16,323 research outputs found
Improved Approximation Algorithms for Stochastic Matching
In this paper we consider the Stochastic Matching problem, which is motivated
by applications in kidney exchange and online dating. We are given an
undirected graph in which every edge is assigned a probability of existence and
a positive profit, and each node is assigned a positive integer called timeout.
We know whether an edge exists or not only after probing it. On this random
graph we are executing a process, which one-by-one probes the edges and
gradually constructs a matching. The process is constrained in two ways: once
an edge is taken it cannot be removed from the matching, and the timeout of
node upper-bounds the number of edges incident to that can be probed.
The goal is to maximize the expected profit of the constructed matching.
For this problem Bansal et al. (Algorithmica 2012) provided a
-approximation algorithm for bipartite graphs, and a -approximation for
general graphs. In this work we improve the approximation factors to
and , respectively.
We also consider an online version of the bipartite case, where one side of
the partition arrives node by node, and each time a node arrives we have to
decide which edges incident to we want to probe, and in which order. Here
we present a -approximation, improving on the -approximation of
Bansal et al.
The main technical ingredient in our result is a novel way of probing edges
according to a random but non-uniform permutation. Patching this method with an
algorithm that works best for large probability edges (plus some additional
ideas) leads to our improved approximation factors
DDSL: Efficient Subgraph Listing on Distributed and Dynamic Graphs
Subgraph listing is a fundamental problem in graph theory and has wide
applications in areas like sociology, chemistry, and social networks. Modern
graphs can usually be large-scale as well as highly dynamic, which challenges
the efficiency of existing subgraph listing algorithms. Recent works have shown
the benefits of partitioning and processing big graphs in a distributed system,
however, there is only few work targets subgraph listing on dynamic graphs in a
distributed environment. In this paper, we propose an efficient approach,
called Distributed and Dynamic Subgraph Listing (DDSL), which can incrementally
update the results instead of running from scratch. DDSL follows a general
distributed join framework. In this framework, we use a Neighbor-Preserved
storage for data graphs, which takes bounded extra space and supports dynamic
updating. After that, we propose a comprehensive cost model to estimate the I/O
cost of listing subgraphs. Then based on this cost model, we develop an
algorithm to find the optimal join tree for a given pattern. To handle dynamic
graphs, we propose an efficient left-deep join algorithm to incrementally
update the join results. Extensive experiments are conducted on real-world
datasets. The results show that DDSL outperforms existing methods in dealing
with both static dynamic graphs in terms of the responding time
Understanding Search Trees via Statistical Physics
We study the random m-ary search tree model (where m stands for the number of
branches of a search tree), an important problem for data storage in computer
science, using a variety of statistical physics techniques that allow us to
obtain exact asymptotic results. In particular, we show that the probability
distributions of extreme observables associated with a random search tree such
as the height and the balanced height of a tree have a traveling front
structure. In addition, the variance of the number of nodes needed to store a
data string of a given size N is shown to undergo a striking phase transition
at a critical value of the branching ratio m_c=26. We identify the mechanism of
this phase transition, show that it is generic and occurs in various other
problems as well. New results are obtained when each element of the data string
is a D-dimensional vector. We show that this problem also has a phase
transition at a critical dimension, D_c= \pi/\sin^{-1}(1/\sqrt{8})=8.69363...Comment: 11 pages, 8 .eps figures included. Invited contribution to
STATPHYS-22 held at Bangalore (India) in July 2004. To appear in the
proceedings of STATPHYS-2
Large Deviations of Extreme Eigenvalues of Random Matrices
We calculate analytically the probability of large deviations from its mean
of the largest (smallest) eigenvalue of random matrices belonging to the
Gaussian orthogonal, unitary and symplectic ensembles. In particular, we show
that the probability that all the eigenvalues of an (N\times N) random matrix
are positive (negative) decreases for large N as \exp[-\beta \theta(0) N^2]
where the parameter \beta characterizes the ensemble and the exponent
\theta(0)=(\ln 3)/4=0.274653... is universal. We also calculate exactly the
average density of states in matrices whose eigenvalues are restricted to be
larger than a fixed number \zeta, thus generalizing the celebrated Wigner
semi-circle law. The density of states generically exhibits an inverse
square-root singularity at \zeta.Comment: 4 pages Revtex, 4 .eps figures included, typos corrected, published
versio
On the distribution of estimators of diffusion constants for Brownian motion
We discuss the distribution of various estimators for extracting the
diffusion constant of single Brownian trajectories obtained by fitting the
squared displacement of the trajectory. The analysis of the problem can be
framed in terms of quadratic functionals of Brownian motion that correspond to
the Euclidean path integral for simple Harmonic oscillators with time dependent
frequencies. Explicit analytical results are given for the distribution of the
diffusion constant estimator in a number of cases and our results are confirmed
by numerical simulations.Comment: 14 pages, 5 figure
Extreme Value Statistics of Eigenvalues of Gaussian Random Matrices
We compute exact asymptotic results for the probability of the occurrence of
large deviations of the largest (smallest) eigenvalue of random matrices
belonging to the Gaussian orthogonal, unitary and symplectic ensembles. In
particular, we show that the probability that all the eigenvalues of an (NxN)
random matrix are positive (negative) decreases for large N as ~\exp[-\beta
\theta(0) N^2] where the Dyson index \beta characterizes the ensemble and the
exponent \theta(0)=(\ln 3)/4=0.274653... is universal. We compute the
probability that the eigenvalues lie in the interval [\zeta_1,\zeta_2] which
allows us to calculate the joint probability distribution of the minimum and
the maximum eigenvalue. As a byproduct, we also obtain exactly the average
density of states in Gaussian ensembles whose eigenvalues are restricted to lie
in the interval [\zeta_1,\zeta_2], thus generalizing the celebrated Wigner
semi-circle law to these restricted ensembles. It is found that the density of
states generically exhibits an inverse square-root singularity at the location
of the barriers. These results are confirmed by numerical simulations.Comment: 17 pages Revtex, 5 .eps figures include
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The effect of host structure on the selectivity and mechanism of supramolecular catalysis of Prins cyclizations.
The effect of host structure on the selectivity and mechanism of intramolecular Prins reactions is evaluated using K12Ga4L6 tetrahedral catalysts. The host structure was varied by modifying the structure of the chelating moieties and the size of the aromatic spacers. While variation in chelator substituents was generally observed to affect changes in rate but not selectivity, changing the host spacer afforded differences in efficiency and product diastereoselectivity. An extremely high number of turnovers (up to 840) was observed. Maximum rate accelerations were measured to be on the order of 105, which numbers among the largest magnitudes of transition state stabilization measured with a synthetic host-catalyst. Host/guest size effects were observed to play an important role in host-mediated enantioselectivity
Wind and boundary layers in Rayleigh-Benard convection. Part 2: boundary layer character and scaling
The effect of the wind of Rayleigh-Benard convection on the boundary layers
is studied by direct numerical simulation of an L/H=4 aspect-ratio domain with
periodic side boundary conditions for Ra={10^5, 10^6, 10^7} and Pr=1. It is
shown that the kinetic boundary layers on the top- and bottom plate have some
features of both laminar and turbulent boundary layers. A continuous spectrum,
as well as significant forcing due to Reynolds stresses indicates undoubtedly a
turbulent character, whereas the classical integral boundary layer parameters
-- the shape factor and friction factor (the latter is shown to be dominated by
the pressure gradient) -- scale with Reynolds number more akin to laminar
boundary layers. This apparent dual behavior is caused by the large influence
of plumes impinging onto and detaching from the boundary layer. The
plume-generated Reynolds stresses have a negligible effect on the friction
factor at the Rayleigh numbers we consider, which indicates that they are
passive with respect to momentum transfer in the wall-parallel direction.
However, the effect of Reynolds stresses cannot be neglected for the thickness
of the kinetic boundary layer. Using a conceptual wind model, we find that the
friction factor C_f should scale proportional to the thermal boundary layer
thickness as C_f ~ lambda_Theta, while the kinetic boundary layer thickness
lambda_u scales inversely proportional to the thermal boundary layer thickness
and wind Reynolds number lambda_u ~ lambda_Theta^{-1} Re^{-1}. The predicted
trends for C_f and \lambda_u are in agreement with DNS results
The statistical mechanics of combinatorial optimization problems with site disorder
We study the statistical mechanics of a class of problems whose phase space
is the set of permutations of an ensemble of quenched random positions.
Specific examples analyzed are the finite temperature traveling salesman
problem on several different domains and various problems in one dimension such
as the so called descent problem. We first motivate our method by analyzing
these problems using the annealed approximation, then the limit of a large
number of points we develop a formalism to carry out the quenched calculation.
This formalism does not require the replica method and its predictions are
found to agree with Monte Carlo simulations. In addition our method reproduces
an exact mathematical result for the Maximum traveling salesman problem in two
dimensions and suggests its generalization to higher dimensions. The general
approach may provide an alternative method to study certain systems with
quenched disorder.Comment: 21 pages RevTex, 8 figure
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