Given a set of points labeled with $k$ labels, we introduce the heat map sorting problem as reordering and merging the points and dimensions while preserving the clusters (labels). A cluster is preserved if it remains connected, i.e., if it is not split into several clusters and no two clusters are merged. We prove the problem is NP-hard and we give a fixed-parameter algorithm with a constant number of rounds in the massively parallel computation model, where each machine has a sublinear memory and the total memory of the machines is linear. We give an approximation algorithm for a NP-hard special case of the problem. We empirically compare our algorithm with k-means and density-based clustering (DBSCAN) using a dimensionality reduction via locality-sensitive hashing on several directed and undirected graphs of email and computer networks.

翻译：给定一个包含$k$个标签的点集，我们提出热图排序问题，即在保持簇结构（标签）的前提下，对点和维度进行重排与合并。如果一个簇保持连通性，即不被分割成多个簇且没有两个簇被合并，则该簇被保留。我们证明该问题是NP难的，并给出一个固定参数算法，该算法在大规模并行计算模型中以常数轮次运行，其中每个机器具有次线性内存，且所有机器的总内存为线性。针对该问题的一个NP难特例，我们给出一个近似算法。我们通过局部敏感哈希对多个有向和无向的电子邮件及计算机网络图进行降维，并将我们的算法与k-means和基于密度的聚类（DBSCAN）进行了实验比较。