The standard graph partitioning (GP) problem asks for a partition of the vertices of an undirected graph into a number of parts. The objective and the constraint of this well-known problem are to minimize the number of edges having vertices in two different parts and to equitably partition the vertices among the parts. The GP problem is NP-complete. We investigate a variant of this problem, called acyclic partitioning, for directed acyclic graphs. In this variant, we have one more constraint: the partition should be acyclic. In other words, for a suitable numbering of the parts, all edges should be directed from a vertex in a part p to another vertex in a part q where p ≤ q.
The directed acyclic graph partitioning (DAGP) problem arises in many applications. The stated variant of the DAGP problem arises in exposing parallelism in automatic differentiation, and particularly in the computation of the Newton step for solving nonlinear systems. The DAGP problem with some additional constraints is used to reason about the parallel data movement complexity and to dynamically analyze the data locality potential. Other important applications of the DAGP problem include (i) fusing loops for improving temporal locality, and enabling streaming and array contractions in runtime systems, such as Bohrium; (ii) analysis of cache efficient execution of streaming applications on uniprocessors; (iii) a number of circuit design applications in which the signal directions impose acyclic partitioning requirement.
In dagP, we adopt the multilevel approach with coarsening, initial partitioning, and refinement phases for acyclic partitioning of directed acyclic graphs. We focus on two-way partitioning (sometimes called bisection), as this scheme can be used in a recursive way for multi-way partitioning. To ensure the acyclicity of the partition at all times, we propose novel and efficient coarsening and refinement heuristics. The quality of the computed acyclic partitions is assessed by computing the edge cut.
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dagP (Updated 09/18/2019)