Automatic Inference of Loop Complexity Through Polynomial Interpolation

Complexity analysis is an important activity for software engineers. Such an analysis can be specially useful in the identification of performance bugs. Although the research community has made significant progress in this field, existing techniques still show limitations. Purely static methods may be imprecise due to their inability to capture the dynamic behaviour of programs. On the other hand, dynamic approaches usually need user intervention and/or are not effective to relate complexity bounds with the symbols in the program code. In this paper, we present a hybrid technique that solves these shortcomings. Our technique uses a numeric method based on polynomial interpolation to precisely determine a complexity function for loops. Statically, we determine: (i) the inputs of a loop, i.e., the variables that control its iterations; and (ii) an algebraic equation relating the loops within a function. We then instrument the program to plot a curve relating inputs and number of operations executed. By running the program over different inputs, we generate sufficient points for our interpolator. In the end, the complexity function for each loop is combined using an algebra of our own craft. We have implemented our technique in the LLVM compiler, being able to analyse 99.7 % of all loops available in the Polybench benchmark suite, and most of the loops in Rodinia. These results indicate that our technique is an effective and useful way to find the complexity of loops in high-performance applications.