We describe a scalable and general-purpose framework for auto-tuning compiler-generated code. We combine Active Harmony's parallel search backend with the CHiLL compiler transformation framework to generate in parallel a set of alternative implementations of computation kernels and automatically select the one with the best-performing implementation. The resulting system achieves performance of compiler-generated code comparable to the fully automated version of the ATLAS library for the tested kernels. Performance for various kernels is 1.4 to 3.6 times faster than the native Intel compiler without search. Our search algorithm simultaneously evaluates different combinations of compiler optimizations and converges to solutions in only a few tens of search-steps.

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A scalable auto-tuning framework for compiler optimization

We develop from first principles an exact model of the behavior of loop nests executing in a memory hicrarchy, by using a nontraditional classification of misses that has the key property of composability. We use Presburger formulas to express various kinds of misses as well as the state of the cache at the end of the loop nest. We use existing tools to simplify these formulas and to count cache misses. The model is powerful enough to handle imperfect loop nests and various flavors of non-linear array layouts based on bit interleaving of array indices. We also indicate how to handle modest levels of associativity, and how to perform limited symbolic analysis of cache behavior. The complexity of the formulas relates to the static structure of the loop nest rather than to its dynamic trip count, allowing our model to gain efficiency in counting cache misses by exploiting repetitive patterns of cache behavior. Validation against cache simulation confirms the exactness of our formulation. Our method can serve as the basis for a static performance predictor to guide program and data transformations to improve performance.

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Exact analysis of the cache behavior of nested loops

Standard array data dependence testing algorithms give information about the aliasing of array references. If statement 1 writes a[5], and statement 2 later reads a [5], standard techniques described this as a flow dependence, even if there was an intervening write. We call a dependence between two references to the same memory location a memory-based dependence. In contrast, if there are no intervening writes, the references touch the same value and we call the dependence a value-based dependence.

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An Exact Method for Analysis of Value-based Array Data Dependences

We describe methods that are able to count the number of integer solutions to selected free variables of a Presburger formula, or sum a polynomial over all integer solutions of selected free variables of a Presburger formula. This answer is given symbolically, in terms of symbolic constants (the remaining free variables in the Presburger formula).For example, we can create a Presburger formula who's solutions correspond to the iterations of a loop. By counting these, we obtain an estimate of the execution time of the loop.In more complicated applications, we can create Presburger formulas who's solutions correspond to the distinct memory locations or cache lines touched by a loop, the flops executed by a loop, or the array elements that need to be communicated at a particular point in a distributed computation. By counting the number of solutions, we can evaluate the computation/memory balance of a computation, determine if a loop is load balanced and evaluate message traffic and allocate message buffers.

Counting solutions to Presburger formulas: how and why

On modern computer systems, the memory performance of an application depends on its locality. For a single execution, locality-correlated measures like average miss rate or working-set size have long been analyzed using reuse distance—the number of distinct locations accessed between consecutive accesses to a given location. This article addresses the analysis problem at the program level, where the size of data and the locality of execution may change significantly depending on the input.The article presents two techniques that predict how the locality of a program changes with its input. The first is approximate reuse-distance measurement, which is asymptotically faster than exact methods while providing a guaranteed precision. The second is statistical prediction of locality in all executions of a program based on the analysis of a few executions. The prediction process has three steps: dividing data accesses into groups, finding the access patterns in each group, and building parameterized models. The resulting prediction may be used on-line with the help of distance-based sampling. When evaluated on fifteen benchmark applications, the new techniques predicted program locality with good accuracy, even for test executions that are orders of magnitude larger than the training executions.The two techniques are among the first to enable quantitative analysis of whole-program locality in general sequential code. These findings form the basis for a unified understanding of program locality and its many facets. Concluding sections of the article present a taxonomy of related literature along five dimensions of locality and discuss the role of reuse distance in performance modeling, program optimization, cache and virtual memory management, and network traffic analysis.

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Program locality analysis using reuse distance

The Omega Library interface guide

Automatic parallelization of real FORTRAN programs does not live up to users expectations yet, and dependence analysis algorithms which either produce too many false dependences or are too slow to contribute significantly to this. In this paper we introduce dataflow dependence analysis algorithm which exactly computes value-based dependence relations for program fragments in which all subscripts, loop bound and IF conditions are affine. Our algorithm also computes good affine approximations of dependence relations for non-affine program fragments. Actually, we do not know about any other algorithm which can compute better approximations.And our algorithm is efficient too, because it is lazy. When searching for write statements that supply values used by a given read statement, it starts with statements which are lexicographically close to the read statement in iteration space. Then if some of the read statement instances are not “satisfied” with these close writes, the algorithm broadens its search scope by looking into more distant writes. The search scope keeps broadening until all read instances are satisfied or no write candidates are left.We timed our algorithm on several benchmark programs and the timing results suggest that our algorithm is fast enough to be used in commercial compilers—it usually takes 5 to 15 percent of f77 -02 compilation time to analyze a program. Most programs in the 100-line range take less than 1 second to analyze on a SUN SparcStation IPX.

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Lazy array data-flow dependence analysis

Why do existing parallelizing compilers and environments fail to parallelize many realistic FORTRAN programs? One of the reasons is that these programs contain a number of linearized array references, such as A(M*N*i+N*j+k) or A(i*(i+1)/2+j). Performing exact dependence analysis for these references requires testing polynomial constraints for integer solutions. Most existing dependence analysis systems, however, restrict themselves to solving affine constraints only, so they have to make worst-case assumptions whenever they encounter a polynomial constraint. In this paper we introduce an algorithm that exactly and efficiently simplifies a class of polynomial constraints which arise in dependence testing.

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Vadim Maslov

Papers

The Omega Library interface guide

Lazy array data-flow dependence analysis

Simplifying Polynominal Constraints Over Integers to Make Dependence Analysis More Precise