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Abstract:
In this paper we study the problem of assigning unit-size tasks to
related machines when only limited online information is provided
to each task. This is a general framework whose special cases are
the classical multiple-choice games for the assignment of
unit-size tasks to identical machines. The latter case was the
subject of intensive research for the last decade. The problem is
intriguing in the sense that the natural extensions of the greedy
oblivious schedulers, which are known to achieve near-optimal
performance in the case of identical machines, are proved to
perform quite poorly in the case of the related machines.
In this work we present a rather surprising lower bound stating
that any oblivious scheduler that assigns an arbitrary number of
tasks to $n$ related machines would need $\Omega\left(\frac{\log
n}{\log\!\log n}\right)$ polls of machine loads per task, in order to
achieve a constant competitive ratio versus the optimum offline
assignment of the same input sequence to these machines. On the
other hand, we prove that the missing information for an oblivious
scheduler to perform almost optimally, is the amount of tasks to
be inserted into the system. In particular, we provide an
oblivious scheduler that only uses $\cal{O}(\log\!\log n)$ polls, along with
the additional information of the size of the input sequence, in
order to achieve a constant competitive ratio vs. the optimum
offline assignment. The philosophy of this scheduler is based on
an interesting exploitation of the {\sc slowfit} concept
([AAFPW97,BFN00,BCK97]; for a survey see
[BY98,Azar98,Sgall98]) for the assignment of the tasks to the
related machines despite the restrictions on the provided online
information, in combination with a layered induction argument for
bounding the tails of the number of tasks passing from slower to
faster machines. We finally use this oblivious scheduler as the
core of an adaptive scheduler that does not demand the knowledge
of the input sequence and yet achieves almost the same
performance.
