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
An algorithm based on the construction of a spanning subgraph of the pipeline is proposed for determining streams in pipelines. ~In this paper, we examine some questions of the mathematical model of pipelined computations proposed in [I]. It was shown in [i] that the operation of an (unconstrained) pipeline may be represented as a finite or infinite collection of similar and independent computational processes (streams). This means that i) these processes solve different variants of the same problem, 2) change of data in one of the variants has no effect on the solution of another variant. This stream decomposition is unique. For the finite case, the maximum number of streams was expressed in [i] in terms of certain parameters of the pipeline graph (in what follows, the terms "pipeline graph" and "pipeline" are used interchangeably). The corresponding streams were called minimal independent streams (MIS). The MIS decomposition suggests the problem of determining the MIS from the pipeline graph. An algorithm for the solution of this problem was proposed in [2]. In this paper, we propose a different algorithm, which we call "spanning-tree algorithm." It has the same complexity as the algorithm of [2] [0(m), where m is the number of pipeline arcs], but does not require preliminary knowledge of the number of MIS, as in [2]. The spanningtree algorithm is based on the following fact. We define the weight of a route ~ in the pipeline as the difference between the number of arcs traversed in ~ from beginning to end and the number of arcs traversed in ~ in the reverse direction. Then the weight of any closed route is a linear combination of the weights of the basis cycles of the pipeline graph (we established this property independently of [3]). 2 o . We start with some definitions and auxiliary propositions. Let G = (V, E) be an arbitrary finite connected graph (here and in what follows, graphs are digraphs); by v, e, with indices and primes, we denote the vertices and edges of this graph. Let i) Mvv r be the set of all routes in G from v to v'; 2) if M is a set of routes, then ~ is the set of weights of the routes from M; 3) SM is the set of all closed routes in G; 3M may satisfy or not satisfy the condition {01 ;(,) 4) k is the number defined by the expression
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