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. Two communication cost minimization problems are defined based on different pattern of communication, which are the Data Redistribution Problem and Data Allocation Problem. 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 two problems in different communication pattern with the four kinds of objective cost functions, 8 problems are obtained. The hardness results of the 8 problems make up the content of this paper. With rigorous proof, we prove that some of the 8 problems are in P, some FPT, some NP-complete, and some W[1]-complete.

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，该模型通过感知网络拓扑增强了经典MPC模型。Hu等人在拓扑感知MPC模型上的工作仅考虑了树形拓扑结构。本文考虑了更一般的情况，其中底层网络为加权完全图。我们称该模型为权重大规模并行计算（WMPC）模型，并研究在此模型下最小化通信代价的问题。基于不同的通信模式，定义了两种通信代价最小化问题：数据重分配问题和数据分配问题。我们还针对通信代价定义了四种目标函数，分别考虑总代价、瓶颈代价、发送与接收代价的最大值以及发送与接收代价之和。将两种不同通信模式的问题与四种目标代价函数组合，共得到8个问题。这8个问题的难度结果构成了本文的主要内容。通过严谨的证明，我们证明了其中部分问题属于P类、部分属于FPT类、部分为NP完全问题、部分为W[1]完全问题。