Ay family

The Ay family is a term used in computer science, specifically in the context of compiler design and data dependency analysis, to describe a family of data dependence vectors associated with a particular data dependence. It is used to represent the distance and direction of data flow between different iterations of a loop, usually in nested loops.

In essence, a data dependence exists when one iteration of a loop needs data produced by a previous iteration. The Ay family provides a way to compactly represent all possible dependence vectors that could occur for a specific data dependence, taking into account loop bounds and array access patterns.

The Ay family is often represented as a set of inequalities or a polyhedron in a multi-dimensional space, where each dimension corresponds to a loop variable. Each point within this polyhedron represents a valid data dependence vector. Techniques from linear algebra and integer programming can be applied to analyze and manipulate Ay families, enabling compiler optimizations such as loop transformations and parallelization. The information encoded within the Ay family helps the compiler determine whether it is safe to reorder or execute loop iterations concurrently without violating data dependencies.