465,869 research outputs found
Recursive Sketching For Frequency Moments
In a ground-breaking paper, Indyk and Woodruff (STOC 05) showed how to
compute (for ) in space complexity O(\mbox{\em poly-log}(n,m)\cdot
n^{1-\frac2k}), which is optimal up to (large) poly-logarithmic factors in
and , where is the length of the stream and is the upper bound on
the number of distinct elements in a stream. The best known lower bound for
large moments is . A follow-up work of
Bhuvanagiri, Ganguly, Kesh and Saha (SODA 2006) reduced the poly-logarithmic
factors of Indyk and Woodruff to . Further reduction of poly-log factors has been an elusive
goal since 2006, when Indyk and Woodruff method seemed to hit a natural
"barrier." Using our simple recursive sketch, we provide a different yet simple
approach to obtain a algorithm for constant (our bound is, in fact, somewhat
stronger, where the term can be replaced by any constant number
of iterations instead of just two or three, thus approaching .
Our bound also works for non-constant (for details see the body of
the paper). Further, our algorithm requires only -wise independence, in
contrast to existing methods that use pseudo-random generators for computing
large frequency moments
Element Distinctness, Frequency Moments, and Sliding Windows
We derive new time-space tradeoff lower bounds and algorithms for exactly
computing statistics of input data, including frequency moments, element
distinctness, and order statistics, that are simple to calculate for sorted
data. We develop a randomized algorithm for the element distinctness problem
whose time T and space S satisfy T in O (n^{3/2}/S^{1/2}), smaller than
previous lower bounds for comparison-based algorithms, showing that element
distinctness is strictly easier than sorting for randomized branching programs.
This algorithm is based on a new time and space efficient algorithm for finding
all collisions of a function f from a finite set to itself that are reachable
by iterating f from a given set of starting points. We further show that our
element distinctness algorithm can be extended at only a polylogarithmic factor
cost to solve the element distinctness problem over sliding windows, where the
task is to take an input of length 2n-1 and produce an output for each window
of length n, giving n outputs in total. In contrast, we show a time-space
tradeoff lower bound of T in Omega(n^2/S) for randomized branching programs to
compute the number of distinct elements over sliding windows. The same lower
bound holds for computing the low-order bit of F_0 and computing any frequency
moment F_k, k neq 1. This shows that those frequency moments and the decision
problem F_0 mod 2 are strictly harder than element distinctness. We complement
this lower bound with a T in O(n^2/S) comparison-based deterministic RAM
algorithm for exactly computing F_k over sliding windows, nearly matching both
our lower bound for the sliding-window version and the comparison-based lower
bounds for the single-window version. We further exhibit a quantum algorithm
for F_0 over sliding windows with T in O(n^{3/2}/S^{1/2}). Finally, we consider
the computations of order statistics over sliding windows.Comment: arXiv admin note: substantial text overlap with arXiv:1212.437
Measuring mass moments and electromagnetic moments of a massive, axisymmetric body, through gravitational waves
The electrovacuum around a rotating massive body with electric charge density
is described by its multipole moments (mass moments, mass-current moments,
electric moments, and magnetic moments). A small uncharged test particle
orbiting around such a body moves on geodesics if gravitational radiation is
ignored. The waves emitted by the small body carry information about the
geometry of the central object, and hence, in principle, we can infer all its
multipole moments. Due to its axisymmetry the source is characterized now by
four families of scalar multipole moments: its mass moments , its
mass-current moments , its electrical moments and its magnetic
moments , where . Four measurable quantities, the energy
emitted by gravitational waves per logarithmic interval of frequency, the
precession of the periastron (assuming almost circular orbits), the precession
of the orbital plane (assuming almost equatorial orbits), and the number of
cycles emitted per logarithmic interval of frequency, are presented as power
series of the newtonian orbital velocity of the test body. The power series
coefficients are simple polynomials of the various moments.Comment: Talk given by T. A. A. at Recent Advances in Astronomy and
Astrophysics, Lixourion, Kefallinia island, Greece, 8-11 Sep 200
The quantum complexity of approximating the frequency moments
The 'th frequency moment of a sequence of integers is defined as , where is the number of times that occurs in the
sequence. Here we study the quantum complexity of approximately computing the
frequency moments in two settings. In the query complexity setting, we wish to
minimise the number of queries to the input used to approximate up to
relative error . We give quantum algorithms which outperform the best
possible classical algorithms up to quadratically. In the multiple-pass
streaming setting, we see the elements of the input one at a time, and seek to
minimise the amount of storage space, or passes over the data, used to
approximate . We describe quantum algorithms for , and
in this model which substantially outperform the best possible
classical algorithms in certain parameter regimes.Comment: 22 pages; v3: essentially published versio
On the theory of microwave absorption by the spin-1/2 Heisenberg-Ising magnet
We analyze the problem of microwave absorption by the Heisenberg-Ising magnet
in terms of shifted moments of the imaginary part of the dynamical
susceptibility. When both, the Zeeman field and the wave vector of the incident
microwave, are parallel to the anisotropy axis, the first four moments
determine the shift of the resonance frequency and the line width in a
situation where the frequency is varied for fixed Zeeman field. For the
one-dimensional model we can calculate the moments exactly. This provides exact
data for the resonance shift and the line width at arbitrary temperatures and
magnetic fields. In current ESR experiments the Zeeman field is varied for
fixed frequency. We show how in this situation the moments give perturbative
results for the resonance shift and for the integrated intensity at small
anisotropy as well as an explicit formula connecting the line width with the
anisotropy parameter in the high-temperature limit.Comment: 4 page
Single electron magneto-conductivity of a nondegenerate 2D electron system in a quantizing magnetic field
We study transport properties of a non-degenerate two-dimensional system of
non-interacting electrons in the presence of a quantizing magnetic field and a
short-range disorder potential. We show that the low-frequency
magnetoconductivity displays a strongly asymmetric peak at a nonzero frequency.
The shape of the peak is restored from the calculated 14 spectral moments, the
asymptotic form of its high-frequency tail, and the scaling behavior of the
conductivity for omega -> 0. We also calculate 10 spectral moments of the
cyclotron resonance absorption peak and restore the corresponding
(non-singular) frequency dependence using the continuous fraction expansion.
Both expansions converge rapidly with increasing number of included moments,
and give numerically accurate results throughout the region of interest. We
discuss the possibility of experimental observation of the predicted effects
for electrons on helium.Comment: RevTeX 3.0, 14 pages, 8 eps figures included with eps
- …