Sub-polyhedral scheduling using (unit-)two-variable-per-inequality polyhedra

Abstract

International audiencePolyhedral compilation has been successful in the design and implementation of complex loop nest optimizers and parallelizing compilers. The algorithmic complexity and scalability limitations remain one important weakness. We address it using sub-polyhedral under-aproximations of the systems of constraints resulting from affine scheduling problems. We propose a sub-polyhedral scheduling technique using (Unit-)Two-Variable-Per-Inequality or (U)TVPI Polyhedra. This technique relies on simple polynomial time algorithms to under-approximate a general polyhedron into (U)TVPI polyhedra. We modify the state-of-the-art PLuTo compiler using our scheduling technique, and show that for a majority of the Polybench (2.0) kernels, the above under-approximations yield polyhedra that are non-empty. Solving the under-approximated system leads to asymptotic gains in complexity, and shows practically significant improvements when compared to a traditional LP solver. We also verify that code generated by our sub-polyhedral parallelization prototype matches the performance of PLuTo-optimized code when the under-approximation preserves feasibility

Full runtime polyhedral optimizing loop transformations with the generation, instantiation, and scheduling of code‐bones

“…However, it would yield a huge number of polyhedral statements to be scheduled and scanned, which is incompatible with the complexity handled by current polyhedral model tools. Upadrasta and Cohen have shown that the execution time of Pluto increases in a roughly n 5 complexity in the number of statements in the system. In consequence, Pluto would introduce a high time overhead, which is inadequate for a runtime usage.…”

A Survey of Loop Parallelization: Models, Approaches, and Recent Developments

“…Some atomic options can be used to achieve more fine-grained control, such as the control of code size, the application of user-directives. Upadrasta et al [43] proposed a sub-polyhedral scheduling algorithm, which is based on TVPI (Two Variables Per Inequality) polyhedra with form of constraints:…”

Improving compiler scalability: optimizing large programs at small price

“…Let Z(m, n) be the complexity of ILP using the Simplex algorithm with m constraints and n variables. Then, on a typical input, Z(m, n) = O((m+n)mn) on average 2 [26]. In our ILP, this amounts to a complexity of (|E|+|V|)|E||V|, or |E| 2 |V|, or |V| 5 , i.e.…”

“…This equips Polly also to achieve scalability as scalefuse, but the optimizer currently suffers from the same limitation (as above) of constructing separate schedules for basic blocks. Upadrasta et al [26] have identified this problem and have proposed a sub-polyhedral technique using (Unit-)Two-Variable-Per-Inequality or (U)TVPI Polyhedra. Their technique is an under-approximation of a general polyhedron, and thus does not guarantee to find best solutions.…”

“…However, while the polyhedral compilers have shown promise in effectively optimizing large scopes [19], the compile time and memory requirement is prohibitively expensive. The unscalability in polyhedral compilers stems from the massive (fifth degree polynomial) increase in compile time with the number of statements [26]. This unscalability of the employed techniques in polyhedral compilers has thus been recognized as a key problem, particularly as polyhedral compilers are becoming increasingly useful [3,13,23,28].…”