1,222 research outputs found
Polylogarithmic Approximation for Generalized Minimum Manhattan Networks
Given a set of terminals, which are points in -dimensional Euclidean
space, the minimum Manhattan network problem (MMN) asks for a minimum-length
rectilinear network that connects each pair of terminals by a Manhattan path,
that is, a path consisting of axis-parallel segments whose total length equals
the pair's Manhattan distance. Even for , the problem is NP-hard, but
constant-factor approximations are known. For , the problem is
APX-hard; it is known to admit, for any \eps > 0, an
O(n^\eps)-approximation.
In the generalized minimum Manhattan network problem (GMMN), we are given a
set of terminal pairs, and the goal is to find a minimum-length
rectilinear network such that each pair in is connected by a Manhattan
path. GMMN is a generalization of both MMN and the well-known rectilinear
Steiner arborescence problem (RSA). So far, only special cases of GMMN have
been considered.
We present an -approximation algorithm for GMMN (and, hence,
MMN) in dimensions and an -approximation algorithm for 2D.
We show that an existing -approximation algorithm for RSA in 2D
generalizes easily to dimensions.Comment: 14 pages, 5 figures; added appendix and figure
Two-Level Rectilinear Steiner Trees
Given a set of terminals in the plane and a partition of into
subsets , a two-level rectilinear Steiner tree consists of a
rectilinear Steiner tree connecting the terminals in each set
() and a top-level tree connecting the trees . The goal is to minimize the total length of all trees. This problem
arises naturally in the design of low-power physical implementations of parity
functions on a computer chip.
For bounded we present a polynomial time approximation scheme (PTAS) that
is based on Arora's PTAS for rectilinear Steiner trees after lifting each
partition into an extra dimension. For the general case we propose an algorithm
that predetermines a connection point for each and
().
Then, we apply any approximation algorithm for minimum rectilinear Steiner
trees in the plane to compute each and independently.
This gives us a -factor approximation with a running time of
suitable for fast practical computations. The
approximation factor reduces to by applying Arora's approximation scheme
in the plane
FPC: A New Approach to Firewall Policies Compression
Firewalls are crucial elements that enhance network security by examining the field values of every packet and deciding whether to accept or discard a packet according to the firewall policies. With the development of networks, the number of rules in firewalls has rapidly increased, consequently degrading network performance. In addition, because most real-life firewalls have been plagued with policy conflicts, malicious traffics can be allowed or legitimate traffics can be blocked. Moreover, because of the complexity of the firewall policies, it is very important to reduce the number of rules in a firewall while keeping the rule semantics unchanged and the target firewall rules conflict-free. In this study, we make three major contributions. First, we present a new approach in which a geometric model, multidimensional rectilinear polygon, is constructed for the firewall rules compression problem. Second, we propose a new scheme, Firewall Policies Compression (FPC), to compress the multidimensional firewall rules based on this geometric model. Third, we conducted extensive experiments to evaluate the performance of the proposed method. The experimental results demonstrate that the FPC method outperforms the existing approaches, in terms of compression ratio and efficiency while maintaining conflict-free firewall rules
FPC: A New Approach to Firewall Policies Compression
Firewalls are crucial elements that enhance network security by examining the field values of every packet and deciding whether to accept or discard a packet according to the firewall policies. With the development of networks, the number of rules in firewalls has rapidly increased, consequently degrading network performance. In addition, because most real-life firewalls have been plagued with policy conflicts, malicious traffics can be allowed or legitimate traffics can be blocked. Moreover, because of the complexity of the firewall policies, it is very important to reduce the number of rules in a firewall while keeping the rule semantics unchanged and the target firewall rules conflict-free. In this study, we make three major contributions. First, we present a new approach in which a geometric model, multidimensional rectilinear polygon, is constructed for the firewall rules compression problem. Second, we propose a new scheme, Firewall Policies Compression (FPC), to compress the multidimensional firewall rules based on this geometric model. Third, we conducted extensive experiments to evaluate the performance of the proposed method. The experimental results demonstrate that the FPC method outperforms the existing approaches, in terms of compression ratio and efficiency while maintaining conflict-free firewall rules
The State of the Art in Cartograms
Cartograms combine statistical and geographical information in thematic maps,
where areas of geographical regions (e.g., countries, states) are scaled in
proportion to some statistic (e.g., population, income). Cartograms make it
possible to gain insight into patterns and trends in the world around us and
have been very popular visualizations for geo-referenced data for over a
century. This work surveys cartogram research in visualization, cartography and
geometry, covering a broad spectrum of different cartogram types: from the
traditional rectangular and table cartograms, to Dorling and diffusion
cartograms. A particular focus is the study of the major cartogram dimensions:
statistical accuracy, geographical accuracy, and topological accuracy. We
review the history of cartograms, describe the algorithms for generating them,
and consider task taxonomies. We also review quantitative and qualitative
evaluations, and we use these to arrive at design guidelines and research
challenges
Structural Properties of the Caenorhabditis elegans Neuronal Network
Despite recent interest in reconstructing neuronal networks, complete wiring
diagrams on the level of individual synapses remain scarce and the insights
into function they can provide remain unclear. Even for Caenorhabditis elegans,
whose neuronal network is relatively small and stereotypical from animal to
animal, published wiring diagrams are neither accurate nor complete and
self-consistent. Using materials from White et al. and new electron micrographs
we assemble whole, self-consistent gap junction and chemical synapse networks
of hermaphrodite C. elegans. We propose a method to visualize the wiring
diagram, which reflects network signal flow. We calculate statistical and
topological properties of the network, such as degree distributions, synaptic
multiplicities, and small-world properties, that help in understanding network
signal propagation. We identify neurons that may play central roles in
information processing and network motifs that could serve as functional
modules of the network. We explore propagation of neuronal activity in response
to sensory or artificial stimulation using linear systems theory and find
several activity patterns that could serve as substrates of previously
described behaviors. Finally, we analyze the interaction between the gap
junction and the chemical synapse networks. Since several statistical
properties of the C. elegans network, such as multiplicity and motif
distributions are similar to those found in mammalian neocortex, they likely
point to general principles of neuronal networks. The wiring diagram reported
here can help in understanding the mechanistic basis of behavior by generating
predictions about future experiments involving genetic perturbations, laser
ablations, or monitoring propagation of neuronal activity in response to
stimulation
Rectilinear crossing number of the double circular complete bipartite graph
In this work, we study a mathematically rigorous metric of a graph
visualization quality under conditions that relate to visualizing a bipartite
graph. Namely we study rectilinear crossing number in a special arrangement of
the complete bipartite graph where the two parts are placed on two concentric
circles. For this purpose, we introduce a combinatorial formulation to count
the number of crossings. We prove a proposition about the rectilinear crossing
number of the complete bipartite graph. Then, we introduce a geometric
optimization problem whose solution gives the optimum radii ratio in the case
that the number of crossings for them is minimized. Later on, we study the
magnitude of change in the number of crossings upon change in the radii of the
circles. In this part, we present and prove a lemma on bounding the changes in
the number of crossings of that is followed by a theorem on asymptotics of the
bounds.Comment: 15 pages, 11 figure
- …