Achieving Scalability in Parallel Reachability Analysis of Very Large Circuits

Abstract

This paper presents a scalable method for parallel symbolic reachability analysis on a distributed-memory environment of workstations. Our method makes use of an adaptive partitioning algorithm which achieves high reduction of space requirements. The memory balance is maintained by dynamically repartitioning the state space throughout the computation. A compact BDD representation allows coordination by shipping BDDs from one machine to another, where different variable orders are allowed. The algorithm uses a distributed termination protocol with none of the memory modules preserving a complete image of the set of reachable states. No external storage is used on the disk; rather, we make use of the network which is much faster. We implemented our method on a standard, loosely-connected environment of workstations, using a high-performance model checker. Our initial performance evaluation using several large circuits shows that our method can handle models that are too large to fit in the memory of a single node. The efficiency of the partitioning algorithm is linear in the number of workstations employed, with a 40-60% efficiency. A corresponding decrease of space requirements is measured throughout the reachability analysis. Our results show that the relatively-slow network does not become a bottleneck, and that computation time is kept reasonably small.

Speculative Image Computation for Distributed Symbolic Reachability Analysis

“…Most parallel or distributed work on symbolic reachability analysis employs a vertical slicing scheme to parallelize BDD manipulations, where image computation jobs are spawned and queued on the individual workstations of the NOW [19,21,29], allowing the algorithm to overlap image computation. Figure 10 shows how an MDDs can be decomposed into three portions where x 5 and x 4 are selected as slicing variables (the corresponding MDD nodes are indicated with solid boxes).…”

“…For explicit reachability analysis or model checking, references [1,24,27] introduce algorithms that use the overall resources of a network of workstations (NOW). For symbolic reachability analysis or model checking, most works employ a vertical slicing scheme to parallelize BDD manipulations by spawning multiple image computations over the NOW, corresponding to distinct sets of paths through the BDD [19,21,28]. Algorithms following this scheme can overlap the application of the next-state function to sets of states encoded by different decision diagram node, i.e.…”

A Work-Efficient Distributed Algorithm for Reachability Analysis

“…Duplication is the amount of sharing in a BDD structure that is lost due to partitioning. The notion of duplication and its implications are discussed in detail in [13] and will not be addressed in this paper. Finding such an effective slicing is a nontrivial problem [8,17,16,13].…”

“…Since the chosen BDDs are large, slicing is always very effective. Furthermore, slicing affects the performance of the new algorithm much less than it affects the one from [13] because, in the case of a high work load at one of the co-workers, the new algorithm can simply split again. These features provide the new algorithm with strength and flexibility, and allow to reduce the slicing complexity.…”

“…In [13], each process iteratively applies image computation to its set of new states N , exchanging non-owned states with other processes, and collecting owned states in its set of reachable states R. Load balance is available at the end of each iteration. It balances the sizes of the sets of reachable states in the different processes.…”