Graph distance (or similarity) scores are used in several graph mining tasks, including anomaly detection, nearest neighbor and similarity search, pattern recognition, transfer learning, and clustering. Graph distances that are metrics and, in particular, satisfy the triangle inequality, have theoretical and empirical advantages. Well-known graph distances that are metrics include the chemical or the Chartrand-Kubiki-Shultz (CKS) distances. Unfortunately, both are computationally intractable. Recent efforts propose using convex relaxations of the chemical and CKS distances. Though distance computation becomes a convex optimization problem under these relaxations, the number of variables is quadratic in the graph size; this makes traditional optimization algorithms prohibitive even for small graphs. We propose a distributed method for massively parallelizing this problem using the Alternating Directions Method of Multipliers (ADMM). Our solution uses a novel, distributed bisection algorithm for computing a $p$ -norm proximal operator as a building block. We demonstrate its scalability by conducting experiments over multiple parallel environments.

中文翻译：

---

大规模分布图距离

图距离（或相似度）分数用于几种图挖掘任务，包括异常检测，最近邻居和相似度搜索，模式识别，转移学习和聚类。作为度量的图形距离，尤其是满足三角形不等式的图形距离，具有理论和经验上的优势。作为度量标准的众所周知的图形距离包括化学距离或Chartrand-Kubiki-Shultz（CKS）距离。不幸的是，两者在计算上都是棘手的。最近的努力建议使用凸松弛和CKS距离的关系。尽管在这些松弛下距离计算成为一个凸优化问题，但是在图形大小上变量的数量是二次的；这使得传统的优化算法即使对于小的图形也无法实现。我们提出了一种使用乘法器交替方向法（ADMM）大规模并行化此问题的分布式方法。我们的解决方案使用新颖的分布式二等分算法来计算$ p $ -将近端运算符作为构建块。我们通过在多个并行环境上进行实验来证明其可伸缩性。