13,452 research outputs found
An information-theoretic view of network management
We present an information-theoretic framework for network management for recovery from nonergodic link failures. Building on recent work in the field of network coding, we describe the input-output relations of network nodes in terms of network codes. This very general concept of network behavior as a code provides a way to quantify essential management information as that needed to switch among different codes (behaviors) for different failure scenarios. We compare two types of recovery schemes, receiver-based and network-wide, and consider two formulations for quantifying network management. The first is a centralized formulation where network behavior is described by an overall code determining the behavior of every node, and the management requirement is taken as the logarithm of the number of such codes that the network may switch among. For this formulation, we give bounds, many of which are tight, on management requirements for various network connection problems in terms of basic parameters such as the number of source processes and the number of links in a minimum source-receiver cut. Our results include a lower bound for arbitrary connections and an upper bound for multitransmitter multicast connections, for linear receiver-based and network-wide recovery from all single link failures. The second is a node-based formulation where the management requirement is taken as the sum over all nodes of the logarithm of the number of different behaviors for each node. We show that the minimum node-based requirement for failures of links adjacent to a single receiver is achieved with receiver-based schemes
Quadrisecants give new lower bounds for the ropelength of a knot
Using the existence of a special quadrisecant line, we show the ropelength of
any nontrivial knot is at least 15.66. This improves the previously known lower
bound of 12. Numerical experiments have found a trefoil with ropelength less
than 16.372, so our new bounds are quite sharp.Comment: v3 is the version published by Geometry & Topology on 25 February
200
-Stars or On Extending a Drawing of a Connected Subgraph
We consider the problem of extending the drawing of a subgraph of a given
plane graph to a drawing of the entire graph using straight-line and polyline
edges. We define the notion of star complexity of a polygon and show that a
drawing of an induced connected subgraph can be extended with at
most bends per edge, where is the
largest star complexity of a face of and is the size of the
largest face of . This result significantly improves the previously known
upper bound of [5] for the case where is connected. We also show
that our bound is worst case optimal up to a small additive constant.
Additionally, we provide an indication of complexity of the problem of testing
whether a star-shaped inner face can be extended to a straight-line drawing of
the graph; this is in contrast to the fact that the same problem is solvable in
linear time for the case of star-shaped outer face [9] and convex inner face
[13].Comment: Appears in the Proceedings of the 26th International Symposium on
Graph Drawing and Network Visualization (GD 2018
Triangle areas in line arrangements
A widely investigated subject in combinatorial geometry, originated from
Erd\H{o}s, is the following. Given a point set of cardinality in the
plane, how can we describe the distribution of the determined distances? This
has been generalized in many directions. In this paper we propose the following
variants. Consider planar arrangements of lines. Determine the maximum
number of triangles of unit area, maximum area or minimum area, determined by
these lines. Determine the minimum size of a subset of these lines so that
all triples determine distinct area triangles.
We prove that the order of magnitude for the maximum occurrence of unit areas
lies between and . This result is strongly connected
to both additive combinatorial results and Szemer\'edi--Trotter type incidence
theorems. Next we show a tight bound for the maximum number of minimum area
triangles. Finally we present lower and upper bounds for the maximum area and
distinct area problems by combining algebraic, geometric and combinatorial
techniques.Comment: Title is shortened. Some typos and small errors were correcte
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