Code generation for multiple mappings

Abstract

There has been a great amount of recent work toward unifying iteration reordering transformations. Many of these approaches represent transformations as a ne mappings from the original iteration space to a new iteration space. These approaches show a great deal of promise, but they all rely on the ability to generate code that iterates over the points in these new iteration spaces in the appropriate order. This problem has been fairly well-studied in the case where all statements use the same mapping. We have developed an algorithm for the less well-studied case where each statement uses a potentially di erent mapping. Unlike many other approaches, our algorithm can also generate code from mappings corresponding to loop blocking. We address the important trade-o between reducing control overhead and duplicating code. AbstractThere has been a great amount of recent work toward unifying iteration reordering transformations. Many of these approaches represent transformations as a ne mappings from the original iteration space to a new iteration space. These approaches show a great deal of promise, but they all rely on the ability to generate code that iterates over the points in these new iteration spaces in the appropriate order. This problem has been fairly well-studied in the case where all statements use the same mapping. We have developed an algorithm for the less well-studied case where each statement uses a potentially di erent mapping. Unlike many other approaches, our algorithm can also generate code from mappings corresponding to loop blocking. We address the important trade-o between reducing control overhead and duplicating code.

Polyhedral Code Generation in the Real World

“…Several works succeeded in relaxing the unimodularity constraint to invertibility (the T matrix has to be invertible), enlarging the set of possible transformations [7,19]. A further step has been achieved by Kelly et al by considering more than one domain and multiple scheduling functions at the same time [15]. All these methods relied on the Fourier-Motzkin elimination method [26] to build the target code.…”

“…The first idea was to compute an appropriate loop stride. At first it was done using the Hermite Normal Form [19,28,7,25], but this was limited to only one domain, then by considering the transformation expression itself [15,2], but some guards cannot be removed in this way. More recent methods suggest to use strip-mining for one domain [10], or to find equivalent transformations with convenient additional dimensions when this is possible [11], or to unroll the loops according to a convenient unroll factor in the case where modulo guards depend on only one loop counter [11].…”

“…Many advances in program restructuring have been achieved through this model which considers each instance of a statement as an integer point in a convenient space [16]. Most of the underlying methods, as data dependence analysis [8,22], transformation computation [20,11] or code generation [15,24] exhibit worst-case exponential complexity.…”

Loop-Carried Code Placement

“…The solid arrows represent the dependences in E. The approach taken in the polyhedron model to represent such structures that are unbounded in size is to use set representations to combine different instances to a single set. E.g., the Omega tool [4] provides this functionality.…”

“…The resulting index space description defines a legal execution order and places lookups as early as possible. From this description, we can generate code using scanning techniques as the one employed by Omega [4] or the one of Quilleré and Rajopadhye [8]; this code is then rewritten to use newly introduced arrays for storing the value of scheduled occurrence instances.…”