Closed-form mapping conditions for the synthesis of linear processor arrays

Abstract

Abstract. This paper addresses the problem of mapping algorithms with constant data dependences to linear processor arrays. The closed-form necessary and sufficient mapping conditions are derived to identify mappings without computational conflicts and data link collisions. These mapping conditions depend on the space-time mapping matrix and the problem size parameters only. Their correctness can be verified in constant time that is independent of problem size. The design of optimal linear processor arrays is formulated as a mathematic programming problem, which can be solved efficiently by a systematic enumeration of a polynomial search space.

An approach to checking link conflicts in the mapping of uniform dependence algorithms into lower dimensional processor arrays

“…Since the array model, described in the next paragraph, considered in this paper is the same as that in [13], [17], [18], we also impose the same restriction on the uniform dependence algorithms as in their works. The restriction is stated as follows.…”

“…Most researches on synthesizing [13], [14], [15], [16], [17], [18] processor arrays from the uniform dependence algorithms focus on finding a space-time mapping (linear transformation) to the algorithm such that the transformed algorithm represents a regular processor array. The space-time mapping is, in general, represented as a transformation matrix.…”

“…There have been several attempts on the problem of checking the link conflicts [13], [14], [16], [17], [18], [19], [20]. One of these methods [13], [14] has to examine the whole index set of size yx Pn , where x is the problem size.…”

“…One of these methods [13], [14] has to examine the whole index set of size yx Pn , where x is the problem size. Other methods [16], [17], [18], [19], [20] use the I/O spaces concept to check link conflicts. Each I/O space is associated with a data dependence vector d. An I/O space can be an input space or an output space and is defined as the set fi À d j i P t nd i À d T P tg and fi j i P t nd i d T P tg, respectively, where t is the index set.…”

“…However, an exact link conflict checking cannot be obtained by their methods if the I/O space is a nonconvex one. In [17], [18], a procedure was proposed to map nonconvex I/O space into convex one. But, the projection of nonconvex I/O space may introduce superfluous points which are not needed in the computation of the algorithm, into the projected I/O space.…”