520,532 research outputs found
Crossing the Logarithmic Barrier for Dynamic Boolean Data Structure Lower Bounds
This paper proves the first super-logarithmic lower bounds on the cell probe
complexity of dynamic boolean (a.k.a. decision) data structure problems, a
long-standing milestone in data structure lower bounds.
We introduce a new method for proving dynamic cell probe lower bounds and use
it to prove a lower bound on the operational
time of a wide range of boolean data structure problems, most notably, on the
query time of dynamic range counting over ([Pat07]). Proving an
lower bound for this problem was explicitly posed as one of
five important open problems in the late Mihai P\v{a}tra\c{s}cu's obituary
[Tho13]. This result also implies the first lower bound for the
classical 2D range counting problem, one of the most fundamental data structure
problems in computational geometry and spatial databases. We derive similar
lower bounds for boolean versions of dynamic polynomial evaluation and 2D
rectangle stabbing, and for the (non-boolean) problems of range selection and
range median.
Our technical centerpiece is a new way of "weakly" simulating dynamic data
structures using efficient one-way communication protocols with small advantage
over random guessing. This simulation involves a surprising excursion to
low-degree (Chebychev) polynomials which may be of independent interest, and
offers an entirely new algorithmic angle on the "cell sampling" method of
Panigrahy et al. [PTW10]
Efficient Method for Computing Lower Bounds on the -radius of Switched Linear Systems
This paper proposes lower bounds on a quantity called -norm joint
spectral radius, or in short, -radius, of a finite set of matrices. Despite
its wide range of applications to, for example, stability analysis of switched
linear systems and the equilibrium analysis of switched linear economical
models, algorithms for computing the -radius are only available in a very
limited number of particular cases. The proposed lower bounds are given as the
spectral radius of an average of the given matrices weighted via Kronecker
products and do not place any requirements on the set of matrices. We show that
the proposed lower bounds theoretically extend and also can practically improve
the existing lower bounds. A Markovian extension of the proposed lower bounds
is also presented
Rigorous Bounds on the Free Energy of Electron-Phonon Models
We present a collection of rigorous upper and lower bounds to the free energy of electron-phonon models with linear electron-phonon interaction. These bounds are used to compare different variational approaches. It is shown rigorously that the ground states corresponding to the sharpest bounds do not exhibit Off-Diagonal Long-Range Order in the two-particle density matrix.
Towards Tight Lower Bounds for Range Reporting on the RAM
In the orthogonal range reporting problem, we are to preprocess a set of
points with integer coordinates on a grid. The goal is to support
reporting all points inside an axis-aligned query rectangle. This is one of
the most fundamental data structure problems in databases and computational
geometry. Despite the importance of the problem its complexity remains
unresolved in the word-RAM. On the upper bound side, three best tradeoffs
exists: (1.) Query time with words
of space for any constant . (2.) Query time with words of space. (3.) Query time
with optimal words of space. However, the
only known query time lower bound is , even for linear
space data structures.
All three current best upper bound tradeoffs are derived by reducing range
reporting to a ball-inheritance problem. Ball-inheritance is a problem that
essentially encapsulates all previous attempts at solving range reporting in
the word-RAM. In this paper we make progress towards closing the gap between
the upper and lower bounds for range reporting by proving cell probe lower
bounds for ball-inheritance. Our lower bounds are tight for a large range of
parameters, excluding any further progress for range reporting using the
ball-inheritance reduction
Best possible rates of distribution of dense lattice orbits in homogeneous spaces
The present paper establishes upper and lower bounds on the speed of
approximation in a wide range of natural Diophantine approximation problems.
The upper and lower bounds coincide in many cases, giving rise to optimal
results in Diophantine approximation which were inaccessible previously. Our
approach proceeds by establishing, more generally, upper and lower bounds for
the rate of distribution of dense orbits of a lattice subgroup in a
connected Lie (or algebraic) group , acting on suitable homogeneous spaces
. The upper bound is derived using a quantitative duality principle for
homogeneous spaces, reducing it to a rate of convergence in the mean ergodic
theorem for a family of averaging operators supported on and acting on
. In particular, the quality of the upper bound on the rate of
distribution we obtain is determined explicitly by the spectrum of in the
automorphic representation on . We show that the rate
is best possible when the representation in question is tempered, and show that
the latter condition holds in a wide range of examples
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