28,546 research outputs found
Solving Medium-Density Subset Sum Problems in Expected Polynomial Time: An Enumeration Approach
The subset sum problem (SSP) can be briefly stated as: given a target integer
and a set containing positive integer , find a subset of
summing to . The \textit{density} of an SSP instance is defined by the
ratio of to , where is the logarithm of the largest integer within
. Based on the structural and statistical properties of subset sums, we
present an improved enumeration scheme for SSP, and implement it as a complete
and exact algorithm (EnumPlus). The algorithm always equivalently reduces an
instance to be low-density, and then solve it by enumeration. Through this
approach, we show the possibility to design a sole algorithm that can
efficiently solve arbitrary density instance in a uniform way. Furthermore, our
algorithm has considerable performance advantage over previous algorithms.
Firstly, it extends the density scope, in which SSP can be solved in expected
polynomial time. Specifically, It solves SSP in expected time
when density , while the previously best
density scope is . In addition, the overall
expected time and space requirement in the average case are proven to be
and respectively. Secondly, in the worst case, it
slightly improves the previously best time complexity of exact algorithms for
SSP. Specifically, the worst-case time complexity of our algorithm is proved to
be , while the previously best result is .Comment: 11 pages, 1 figur
Complexity transitions in global algorithms for sparse linear systems over finite fields
We study the computational complexity of a very basic problem, namely that of
finding solutions to a very large set of random linear equations in a finite
Galois Field modulo q. Using tools from statistical mechanics we are able to
identify phase transitions in the structure of the solution space and to
connect them to changes in performance of a global algorithm, namely Gaussian
elimination. Crossing phase boundaries produces a dramatic increase in memory
and CPU requirements necessary to the algorithms. In turn, this causes the
saturation of the upper bounds for the running time. We illustrate the results
on the specific problem of integer factorization, which is of central interest
for deciphering messages encrypted with the RSA cryptosystem.Comment: 23 pages, 8 figure
Goal-based h-adaptivity of the 1-D diamond difference discrete ordinate method.
The quantity of interest (QoI) associated with a solution of a partial differential equation (PDE) is not, in general, the solution itself, but a functional of the solution. Dual weighted residual (DWR) error estimators are one way of providing an estimate of the error in the QoI resulting from the discretisation of the PDE. This paper aims to provide an estimate of the error in the QoI due to the spatial discretisation, where the discretisation scheme being used is the diamond difference (DD) method in space and discrete ordinate (SNSN) method in angle. The QoI are reaction rates in detectors and the value of the eigenvalue (Keff)(Keff) for 1-D fixed source and eigenvalue (KeffKeff criticality) neutron transport problems respectively. Local values of the DWR over individual cells are used as error indicators for goal-based mesh refinement, which aims to give an optimal mesh for a given QoI
Spatial Coordination Strategies in Future Ultra-Dense Wireless Networks
Ultra network densification is considered a major trend in the evolution of
cellular networks, due to its ability to bring the network closer to the user
side and reuse resources to the maximum extent. In this paper we explore
spatial resources coordination as a key empowering technology for next
generation (5G) ultra-dense networks. We propose an optimization framework for
flexibly associating system users with a densely deployed network of access
nodes, opting for the exploitation of densification and the control of overhead
signaling. Combined with spatial precoding processing strategies, we design
network resources management strategies reflecting various features, namely
local vs global channel state information knowledge exploitation, centralized
vs distributed implementation, and non-cooperative vs joint multi-node data
processing. We apply these strategies to future UDN setups, and explore the
impact of critical network parameters, that is, the densification levels of
users and access nodes as well as the power budget constraints, to users
performance. We demonstrate that spatial resources coordination is a key factor
for capitalizing on the gains of ultra dense network deployments.Comment: An extended version of a paper submitted to ISWCS'14, Special Session
on Empowering Technologies of 5G Wireless Communication
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