17,806 research outputs found
Time and Space Bounds for Reversible Simulation
We prove a general upper bound on the tradeoff between time and space that
suffices for the reversible simulation of irreversible computation. Previously,
only simulations using exponential time or quadratic space were known.
The tradeoff shows for the first time that we can simultaneously achieve
subexponential time and subquadratic space.
The boundary values are the exponential time with hardly any extra space
required by the Lange-McKenzie-Tapp method and the ()th power time with
square space required by the Bennett method. We also give the first general
lower bound on the extra storage space required by general reversible
simulation. This lower bound is optimal in that it is achieved by some
reversible simulations.Comment: 11 pages LaTeX, Proc ICALP 2001, Lecture Notes in Computer Science,
Vol xxx Springer-Verlag, Berlin, 200
Reversibility and Adiabatic Computation: Trading Time and Space for Energy
Future miniaturization and mobilization of computing devices requires energy
parsimonious `adiabatic' computation. This is contingent on logical
reversibility of computation. An example is the idea of quantum computations
which are reversible except for the irreversible observation steps. We propose
to study quantitatively the exchange of computational resources like time and
space for irreversibility in computations. Reversible simulations of
irreversible computations are memory intensive. Such (polynomial time)
simulations are analysed here in terms of `reversible' pebble games. We show
that Bennett's pebbling strategy uses least additional space for the greatest
number of simulated steps. We derive a trade-off for storage space versus
irreversible erasure. Next we consider reversible computation itself. An
alternative proof is provided for the precise expression of the ultimate
irreversibility cost of an otherwise reversible computation without
restrictions on time and space use. A time-irreversibility trade-off hierarchy
in the exponential time region is exhibited. Finally, extreme
time-irreversibility trade-offs for reversible computations in the thoroughly
unrealistic range of computable versus noncomputable time-bounds are given.Comment: 30 pages, Latex. Lemma 2.3 should be replaced by the slightly better
``There is a winning strategy with pebbles and erasures for
pebble games with , for all '' with appropriate
further changes (as pointed out by Wim van Dam). This and further work on
reversible simulations as in Section 2 appears in quant-ph/970300
Reversible Simulation of Irreversible Computation by Pebble Games
Reversible simulation of irreversible algorithms is analyzed in the stylized
form of a `reversible' pebble game. While such simulations incur little
overhead in additional computation time, they use a large amount of additional
memory space during the computation. The reacheable reversible simulation
instantaneous descriptions (pebble configurations) are characterized
completely. As a corollary we obtain the reversible simulation by Bennett and
that among all simulations that can be modelled by the pebble game, Bennett's
simulation is optimal in that it uses the least auxiliary space for the
greatest number of simulated steps. One can reduce the auxiliary storage
overhead incurred by the reversible simulation at the cost of allowing limited
erasing leading to an irreversibility-space tradeoff. We show that in this
resource-bounded setting the limited erasing needs to be performed at precise
instants during the simulation. We show that the reversible simulation can be
modified so that it is applicable also when the simulated computation time is
unknown.Comment: 11 pages, Latex, Submitted to Physica
Quantum Branching Programs and Space-Bounded Nonuniform Quantum Complexity
In this paper, the space complexity of nonuniform quantum computations is
investigated. The model chosen for this are quantum branching programs, which
provide a graphic description of sequential quantum algorithms. In the first
part of the paper, simulations between quantum branching programs and
nonuniform quantum Turing machines are presented which allow to transfer lower
and upper bound results between the two models. In the second part of the
paper, different variants of quantum OBDDs are compared with their
deterministic and randomized counterparts. In the third part, quantum branching
programs are considered where the performed unitary operation may depend on the
result of a previous measurement. For this model a simulation of randomized
OBDDs and exponential lower bounds are presented.Comment: 45 pages, 3 Postscript figures. Proofs rearranged, typos correcte
Nonasymptotic bounds on the mean square error for MCMC estimates via renewal techniques
The Nummellinâs split chain construction allows to decompose a Markov
chain Monte Carlo (MCMC) trajectory into i.i.d. "excursions". Regenerative MCMC
algorithms based on this technique use a random number of samples. They have
been proposed as a promising alternative to usual fixed length simulation [25, 33,
14]. In this note we derive nonasymptotic bounds on the mean square error (MSE)
of regenerative MCMC estimates via techniques of renewal theory and sequential
statistics. These results are applied to costruct confidence intervals. We then focus
on two cases of particular interest: chains satisfying the Doeblin condition and a geometric
drift condition. Available explicit nonasymptotic results are compared for
different schemes of MCMC simulation
Additional material on bounds of -spectral gap for discrete Markov chains with band transition matrices
We analyse the -convergence rate of irreducible and aperiodic
Markov chains with -band transition probability matrix and with
invariant distribution . This analysis is heavily based on: first the
study of the essential spectral radius of
derived from Hennion's quasi-compactness criteria; second
the connection between the spectral gap property (SG) of on
and the -geometric ergodicity of . Specifically, (SG)
is shown to hold under the condition \alpha\_0 := \sum\_{{m}=-N}^N
\limsup\_{i\rightarrow +\infty} \sqrt{P(i,i+{m})\, P^*(i+{m},i)}\ \textless{}\,
1. Moreover . Simple conditions
on asymptotic properties of and of its invariant probability distribution
to ensure that \alpha\_0\textless{}1 are given. In particular this
allows us to obtain estimates of the -geometric convergence rate
of random walks with bounded increments. The specific case of reversible is
also addressed. Numerical bounds on the convergence rate can be provided via a
truncation procedure. This is illustrated on the Metropolis-Hastings algorithm
Sticky Seeding in Discrete-Time Reversible-Threshold Networks
When nodes can repeatedly update their behavior (as in agent-based models
from computational social science or repeated-game play settings) the problem
of optimal network seeding becomes very complex. For a popular
spreading-phenomena model of binary-behavior updating based on thresholds of
adoption among neighbors, we consider several planning problems in the design
of \textit{Sticky Interventions}: when adoption decisions are reversible, the
planner aims to find a Seed Set where temporary intervention leads to long-term
behavior change. We prove that completely converting a network at minimum cost
is -hard to approximate and that maximizing conversion
subject to a budget is -hard to approximate. Optimization
heuristics which rely on many objective function evaluations may still be
practical, particularly in relatively-sparse networks: we prove that the
long-term impact of a Seed Set can be evaluated in operations. For a
more descriptive model variant in which some neighbors may be more influential
than others, we show that under integer edge weights from
objective function evaluation requires only operations. These
operation bounds are based on improvements we give for bounds on
time-steps-to-convergence under discrete-time reversible-threshold updates in
networks.Comment: 19 pages, 2 figure
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