511 research outputs found
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
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
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
Non-Zero Sum Games for Reactive Synthesis
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
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
Power Aware Wireless File Downloading: A Constrained Restless Bandit Approach
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
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
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 different interval lengths, we give a simple algorithm that achieves a
competitive ratio of 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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