32,116 research outputs found
Distributed Connectivity Decomposition
We present time-efficient distributed algorithms for decomposing graphs with
large edge or vertex connectivity into multiple spanning or dominating trees,
respectively. As their primary applications, these decompositions allow us to
achieve information flow with size close to the connectivity by parallelizing
it along the trees. More specifically, our distributed decomposition algorithms
are as follows:
(I) A decomposition of each undirected graph with vertex-connectivity
into (fractionally) vertex-disjoint weighted dominating trees with total weight
, in rounds.
(II) A decomposition of each undirected graph with edge-connectivity
into (fractionally) edge-disjoint weighted spanning trees with total
weight , in
rounds.
We also show round complexity lower bounds of
and
for the above two decompositions,
using techniques of [Das Sarma et al., STOC'11]. Moreover, our
vertex-connectivity decomposition extends to centralized algorithms and
improves the time complexity of [Censor-Hillel et al., SODA'14] from
to near-optimal .
As corollaries, we also get distributed oblivious routing broadcast with
-competitive edge-congestion and -competitive
vertex-congestion. Furthermore, the vertex connectivity decomposition leads to
near-time-optimal -approximation of vertex connectivity: centralized
and distributed . The former moves
toward the 1974 conjecture of Aho, Hopcroft, and Ullman postulating an
centralized exact algorithm while the latter is the first distributed vertex
connectivity approximation
Ultrametric pseudodifferential operators and wavelets for the case of non homogeneous measure
A family of orthonormal bases of ultrametric wavelets in the space of
quadratically integrable with respect to arbitrary measure functions on general
(up to some topological restrictions) ultrametric space is introduced.
Pseudodifferential operators (PDO) on the ultrametric space are investigated.
We prove that these operators are diagonal in the introduced bases of
ultrametric wavelets and compute the corresponding eigenvalues.
Duality between ultrametric spaces and directed trees is discussed. In
particular, a new way of construction of ultrametric spaces by completion of
directed trees is proposed.Comment: 16 pages, LaTeX, some corrections and generalization
Computational convergence of the path integral for real dendritic morphologies
Neurons are characterised by a morphological structure unique amongst biological cells, the core of which is the dendritic tree. The vast number of dendritic geometries, combined with heterogeneous properties of the cell membrane, continue to challenge scientists in predicting neuronal input-output relationships, even in the case of sub-threshold dendritic currents. The Greenâs function obtained for a given dendritic geometry provides this functional relationship for passive or quasi-active dendrites and can be constructed by a sum-over-trips approach based on a path integral formalism. In this paper, we introduce a number of efficient algorithms for realisation of the sum-over-trips framework and investigate the convergence of these algorithms on different dendritic geometries. We demonstrate that the convergence of the trip sampling methods strongly depends on dendritic morphology as well as the biophysical properties of the cell membrane. For real morphologies, the number of trips to guarantee a small convergence error might become very large and strongly affect computational efficiency. As an alternative, we introduce a highly-efficient matrix method which can be applied to arbitrary branching structures
A diffusion model of scheduling control in queueing systems with many servers
This paper studies a diffusion model that arises as the limit of a queueing
system scheduling problem in the asymptotic heavy traffic regime of Halfin and
Whitt. The queueing system consists of several customer classes and many
servers working in parallel, grouped in several stations. Servers in different
stations offer service to customers of each class at possibly different rates.
The control corresponds to selecting what customer class each server serves at
each time. The diffusion control problem does not seem to have explicit
solutions and therefore a characterization of optimal solutions via the
Hamilton-Jacobi-Bellman equation is addressed. Our main result is the existence
and uniqueness of solutions of the equation. Since the model is set on an
unbounded domain and the cost per unit time is unbounded, the analysis requires
estimates on the state process that are subexponential in the time variable. In
establishing these estimates, a key role is played by an integral formula that
relates queue length and idle time processes, which may be of independent
interest.Comment: Published at http://dx.doi.org/10.1214/105051604000000963 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Validation and refinement of allometric equations for roots of northern hardwoods
The allometric equations developed by Whittaker et al. (1974. Ecol. Monogr. 44: 233â252), at the Hubbard Brook Experimental Forest have been used to estimate biomass and productivity in northern hardwood forest systems for over three decades. Few other species-specific allometric estimates of belowground biomass are available because of the difficulty in collecting the data, and such equations are rarely validated. Using previously unpublished data from Whittakerâs sampling effort, we extended the equations to predict the root crown and lateral root components for the three dominant species of the northern hardwood forest: American beech (Fagus grandifolia Ehrh.), yellow birch (Betula alleghaniensis Britt), and sugar maple (Acer saccharum Marsh.). We also refined the allometric models by eliminating the use of very small trees for which the original data were unreliable. We validated these new models of the relationship of tree diameter to the mass of root crowns and lateral roots using root mass data collected from 12 northern hardwood stands of varying age in central New Hampshire. These models provide accurate estimates of lateral roots (diameter) in northern hardwood stands \u3e20 years old (mean error 24%â32%). For the younger stands that we studied, allometric equations substantially underestimated observed root biomass (mean error \u3e60%), presumably due to remnant mature root systems from harvested trees supporting young root-sprouted trees
Random graphs from a block-stable class
A class of graphs is called block-stable when a graph is in the class if and
only if each of its blocks is. We show that, as for trees, for most -vertex
graphs in such a class, each vertex is in at most blocks, and each path passes through at most blocks.
These results extend to `weakly block-stable' classes of graphs
Effects of Lightning on Trees: A Predictive Model Based on in situ Electrical Resistivity
The effects of lightning on trees range from catastrophic death to the absence of observable damage. Such differences may be predictable among tree species, and more generally among plant life history strategies and growth forms. We used fieldâcollected electrical resistivity data in temperate and tropical forests to model how the distribution of power from a lightning discharge varies with tree size and identity, and with the presence of lianas. Estimated heating density (heat generated per volume of tree tissue) and maximum power (maximum rate of heating) from a standardized lightning discharge differed 300% among tree species. Tree size and morphology also were important; the heating density of a hypothetical 10 m tall Alseis blackiana was 49 times greater than for a 30 m tall conspecific, and 127 times greater than for a 30 m tall Dipteryx panamensis. Lianas may protect trees from lightning by conducting electric current; estimated heating and maximum power were reduced by 60% (±7.1%) for trees with one liana and by 87% (±4.0%) for trees with three lianas. This study provides the first quantitative mechanism describing how differences among trees can influence lightningâtree interactions, and how lianas can serve as natural lightning rods for trees
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