A spanning-tree algorithm for constructing streams in pipelines

Abstract

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