The topology-aware Massively Parallel Computation (MPC) model is proposed and studied recently, which enhances the classical MPC model by the awareness of network topology. The work of Hu et. al. on topology-aware MPC model considers only the tree topology. In this paper a more general case is considered, where the underlying network is a weighted complete graph. We then call this model as Weighted Massively Parallel Computation (WMPC) model, and study the problem of minimizing communication cost under it. Three communication cost minimization problems are defined based on different pattern of communication, which are the Data Redistribution Problem, Data Allocation Problem on Continuous data, and Data Allocation Problem on Categorized data. We also define four kinds of objective functions for communication cost, which consider the total cost, bottleneck cost, maximum of send and receive cost, and summation of send and receive cost, respectively. Combining the three problems in different communication pattern with the four kinds of objective cost functions, 12 problems are obtained. The hardness results and algorithms of the 12 problems make up the content of this paper. With rigorous proof, we prove that some of the 12 problems are in P, some FPT, some NP-complete, and some W[1]-complete. Approximate algorithms are proposed for several selected problems.

翻译：拓扑感知的大规模并行计算（MPC）模型近年来被提出并研究，该模型通过引入网络拓扑感知能力增强了经典MPC模型。Hu等人关于拓扑感知MPC模型的工作仅考虑了树形拓扑。本文研究了更一般的情况，其中底层网络为加权完全图。我们将此模型称为加权大规模并行计算（WMPC）模型，并研究其下的通信成本最小化问题。基于不同的通信模式，定义了三个通信成本最小化问题：数据重分配问题、连续数据分配问题和分类数据分配问题。我们还定义了四类通信成本目标函数，分别考虑总成本、瓶颈成本、发送与接收成本最大值以及发送与接收成本之和。将三种不同通信模式的问题与四类目标成本函数组合，共得到12个问题。这些问题的难度结果与算法构成了本文的主要内容。通过严谨证明，我们证明了这12个问题中部分属于P类、部分属于FPT类、部分属于NP完全类，还有部分属于W[1]完全类。针对若干选定问题提出了近似算法。