6 research outputs found
Computing Bounds on Network Capacity Regions as a Polytope Reconstruction Problem
We define a notion of network capacity region of networks that generalizes
the notion of network capacity defined by Cannons et al. and prove its notable
properties such as closedness, boundedness and convexity when the finite field
is fixed. We show that the network routing capacity region is a computable
rational polytope and provide exact algorithms and approximation heuristics for
computing the region. We define the semi-network linear coding capacity region,
with respect to a fixed finite field, that inner bounds the corresponding
network linear coding capacity region, show that it is a computable rational
polytope, and provide exact algorithms and approximation heuristics. We show
connections between computing these regions and a polytope reconstruction
problem and some combinatorial optimization problems, such as the minimum cost
directed Steiner tree problem. We provide an example to illustrate our results.
The algorithms are not necessarily polynomial-time.Comment: Appeared in the 2011 IEEE International Symposium on Information
Theory, 5 pages, 1 figur
On Network Coding Capacity - Matroidal Networks and Network Capacity Regions
One fundamental problem in the field of network coding is to determine the
network coding capacity of networks under various network coding schemes. In
this thesis, we address the problem with two approaches: matroidal networks and
capacity regions.
In our matroidal approach, we prove the converse of the theorem which states
that, if a network is scalar-linearly solvable then it is a matroidal network
associated with a representable matroid over a finite field. As a consequence,
we obtain a correspondence between scalar-linearly solvable networks and
representable matroids over finite fields in the framework of matroidal
networks. We prove a theorem about the scalar-linear solvability of networks
and field characteristics. We provide a method for generating scalar-linearly
solvable networks that are potentially different from the networks that we
already know are scalar-linearly solvable.
In our capacity region approach, we define a multi-dimensional object, called
the network capacity region, associated with networks that is analogous to the
rate regions in information theory. For the network routing capacity region, we
show that the region is a computable rational polytope and provide exact
algorithms and approximation heuristics for computing the region. For the
network linear coding capacity region, we construct a computable rational
polytope, with respect to a given finite field, that inner bounds the linear
coding capacity region and provide exact algorithms and approximation
heuristics for computing the polytope. The exact algorithms and approximation
heuristics we present are not polynomial time schemes and may depend on the
output size.Comment: Master of Engineering Thesis, MIT, September 2010, 70 pages, 10
figure
On network coding capacity : matroidal networks and network capacity regions
Thesis (M. Eng.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2010.Cataloged from PDF version of thesis.Includes bibliographical references (p. 68-70).One fundamental problem in the field of network coding is to determine the network coding capacity of networks under various network coding schemes. In this thesis, we address the problem with two approaches: matroidal networks and capacity regions. In our matroidal approach, we prove the converse of the theorem which states that, if a network is scalar-linearly solvable then it is a matroidal network associated with a representable matroid over a finite field. As a consequence, we obtain a correspondence between scalar-linearly solvable networks and representable matroids over finite fields in the framework of matroidal networks. We prove a theorem about the scalar-linear solvability of networks and field characteristics. We provide a method for generating scalar-linearly solvable networks that are potentially different from the networks that we already know are scalar-linearly solvable. In our capacity region approach, we define a multi-dimensional object, called the network capacity region, associated with networks that is analogous to the rate regions in information theory. For the network routing capacity region, we show that the region is a computable rational polytope and provide exact algorithms and approximation heuristics for computing the region. For the network linear coding capacity region, we construct a computable rational polytope, with respect to a given finite field, that inner bounds the linear coding capacity region and provide exact algorithms and approximation heuristics for computing the polytope. The exact algorithms and approximation heuristics we present are not polynomial time schemes and may depend on the output size.by Anthony Eli Kim.M.Eng
Polytope containment and determination by linear probes
The paper discusses two main problems, the first of which - the containment problem - arose from a question in abstract numerical analysis. Here the goal is to construct a polytope P, that contains C, where this requires precise specification of P's vertices. In the other main problem - the reconstruction problem - it is known only that C is itself a polytope and the problem is to construct C with the aid of a finite number of calls to the ray-oracle. Thatt is accomplished with a number of calls that depends on the combinatoral complexity of CAvailable from TIB Hannover: RO 7722(410) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman
Polytope containment and determination by linear probes
As the terms are used here, a body in R el is a compact convex set with non-empty interior, and a polytope is a body that has only finitely many extreme points. The class of all bodies whose interior includes the origin 0 is denoted by %%. A set X is symmetric if X =-X. The ray-oracle of a body C e "#( { is the function 0c which, accepting as input an arbitrary ray R issuing from 0, produces the point at which R intersects the boundary of C. This paper is concerned with a few central aspects of the following general question: given certain information about C, what additional information can be obtained by questioning the ray-oracle, and how efficiently can it be obtained? It is assumed that infinite-precision real arithmetic and the usual vector operations in U d are available at no cost, so the efficiency of an algorithm is measured solely in terms of its number of calls to the ray-oracle. The paper discusses two main problems, the first of which—the containment problem—arose from a question in abstract numerical analysis. Here the goal is to construct a polytope P (not necessarily in any sense a small one) that contains C, where this requires precise specification of the vertices of P. There are some sharp positive results for the case in which d = 2 and C is known not to be too asymmetric, but the main result on the containment problem is negative. It asserts that when d 2 s 3 and the body is known only to be rotund and symmetric, there is no algorithm for the containment problem. This is the case even when there is available a certain master oracle whose questionanswering power far exceeds that of the ray-oracle. However, it turns out that even when there is no additional information about C, the following relaxation of the containment problem admits an algorithmic solution based solely on the ray-oracle: construct a polytope containing C or conclude that the centred condition number of C exceeds a prescribed bound. In the other main problem—the reconstruction problem — it is known only that C is itself a polytope and the problem is to construct C with the aid of a finite number of calls to the ray-oracle. That is accomplished with a number of calls that depends on the number of faces (and hence on the 'combinatorial complexity') of C