93 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
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
The generalized work function algorithm is competitive for the generalized 2-server problem
The generalized 2-server problem is an online optimization problem where a
sequence of requests has to be served at minimal cost. Requests arrive one by
one and need to be served instantly by at least one of two servers. We consider
the general model where the cost function of the two servers may be different.
Formally, each server moves in its own metric space and a request consists of
one point in each metric space. It is served by moving one of the two servers
to its request point. Requests have to be served without knowledge of the
future requests. The objective is to minimize the total traveled distance. The
special case where both servers move on the real line is known as the
CNN-problem. We show that the generalized work function algorithm is constant
competitive for the generalized 2-server problem
The Generalized Work Function Algorithm Is Competitive for the Generalized 2-Server Problem
The generalized 2-server problem is an online optimization problem where a sequence of requests has to be served at minimal cost. Requests arrive one by one and need to be served instantly by at least one of two servers. We consider the general model where the cost function of the two servers may be different. Formally, each server moves in its own metric space and a request consists of one point in each metric space. It is served by moving one of the two servers to its request point. Requests have to be served without knowledge of future requests. The objective is to minimize the total traveled distance. The special case where both servers move on the real line is known as the CNN problem. We show that the generalized work function algorithm, , is constant competitive for the generalized 2-server problem. Further, we give an outline for a possible extension to servers and discuss the applicability of our techniques and of the work function algorithm in general. We conclude with a discussion on several open problems in online optimization
The randomized server problem
In the k-server problem there are k ≥ 2 identical servers which are located at k points in a metric space M. If there is a request to a point r ∈ M, one of the servers must be moved to the request point in order to serve this request. The cost of this service is the distance between the points where the server resided before the service and after the service. A k-server algorithm A must decide which server should be moved at each step. The goal of A is to minimize the total service cost. Competitiveness makes sense as a concept when A lacks timely access to all input data. We consider the version of the problem where requests must be served online , i.e., the algorithm must decide which server to move without knowledge of future requests. Randomization is a strong tool to derive algorithms with better competitiveness; The main contributions of this thesis are: (1) An explicit detailed proof of the 2-competitiveness of the Random Slack Algorithm, which has never been given before. We note that Random Slack is a trackless algorithm. (2) An essay-style description of a new concept called the knowledge state approach, which has recently been developed by Bein, Larmore, and Reischuk. (3) We give optimally competitive randomized algorithms for 2 and 3 cache paging with few bookmarks. We note that the paging problem is a special case of the server problem, and that it is desirable to minimize the number of bookmarks, as such bookmarks pose a considerable challenge in real world applications such as cache management of pages on the world wide web; Furthermore, the thesis summarizes a number of basic results for both the randomized and the deterministic server problem
On-line algorithms for the K-server problem and its variants.
by Chi-ming Wat.Thesis (M.Phil.)--Chinese University of Hong Kong, 1995.Includes bibliographical references (leaves 77-82).Chapter 1 --- Introduction --- p.1Chapter 1.1 --- Performance analysis of on-line algorithms --- p.2Chapter 1.2 --- Randomized algorithms --- p.4Chapter 1.3 --- Types of adversaries --- p.5Chapter 1.4 --- Overview of the results --- p.6Chapter 2 --- The k-server problem --- p.8Chapter 2.1 --- Introduction --- p.8Chapter 2.2 --- Related Work --- p.9Chapter 2.3 --- The Evolution of Work Function Algorithm --- p.12Chapter 2.4 --- Definitions --- p.16Chapter 2.5 --- The Work Function Algorithm --- p.18Chapter 2.6 --- The Competitive Analysis --- p.20Chapter 3 --- The weighted k-server problem --- p.27Chapter 3.1 --- Introduction --- p.27Chapter 3.2 --- Related Work --- p.29Chapter 3.3 --- Fiat and Ricklin's Algorithm --- p.29Chapter 3.4 --- The Work Function Algorithm --- p.32Chapter 3.5 --- The Competitive Analysis --- p.35Chapter 4 --- The Influence of Lookahead --- p.41Chapter 4.1 --- Introduction --- p.41Chapter 4.2 --- Related Work --- p.42Chapter 4.3 --- The Role of l-lookahead --- p.43Chapter 4.4 --- The LRU Algorithm with l-lookahead --- p.45Chapter 4.5 --- The Competitive Analysis --- p.45Chapter 5 --- Space Complexity --- p.57Chapter 5.1 --- Introduction --- p.57Chapter 5.2 --- Related Work --- p.59Chapter 5.3 --- Preliminaries --- p.59Chapter 5.4 --- The TWO Algorithm --- p.60Chapter 5.5 --- Competitive Analysis --- p.61Chapter 5.6 --- Remarks --- p.69Chapter 6 --- Conclusions --- p.70Chapter 6.1 --- Summary of Our Results --- p.70Chapter 6.2 --- Recent Results --- p.71Chapter 6.2.1 --- The Adversary Models --- p.71Chapter 6.2.2 --- On-line Performance-Improvement Algorithms --- p.73Chapter A --- Proof of Lemma1 --- p.75Bibliography --- p.7
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
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