Decomposition in pipelines

Abstract

The operation of a pipeline is considered as a special graph (the unfolding) associated to the pipeline graph. We investigate the decomposition of the unfolding, a particular case of which is its decomposition into minimal independent streams.i. Statement of the Problem. The aim of this paper is to elucidate some structural questions connected with the operation of pipelines in the framework of [i].We represent the operation of a fully loaded pipeline by its unfolding graph [2].A natural question in this context is, what subgraphs of the unfolding graph have independent computational meaning? In this paper, we consider as such subgraphs (computational processes) subgraphs that are closed in the following sense. Let a vertex have a nonempty set of immediate predecessors. Then if they all belong to the subgraph, the vertex itself also belongs to the subgraph, and conversely. Informally, this definition means the following. If at the moment t the arguments of some operation are ready for computation, then at the moment t + i the process will necessarily compute the result of this operation, and conversely. It is easy to see that the minimal independent streams (MIS) of [1]the connected components of the unfolding -are processes, but not only they.In this paper, we develop a decomposition of an arbitrary process into a union of finitely many processes of special types, which we call simple. This decomposition is more general than the decomposition into MIS. Thus, if the pipeline graph is strongly connected and the process is complete (the operation of a fully loaded pipeline), then this decomposition reduces to a MIS decomposition.It is interesting, however, that MIS may have proper subprocesses, some of which may be independent and some similar. Independence means that the processes have no shared vertices, while similarity means that one process is obtained by a time shift of the other.In addition to decomposition, we also highlight some other properties of these processes.2. Processes in Pipelines. Let G = (V, E) be a finite connected digraph. Its vertices will be denoted by ~ possibly with indices and primes, and the arc from ~ to ~' will be denoted by ~' . Below we only deal with digraphs, and we therefore simply call them graphs. To the graph G we associate another graph R=iV~Z~ { ~, ~%+I) : ~'~[~)-its unfolding. We use nonparenthesized notation for the pairs ~i~V~Z i arithmetic

Lattices of processes for graphs

“…is an infinite bounded lattice and, by [1][2][3], it has a principal boolean ideal I with countable base.…”

“…This is a formalization of the computing process in an arbitrary pipeline. This paper contains some definitions and results from [1][2][3][4] and presents some new results related to the separation of the class of lattices of processes from the set of all distributive lattices.i ~ Definitions and necessary information Let G = (V, E) be a finite digraph with vertex set V and edge set E, and let the graph R(G) = (V x Z, U) be its unfolding, where u = = v., v'. t + 1, t e z, 9 E}.…”