Reachability In Directed Graphs Research Articles

Identifying superword level parallelism with directed graph reachability}{Identifying superword level parallelism with directed graph reachability

SLP (superword level parallelism) is an efficient solution to exploit the parallelism between statements in the basic blocks of SIMD (single instruction multiple data). It has been implemented in almost all the mainstream vectorizing compilers. However, its vectorization ability is limited due to the conservativeness of the parallelism identification process. To solve this problem, this paper proposes an edg (extended dependence graph) approach to identify SLP. First, we extend the adg (array dependence graph) and sdg (statement dependence graph) to construct the edg, which includes both the dependences between each array pair and those between each statement pair. When a statement is represented as an SCC (strong connected component), all of its array references are also constructed in this SCC. We then eliminate the redundant dependence edges between the SCCs from the edg. The dependence information of each statement pair and its SLP vectorization are thus determined by analyzing the reachability between each node pair from the corresponding SCC. We implement this approach to optimize the Open64-5.0 compiler, which improves the compilers identification ability. The evaluation tests on the gcc-vect benchmarks show that the optimized Open64-5.0 compiler can identify more SLP vectorizable loops than the GCC4.9, and that the number of vectorizable loops is comparable to that of ICC14.0. The performance of our generated codes is better than the state-of-the-art for most practical applications.

On the parallel complexity of digraph reachability

We formally show that the directed graph reachability problem can be reduced to several problems using a linear number of processors; hence an efficient parallel algorithm to solve any of these problems would imply an efficient parallel algorithm for the directed graph reachability problem. This formally establishes that all these problems are at least as hard as the s− t reachability problem.

Parallel Algorithms for Term Matching

We present a randomized parallel algorithm for term matching. Let n be the number of nodes of the directed acyclic graphs (dags) representing the terms to be matched. Then our algorithm uses $O(\log ^2 n)$ parallel time and $M(n)$ processors, where $M(n)$ is the complexity of $n \times n$ matrix multiplication. The randomized algorithm is of the Las Vegas type, that is, the answer is always correct, although with small probability the algorithm might fail to produce an answer. The number of processors is a significant improvement over previously known bounds. Under various syntactic restrictions on the form of the input dags, only $O(n^2 )$ processors are required in order to achieve deterministic $O(\log ^2 n)$ parallel time. Furthermore, we reduce directed graph reachability to term matching using constant parallel time and $O(n^2 )$ processors. This is evidence that no deterministic algorithm can significantly beat the processor bound of our randomized algorithm. We also improve the P-completeness result of Dwork, Kanellakis, and Mitchell on the unification problem, showing that unification is P-complete even if both input terms are linear, i.e., no variable appears more than once in each term.