For more than thirty years, the parallel programming community has used the dependence graph as the main abstraction for reasoning about and exploiting parallelism in ``regular'' algorithms that use dense arrays, such as finite-differences and FFTs. In this paper, we argue that the dependence graph is not a suitable abstraction for algorithms in new application areas like machine learning and network analysis in which the key data structures are ``irregular'' data structures like graphs, trees, and sets.

To address the need for better abstractions, we introduce a data-centric formulation of algorithms called the operator formulation in which an algorithm is expressed in terms of its action on data structures. This formulation is the basis for a structural analysis of algorithms that we call tao-analysis. Tao-analysis can be viewed as an abstraction of algorithms that distills out algorithmic properties important for parallelization. It reveals that a generalized form of data-parallelism called amorphous data-parallelism is ubiquitous in algorithms, and that, depending on the tao-structure of the algorithm, this parallelism may be exploited by compile-time, inspector-executor or optimistic parallelization, thereby unifying these seemingly unrelated parallelization techniques. Regular algorithms emerge as a special case of irregular algorithms, and many application-specific optimization techniques can be generalized to a broader context.

These results suggest that the operator formulation and tao-analysis of algorithms can be the foundation of a systematic approach to parallel programming.