95 research outputs found
The Advice Complexity of a Class of Hard Online Problems
The advice complexity of an online problem is a measure of how much knowledge
of the future an online algorithm needs in order to achieve a certain
competitive ratio. Using advice complexity, we define the first online
complexity class, AOC. The class includes independent set, vertex cover,
dominating set, and several others as complete problems. AOC-complete problems
are hard, since a single wrong answer by the online algorithm can have
devastating consequences. For each of these problems, we show that
bits of advice are
necessary and sufficient (up to an additive term of ) to achieve a
competitive ratio of .
The results are obtained by introducing a new string guessing problem related
to those of Emek et al. (TCS 2011) and B\"ockenhauer et al. (TCS 2014). It
turns out that this gives a powerful but easy-to-use method for providing both
upper and lower bounds on the advice complexity of an entire class of online
problems, the AOC-complete problems.
Previous results of Halld\'orsson et al. (TCS 2002) on online independent
set, in a related model, imply that the advice complexity of the problem is
. Our results improve on this by providing an exact formula for
the higher-order term. For online disjoint path allocation, B\"ockenhauer et
al. (ISAAC 2009) gave a lower bound of and an upper bound of
on the advice complexity. We improve on the upper bound by a
factor of . For the remaining problems, no bounds on their advice
complexity were previously known.Comment: Full paper to appear in Theory of Computing Systems. A preliminary
version appeared in STACS 201
Efficient Rebalancing of Chromatic Search Trees
In PODS'91, Nurmi and Soisalon-Soininen presented a new type of binary search tree for databases, which they call a chromatic tree. The aim is to improve runtime performance by allowing a greater degree of concurrency, which, in turn, is obtained by uncoupling updating from rebalancing. This also allows rebalancing to be postponed completely or partially until after peak working hours.The advantages of the proposal of Nurmi and Soisalon-Soininen are quite significant, but there are definite problems with it. First, they give no explicit upper bound on the complexity of their algorithm. Second, some of their rebalancing operations can be applied many more times than necessary. Third, some of their operations, when removing one problem, create another.We define a new set of rebalancing operations which we prove give rise to at most I_ log_2(N+1) _I - 1 $ rebalancing operations per insertion and at most I_ log_2 (N+1)_I - 2 rebalancing operations per deletion, where N is the maximum'size the tree could ever have, given its initial size and the number of insertions performed. Most of these rebalancing operations, in fact, do no restructuring; they simply move weights around. The number of operations which actually change the structure of the tree is at most one per update
Efficient Rebalancing of Chromatic Search Trees
In PODS'91, Nurmi and Soisalon-Soininen presented a new type of binary search tree for databases, which they call a chromatic tree. The aim is to improve runtime performance by allowing a greater degree of concurrency, which, in turn, is obtained by uncoupling updating from rebalancing. This also allows rebalancing to be postponed completely or partially until after peak working hours.The advantages of the proposal of Nurmi and Soisalon-Soininen are quite significant, but there are definite problems with it. First, they give no explicit upper bound on the complexity of their algorithm. Second, some of their rebalancing operations can be applied many more times than necessary. Third, some of their operations, when removing one problem, create another.We define a new set of rebalancing operations which we prove give rise to at most I_ log_2(N+1) _I - 1 $ rebalancing operations per insertion and at most I_ log_2 (N+1)_I - 2 rebalancing operations per deletion, where N is the maximum'size the tree could ever have, given its initial size and the number of insertions performed. Most of these rebalancing operations, in fact, do no restructuring; they simply move weights around. The number of operations which actually change the structure of the tree is at most one per update
Online Bin Packing with Advice
We consider the online bin packing problem under the advice complexity model
where the 'online constraint' is relaxed and an algorithm receives partial
information about the future requests. We provide tight upper and lower bounds
for the amount of advice an algorithm needs to achieve an optimal packing. We
also introduce an algorithm that, when provided with log n + o(log n) bits of
advice, achieves a competitive ratio of 3/2 for the general problem. This
algorithm is simple and is expected to find real-world applications. We
introduce another algorithm that receives 2n + o(n) bits of advice and achieves
a competitive ratio of 4/3 + {\epsilon}. Finally, we provide a lower bound
argument that implies that advice of linear size is required for an algorithm
to achieve a competitive ratio better than 9/8.Comment: 19 pages, 1 figure (2 subfigures
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