Stable networks and product graphs

• Authors: Donald E. Knuth;Tomás Feder
• Venue: Stable networks and product graphs
• Year: 1992

A network is a collection of gates, each with many inputs andmany outputs, where links join individual outputs to individualinputs of gates; the unlinked inputs and outputs of gates areviewed as inputs and outputs of the network. A stable configurationassigns values to inputs, outputs, and links in a network, so as toensure that the gate equations are satisfied. The problem offinding stable configurations in a network is computationally hard,even when all values are boolean and all input values are specifiedin advance; in general, the difficulty of a stability problem seemsto depend on the kinds of mappings associated with the gatespresent in the network. The study can be restricted to gates thatsatisfy a nonexpansiveness condition requiring small perturbationsat the inputs of a gate to have only a small effect at the outputsof the gate. The stability question on the class of networkssatisfying this local nonexpansiveness condition contains stablematching as a main example, and defines the boundary betweentractable and intractable versions of network stability. Thestructural and algorithmic study of stability in nonexpansivenetworks is based on a representation of the possible assignmentsof boolean values for a network as vertices in a boolean cube underthe associated Hamming metric. This global view takes advantage ofthe median properties of the cube, and extends to metric networks,where individual values are now chosen from finite metric spacesand combined by means of an additive product operation. Therelationship between products of metric spaces and products ofgraphs establishes then a connection between isometricrepresentations in graphs and nonexpansiveness in metricnetworks.