem is a far-reaching extension of the k-server problem with several applications. Here, each server si lies in its own metric space Mi. A request is a k-tuple r = (r1, r2, … , rk), which is served by moving some server si to the point ri ∈ Mi, and the goal is to minimize the total distance traveled by the servers. Despite much work, no f(k)-competitive algorithm is known for the problem for k > 2 servers, even for special cases such as uniform metrics and lines. Here, we consider the problem in uniform metrics and give the first f(k)-competitive algorithms for general k. In particular, we obtain deterministic and randomized algorithms with competitive ratio k · 2k and O(k3log k) respectively. Our deterministic bound is based on a novel application of the polynomial method to online algorithms, and essentially matches the long-known lower bound of 2k − 1. We also give a (2^{2^{O(k)}} ) -competitive deterministic algorithm for weighted uniform metrics, which also essentially matches the recent doubly exponential lower bound for the problem
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