3,142 research outputs found
The Gilbert Arborescence Problem
We investigate the problem of designing a minimum cost flow network
interconnecting n sources and a single sink, each with known locations in a
normed space and with associated flow demands. The network may contain any
finite number of additional unprescribed nodes from the space; these are known
as the Steiner points. For concave increasing cost functions, a minimum cost
network of this sort has a tree topology, and hence can be called a Minimum
Gilbert Arborescence (MGA). We characterise the local topological structure of
Steiner points in MGAs, showing, in particular, that for a wide range of
metrics, and for some typical real-world cost-functions, the degree of each
Steiner point is 3.Comment: 19 pages, 7 figures. arXiv admin note: text overlap with
arXiv:0903.212
Maximizing Maximal Angles for Plane Straight-Line Graphs
Let be a plane straight-line graph on a finite point set
in general position. The incident angles of a vertex
of are the angles between any two edges of that appear consecutively in
the circular order of the edges incident to .
A plane straight-line graph is called -open if each vertex has an
incident angle of size at least . In this paper we study the following
type of question: What is the maximum angle such that for any finite set
of points in general position we can find a graph from a certain
class of graphs on that is -open? In particular, we consider the
classes of triangulations, spanning trees, and paths on and give tight
bounds in most cases.Comment: 15 pages, 14 figures. Apart of minor corrections, some proofs that
were omitted in the previous version are now include
Scalable Parameterised Algorithms for two Steiner Problems
In the Steiner Problem, we are given as input (i) a connected graph with nonnegative integer weights associated with the edges; and (ii) a subset of vertices called terminals. The task is to find a minimum-weight subgraph connecting all the terminals. In the Group Steiner Problem, we are given as input (i) a connected graph with nonnegative integer weights associated with the edges; and (ii) a collection of subsets of vertices called groups. The task is to find a minimum-weight subgraph that contains at least one vertex from each group. Even though the Steiner Problem and the Group Steiner Problem are NP-complete, they are known to admit parameterised algorithms that run in linear time in the size of the input graph and the exponential part can be restricted to the number of terminals and the number of groups, respectively.
In this thesis, we discuss two parameterised algorithms for solving the Steiner Problem, and by reduction, the Group Steiner Problem: (a) a dynamic programming algorithm presented by Dreyfus and Wagner in 1971; and (b) an improvement of the Dreyfus-Wagner algorithm presented by Erickson, Monma and Veinott in 1987 that runs in linear time in the size of the input graph. We develop a parallel implementation of the Erickson-Monma-Veinott algorithm, and carry out extensive experiments to study the scalability of our implementation with respect to its runtime, memory bandwidth, and memory usage. Our experimental results demonstrate that the implementation can scale up to a billion edges on a single modern compute node provided that the number of terminals is small. For example, using our parallel implementation a Steiner tree for a graph with hundred million edges and ten terminals can be found in approximately twenty minutes. For an input graph with one hundred million edges and ten terminals, our parallel implementation is at least fifteen times faster than its serial counterpart on a Haswell compute node with two processors and twelve cores in each processor. Our implementation of the Erickson-Monma-Veinott algorithm is available as open source
Networking - A Statistical Physics Perspective
Efficient networking has a substantial economic and societal impact in a
broad range of areas including transportation systems, wired and wireless
communications and a range of Internet applications. As transportation and
communication networks become increasingly more complex, the ever increasing
demand for congestion control, higher traffic capacity, quality of service,
robustness and reduced energy consumption require new tools and methods to meet
these conflicting requirements. The new methodology should serve for gaining
better understanding of the properties of networking systems at the macroscopic
level, as well as for the development of new principled optimization and
management algorithms at the microscopic level. Methods of statistical physics
seem best placed to provide new approaches as they have been developed
specifically to deal with non-linear large scale systems. This paper aims at
presenting an overview of tools and methods that have been developed within the
statistical physics community and that can be readily applied to address the
emerging problems in networking. These include diffusion processes, methods
from disordered systems and polymer physics, probabilistic inference, which
have direct relevance to network routing, file and frequency distribution, the
exploration of network structures and vulnerability, and various other
practical networking applications.Comment: (Review article) 71 pages, 14 figure
The rectilinear Steiner tree problem with given topology and length restrictions
We consider the problem of embedding the Steiner points of a Steiner tree
with given topology into the rectilinear plane. Thereby, the length of the path
between a distinguished terminal and each other terminal must not exceed given
length restrictions. We want to minimize the total length of the tree.
The problem can be formulated as a linear program and therefore it is
solvable in polynomial time. In this paper we analyze the structure of feasible
embeddings and give a combinatorial polynomial time algorithm for the problem.
Our algorithm combines a dynamic programming approach and binary search and
relies on the total unimodularity of a matrix appearing in a sub-problem.Comment: 14 page
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