labelled digraph

A triple (V,E,w) is called a labelled digraph, if (V,E) is a digraphMathworldPlanetmath and w is an association of elements from some set L, the labels, to some of the edges and vertices of the digraph. In other words, w is a mapping from a subset AVE to L. Most often, L is a subset of the real numbers, in which case (V,E,w) is called a weighted digraph and its labels are called weights. Typically, either A=V or A=E, in which case (V,E,w) is called either a vertex-weighted digraph or an edge-weighted digraph, respectively.

Application examples

We give two typical “real life” examples. The first features an edge-weighted digraph, while the second requires the implementation of a vertex-weighted digraph.

Railway network

A railway network consists of railway stations connectedPlanetmathPlanetmath by rails. A train needs a certain time (measured in minutes) to fare from one station to another. In a formalisation, V is the set of train stations, E the set of direct connexions between them and w:E a weighting corresponding to the journey times, so (V,E,w) is an edge-weighted digraph. Although typically (V,E) is a digraph, w does not need to be symmetric: for example, the journey from a to b might take longer than the return journey because b is located on a mountain.

An important optimisation problem is the efficient determination of the fastest way from one station to another. An even harder problem is to find the fastest round trip (usually called a tour) via a given number of stations. This is the travelling salesman problem.

Dependency graph

A software bundle consists of a number of packages each of which is either installed or not. An installed package occupies a certain amount of bytes on a storage medium. Packages may depend on other packages, that is installation of a package may require other packages to be installed first, which in turn may require still other packages and so forth. One is interested in the completePlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath storage requirement incurred by the installation of one package and all its dependencies.

In a formalisation, the packages are vertices of a digraph (V,E), and an edge (a,b)E means “a depends on b”. Such a digraph is typically not symmetric. The weighting w:V associates sizes to packages. A subset W of V is dependency-closed, if for any wW, all dependencies of w are in W. Given a to-be-installed package v, the storage requirement incurred by the installation of v and all its dependencies is the sum of the vertex weights of the smallest dependency-closed subset of V containing v.

Title labelled digraph
Canonical name LabelledDigraph
Date of creation 2013-03-22 15:15:07
Last modified on 2013-03-22 15:15:07
Owner GrafZahl (9234)
Last modified by GrafZahl (9234)
Numerical id 4
Author GrafZahl (9234)
Entry type Definition
Classification msc 05C20
Classification msc 05C12
Classification msc 05C78
Classification msc 05C90
Defines label
Defines weighted digraph
Defines weight
Defines vertex-weighted digraph
Defines edge-weighted digraph