475 research outputs found
The Range of Topological Effects on Communication
We continue the study of communication cost of computing functions when
inputs are distributed among processors, each of which is located at one
vertex of a network/graph called a terminal. Every other node of the network
also has a processor, with no input. The communication is point-to-point and
the cost is the total number of bits exchanged by the protocol, in the worst
case, on all edges.
Chattopadhyay, Radhakrishnan and Rudra (FOCS'14) recently initiated a study
of the effect of topology of the network on the total communication cost using
tools from embeddings. Their techniques provided tight bounds for simple
functions like Element-Distinctness (ED), which depend on the 1-median of the
graph. This work addresses two other kinds of natural functions. We show that
for a large class of natural functions like Set-Disjointness the communication
cost is essentially times the cost of the optimal Steiner tree connecting
the terminals. Further, we show for natural composed functions like and , the naive protocols
suggested by their definition is optimal for general networks. Interestingly,
the bounds for these functions depend on more involved topological parameters
that are a combination of Steiner tree and 1-median costs.
To obtain our results, we use some new tools in addition to ones used in
Chattopadhyay et. al. These include (i) viewing the communication constraints
via a linear program; (ii) using tools from the theory of tree embeddings to
prove topology sensitive direct sum results that handle the case of composed
functions and (iii) representing the communication constraints of certain
problems as a family of collection of multiway cuts, where each multiway cut
simulates the hardness of computing the function on the star topology
Predictable arguments of knowledge
We initiate a formal investigation on the power of predictability for argument of knowledge systems for NP. Specifically, we consider private-coin argument systems where the answer of the prover can be predicted, given the private randomness of the verifier; we call such protocols Predictable Arguments of Knowledge (PAoK).
Our study encompasses a full characterization of PAoK, showing that such arguments can be made extremely laconic, with the prover sending a single bit, and assumed to have only one round (i.e., two messages) of communication without loss of generality.
We additionally explore PAoK satisfying additional properties (including zero-knowledge and the possibility of re-using the same challenge across multiple executions with the prover), present several constructions of PAoK relying on different cryptographic tools, and discuss applications to cryptography
How Long It Takes for an Ordinary Node with an Ordinary ID to Output?
In the context of distributed synchronous computing, processors perform in
rounds, and the time-complexity of a distributed algorithm is classically
defined as the number of rounds before all computing nodes have output. Hence,
this complexity measure captures the running time of the slowest node(s). In
this paper, we are interested in the running time of the ordinary nodes, to be
compared with the running time of the slowest nodes. The node-averaged
time-complexity of a distributed algorithm on a given instance is defined as
the average, taken over every node of the instance, of the number of rounds
before that node output. We compare the node-averaged time-complexity with the
classical one in the standard LOCAL model for distributed network computing. We
show that there can be an exponential gap between the node-averaged
time-complexity and the classical time-complexity, as witnessed by, e.g.,
leader election. Our first main result is a positive one, stating that, in
fact, the two time-complexities behave the same for a large class of problems
on very sparse graphs. In particular, we show that, for LCL problems on cycles,
the node-averaged time complexity is of the same order of magnitude as the
slowest node time-complexity.
In addition, in the LOCAL model, the time-complexity is computed as a worst
case over all possible identity assignments to the nodes of the network. In
this paper, we also investigate the ID-averaged time-complexity, when the
number of rounds is averaged over all possible identity assignments. Our second
main result is that the ID-averaged time-complexity is essentially the same as
the expected time-complexity of randomized algorithms (where the expectation is
taken over all possible random bits used by the nodes, and the number of rounds
is measured for the worst-case identity assignment).
Finally, we study the node-averaged ID-averaged time-complexity.Comment: (Submitted) Journal versio
Probing Solar Convection
In the solar convection zone acoustic waves are scattered by turbulent sound
speed fluctuations. In this paper the scattering of waves by convective cells
is treated using Rytov's technique. Particular care is taken to include
diffraction effects which are important especially for high-degree modes that
are confined to the surface layers of the Sun. The scattering leads to damping
of the waves and causes a phase shift. Damping manifests itself in the width of
the spectral peak of p-mode eigenfrequencies. The contribution of scattering to
the line widths is estimated and the sensitivity of the results on the assumed
spectrum of the turbulence is studied. Finally the theoretical predictions are
compared with recently measured line widths of high-degree modes.Comment: 26 pages, 7 figures, accepted by MNRA
Dynamics of Circumstellar Disks II: Heating and Cooling
We present a series of 2-d () hydrodynamic simulations of marginally
self gravitating disks around protostars using an SPH code. We implement simple
dynamical heating and we cool each location as a black body, using a
photosphere temperature obtained from the local vertical structure. We
synthesize SEDs from our simulations and compare them to fiducial SEDs derived
from observed systems. These simulations produce less distinct spiral structure
than isothermally evolved systems, especially in the inner third of the disk.
Pattern are similar further from the star but do not collapse into condensed
objects. The photosphere temperature is well fit to a power law in radius with
index , which is very steep. Far from the star, internal heating
( work and shocks) are not responsible for generating a large fraction of
the thermal energy contained in the disk matter. Gravitational torques
responsible for such shocks cannot transport mass and angular momentum
efficiently in the outer disk. Within 5--10 AU of the star, rapid break
up and reformation of spiral structure causes shocks, which provide sufficient
dissipation to power a larger fraction of the near IR energy output. The
spatial and size distribution of grains can have marked consequences on the
observed near IR SED and can lead to increased emission and variability on
year time scales. When grains are vaporized they do not reform
into a size distribution similar to that from which most opacity calculations
are based. With rapid grain reformation into the original size distribution,
the disk does not emit near infrared photons. With a plausible modification to
the opacity, it contributes much more.Comment: Accepted by ApJ, 60pg incl 24 figure
Distributed Testing of Excluded Subgraphs
We study property testing in the context of distributed computing, under the
classical CONGEST model. It is known that testing whether a graph is
triangle-free can be done in a constant number of rounds, where the constant
depends on how far the input graph is from being triangle-free. We show that,
for every connected 4-node graph H, testing whether a graph is H-free can be
done in a constant number of rounds too. The constant also depends on how far
the input graph is from being H-free, and the dependence is identical to the
one in the case of testing triangles. Hence, in particular, testing whether a
graph is K_4-free, and testing whether a graph is C_4-free can be done in a
constant number of rounds (where K_k denotes the k-node clique, and C_k denotes
the k-node cycle). On the other hand, we show that testing K_k-freeness and
C_k-freeness for k>4 appear to be much harder. Specifically, we investigate two
natural types of generic algorithms for testing H-freeness, called DFS tester
and BFS tester. The latter captures the previously known algorithm to test the
presence of triangles, while the former captures our generic algorithm to test
the presence of a 4-node graph pattern H. We prove that both DFS and BFS
testers fail to test K_k-freeness and C_k-freeness in a constant number of
rounds for k>4
Spatial Structure and Coherent Motion in Dense Planetary Rings Induced by Self-Gravitational Instability
We investigate the formation of spatial structure in dense, self-gravitating
particle systems such as Saturn's B-ring through local -body simulations to
clarify the intrinsic physics based on individual particle motion. In such a
system, Salo (1995) showed that the formation of spatial structure such as
wake-like structure and particle grouping (clump) arises spontaneously due to
gravitational instability and the radial velocity dispersion increases as the
formation of the wake structure. However, intrinsic physics of the phenomena
has not been clarified. We performed local -body simulations including
mutual gravitational forces between ring particles as well as direct
(inelastic) collisions with identical (up to ) particles. In the
wake structure particles no longer move randomly but coherently. We found that
particle motion was similar to Keplerian motion even in the wake structure and
that the coherent motion was produced since the particles in a clump had
similar eccentricity and longitude of perihelion. This coherent motion causes
the increase and oscillation in the radial velocity dispersion. The mean
velocity dispersion is rather larger in a more dissipative case with a smaller
restitution coefficient and/or a larger surface density since the coherence is
stronger in the more dissipative case. Our simulations showed that the
wavelength of the wake structure was approximately given by the longest
wavelength \hs{\lambda}{cr} = 4\pi^2 G\Sigma/\kappa^2 in the linear theory of
axisymmetric gravitational instability in a thin disk, where , , and
are the gravitational constant, surface density, and a epicyclic
frequency.Comment: Accepted by Earth, Planets, and Space. 39 pages, 20 figures.
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The Thermal Regulation of Gravitational Instabilities in Protoplanetary Disks II. Extended Simulations with Varied Cooling Rates
In order to investigate mass transport and planet formation by gravitational
instabilities (GIs), we have extended our 3-D hydrodynamic simulations of
protoplanetary disks from a previous paper. Our goal is to determine the
asymptotic behavior of GIs and how it is affected by different constant cooling
times. Initially, Rdisk = 40 AU, Mdisk = 0.07 Mo, M* = 0.5 Mo, and Qmin = 1.8.
Sustained cooling, with tcool = 2 orps (outer rotation periods, 1 orp ~ 250
yrs), drives the disk to instability in ~ 4 orps. This calculation is followed
for 23.5 orps. After 12 orps, the disk settles into a quasi-steady state with
sustained nonlinear instabilities, an average Q = 1.44 over the outer disk, a
well-defined power-law Sigma(r), and a roughly steady Mdot ~ 5(-7) Mo/yr. The
transport is driven by global low-order spiral modes. We restart the
calculation at 11.2 orps with tcool = 1 and 1/4 orp. The latter case is also
run at high azimuthal resolution. We find that shorter cooling times lead to
increased Mdots, denser and thinner spiral structures, and more violent dynamic
behavior. The asymptotic total internal energy and the azimuthally averaged
Q(r) are insensitive to tcool. Fragmentation occurs only in the high-resolution
tcool = 1/4 orp case; however, none of the fragments survive for even a quarter
of an orbit. Ring-like density enhancements appear and grow near the boundary
between GI active and inactive regions. We discuss the possible implications of
these rings for gas giant planet formation.Comment: Due to document size restrictions, the complete manuscript could not
be posted on astroph. Please go to http://westworld.astro.indiana.edu to
download the full document including figure
Disk Planet Interactions and Early Evolution in Young Planetary Systems
We study and review disk protoplanet interactions using local shearing box
simulations. These suffer the disadvantage of having potential artefacts
arising from periodic boundary conditions but the advantage, when compared to
global simulations, of being able to capture much of the dynamics close to the
protoplanet at high resolution for low computational cost. Cases with and
without self sustained MHD turbulence are considered. The conditions for gap
formation and the transition from type I migration are investigated and found
to depend on whether the single parameter M_p R^3/(M_* H^3), with M_p, M_*, R
and H being the protoplanet mass, the central mass, the orbital radius and the
disk semi-thickness respectively exceeds a number of order unity. We also
investigate the coorbital torques experienced by a moving protoplanet in an
inviscid disk. This is done by demonstrating the equivalence of the problem for
a moving protoplanet to one where the protoplanet is in a fixed orbit which the
disk material flows through radially as a result of the action of an
appropriate external torque. For sustainable coorbital torques to be realized a
quasi steady state must be realized in which the planet migrates through the
disk without accreting significant mass. In that case although there is
sensitivity to computational parameters, in agreement with earlier work by
Masset & Papaloizou (2003) based on global simulations, the coorbital torques
are proportional to the migration speed and result in a positive feedback on
the migration, enhancing it and potentially leading to a runaway. This could
lead to a fast migration for protoplanets in the Saturn mass range in massive
disks and may be relevant to the mass period correlation for extrasolar planets
which gives a preponderance of sub Jovian masses at short orbital period.Comment: To appear in Celestial Mechanics and Dynamical Astronomy (with higher
resolution figures
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