1,756 research outputs found
On the query complexity of real functionals
Recently Kawamura and Cook developed a framework to define the computational complexity of operators arising in analysis. Our goal is to understand the effects of complexity restrictions on the analytical properties of the operator. We focus on the case of norms over C[0,1] and introduce the notion of dependence of a norm on a point and relate it to the query complexity of the norm. We show that the dependence of almost every point is of the order of the query complexity of the norm. A norm with small complexity depends on a few points but, as compensation, highly depends on them. We characterize the functionals that are computable using one oracle call only and discuss the uniformity of that characterization
Randomized and Quantum Algorithms Yield a Speed-Up for Initial-Value Problems
Quantum algorithms and complexity have recently been studied not only for
discrete, but also for some numerical problems. Most attention has been paid so
far to the integration problem, for which a speed-up is shown by quantum
computers with respect to deterministic and randomized algorithms on a
classical computer. In this paper we deal with the randomized and quantum
complexity of initial-value problems. For this nonlinear problem, we show that
both randomized and quantum algorithms yield a speed-up over deterministic
algorithms. Upper bounds on the complexity in the randomized and quantum
settings are shown by constructing algorithms with a suitable cost, where the
construction is based on integral information. Lower bounds result from the
respective bounds for the integration problem.Comment: LaTeX v. 2.09, 13 page
Bounded time computation on metric spaces and Banach spaces
We extend the framework by Kawamura and Cook for investigating computational
complexity for operators occurring in analysis. This model is based on
second-order complexity theory for functions on the Baire space, which is
lifted to metric spaces by means of representations. Time is measured in terms
of the length of the input encodings and the required output precision. We
propose the notions of a complete representation and of a regular
representation. We show that complete representations ensure that any
computable function has a time bound. Regular representations generalize
Kawamura and Cook's more restrictive notion of a second-order representation,
while still guaranteeing fast computability of the length of the encodings.
Applying these notions, we investigate the relationship between purely metric
properties of a metric space and the existence of a representation such that
the metric is computable within bounded time. We show that a bound on the
running time of the metric can be straightforwardly translated into size bounds
of compact subsets of the metric space. Conversely, for compact spaces and for
Banach spaces we construct a family of admissible, complete, regular
representations that allow for fast computation of the metric and provide short
encodings. Here it is necessary to trade the time bound off against the length
of encodings
Convex Combinatorial Optimization
We introduce the convex combinatorial optimization problem, a far reaching
generalization of the standard linear combinatorial optimization problem. We
show that it is strongly polynomial time solvable over any edge-guaranteed
family, and discuss several applications
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