Abstract

Block-oriented sparse Cholesky factorization decomposes a sparse matrix into rectangular subblocks; each block can then be handled as a computational unit in order to increase data reuse in a hierarchical memory system. Also, the factorization method increases the degree of concurrency and reduces the overall communication volume so that it performs more efficiently on a distributed-memory multiprocessor system than the customary column-oriented factorization method. But until now, mapping of blocks to processors has been designed for load balance with restricted communication patterns. In this paper, we represent tasks using a block dependency DAG that represents the execution behavior of block sparse Cholesky factorization in a distributed-memory system. Since the characteristics of tasks for block Cholesky factorization are different from those of the conventional parallel task model, we propose a new task scheduling algorithm using a block dependency DAG. The proposed algorithm consists of two stages: early-start clustering, and affined cluster mapping (ACM). The early-start clustering stage is used to cluster tasks while preserving the earliest start time of a task without limiting parallelism. After task clustering, the ACM stage allocates clusters to processors considering both communication cost and load balance. Experimental results on a Myrinet cluster system show that the proposed task scheduling approach outperforms other processor mapping methods.

Author Keywords: Task scheduling; Parallel sparse matrix factorization; Block-oriented Cholesky factorization; Directed acyclic graph

Article Outline

1. Introduction

2. Block-oriented sparse Cholesky factorization

2.1. Block decomposition

2.2. Block Cholesky factorization

2.3. Block operations

2.4. Required number of block update operations

3. Task model with communication costs

3.1. Task characteristics

3.2. Task graph

3.3. Task execution behavior on previous block mapping methods

3.3.1. 2-D cyclic mapping

3.3.2. Load balanced mapping

4. Task scheduling using a block dependency DAG

4.1. Task scheduling parameters

4.1.1. Work and parents of subtasks

4.1.2. Earliest start time of a task

4.1.3. Earliest completion time of a task

4.1.4. Level of a task

4.2. Early-start clustering

4.3. Affined cluster mapping

4.4. Running trace of the proposed scheduling algorithm

5. Performance comparison

6. Conclusion

Acknowledgements

References