Louis Nel

Unranking and ranking spanning trees of a graph

Abstract The set S of spanning trees of an n -vertex graph G can be placed in one-to-one correspondence with the integers in the interval [1, s ], where s = | S |. We develop O ( n 3 ) unranking and ranking functions for the spanning trees of an arbitrary graph. The unranking function maps any interval [1, s ] to the corresponding tree, while the ranking function maps a spanning tree to the appropriate index in the interval. The unranking function provides an O ( n 3 ) method for generating a random spanning tree of a graph with uniform distribution.

Combining monte carlo estimates and bounds for network reliability

A simplified model of a communications network is a probabilistic graph in which each edge operates with the same probability. The all-terminal reliability, or probability that all nodes are connected, can be expressed as a polynomial in the edge operation probability. The coefficients of this polynomial are obtained from an interval partitioning of the cographic matroid, and only the first few coefficients can be computed efficiently. One of the best sets of efficiently computable reliability bounds is the Ball-Provan bounds. These bounds are obtained using the efficiently computable coefficients and can be improved substantially if additional coefficients are known. In this paper, we develop a Monte Carlo method for estimating additional coefficients by randomly sampling over spanning trees of the network. Confidence intervals for all-terminal reliability are obtained by using these estimates as additional constraints in the Ball-Provan bounds. This approach has some advantages over conventional Monte Carlo point estimate methods. In particular, the computational complexity does not depend on the reliability of the network.

Locating A Broadcast Facility In An Unreliable Network

AbstractA simple model of an unreliable communications network is a probabilistic graph in which each edge has some independent probability Of being operational. The Network Broadcast Facility location problem is to identify a node for which the expected number of nodes connected to it in the presence of random edge failures is as large as possible. This problem is, in fact, a special case of the stochastic 1-median problem. In this paper both problems are shown to be NP-hard In light of this apparent intractability, a location strategy based on efficiently computable two-terminal reliability bounds is suggested. This location strategy does not guarantee to find the best location but instead limits the choice to a subset of the nodes which contain the most resilient nodes of the network. The success of this approach depends on obtaining sufficiently tight reliability bounds. Edge-packing methods for computing upper and lower two-terminal bounds, as well as global methods for tightening these bounds are di...

Differential calculus founded on an isomorphism

This paper founds calculus on a natural isometric linear isomorphism. Once this Foundational Isomorphism is proved (with elementary Banach space methods) several familiar calculus properties of continuous curves drop out as quick corollaries. Then the calculus can be further developed in very general setting via categorical methods.

Infinite dimensional calculus allowing nonconvex domains with empty interior

We develop differentiation theory based on a new definition of derivative: in terms of integrals of curves. It applies to a very wide class of domains. Some ‘elementary’ maps between Fréchet spaces, excluded from calculus by previous theories, now emerge as smooth maps.

Universal topological algebra needs closed topological categories

Abstract It is shown that a development of universal topological algebra, based in the obvious way on the category of topological spaces, leads in general to a pathological situation. The pathology disappears when the base category is changed to a cartesian closed topological category or to a topological category endowed with a compatible closed symmetric monoidal structure, provided that in the latter case, the algebraic operations are expressed in terms of monoidal powers rather than the usual cartesian powers. With such base categories, universal topological algebra becomes virtually as well-behaved as ordinary (setbased) universal algebra.

Network reliability and the probabilistic estimation of damage from fire spread

An efficient generalization of Shanthikumars upper bound on two-terminal reliability is developed, that leads to efficient methods for the probabilistic assessment of damage from fire spread and other invasive hazards in segmented structures. The methods exploit a basic relationship between the fire spread problem and the probability of reachability in communications networks. The upper bound employs noncrossing cuts of the network.

Infinite dimensional holomorphy via categorical differential calculus

We establish that the category of hological spaces is equipped for calculus with complex scalars. This provides a theory of infinite dimensional holomorphy which allows maps to have nonconvex domains with empty interior. Some relatively elementary functions, hitherto excluded by the restrictive definitions of other theories, emerge as holomorphic maps.

Using and abusing bounds for network reliability

The roles of reliability analysis and connectivity analysis, in the selection of a network topology that can support the intended traffic at acceptable speed and acceptable cost are considered. The use of bounds for network reliability is studied, focusing on the effective use of the bounds in network design and indicating where their use is inappropriate. A number of strategies for bounding network reliability that can be applied in the network design process are studied.<<ETX>>

Holomorphy in convergence spaces

We introduce a new approach to infinite dimensional holomorphy. Cast in the setting of closed-embedded linear convergence spaces and based on a categorical definition of derivative, our theory applies beyond the traditional open domains. It reaches certain domains with empty interior (that arise naturally in Fréchet spaces) and gives a fully fledged differential calculus. On open domains our approach provides a new characterization of holomorphic maps. Thus earlier theories become expanded as well as strengthened.