We introduce a novel concept termed "stochastic distance" for property testing. Diverging from the traditional definition of distance, where a distance $t$ implies that there exist $t$ edges that can be added to ensure a graph possesses a certain property (such as $k$-edge-connectivity), our new notion implies that there is a high probability that adding $t$ random edges will endow the graph with the desired property. While formulating testers based on this new distance proves challenging in a sequential environment, it is much easier in a distributed setting. Taking $k$-edge-connectivity as a case study, we design ultra-fast testing algorithms in the CONGEST model. Our introduction of stochastic distance offers a more natural fit for the distributed setting, providing a promising avenue for future research in emerging models of computation.

翻译：本文引入了一种称为“随机距离”的新概念，用于属性测试。与传统的距离定义（即距离$t$意味着存在$t$条边可以添加以确保图具有特定属性，如$k$-边连通性）不同，我们的新概念意味着以高概率添加$t$条随机边将使图获得所需属性。虽然基于这种新距离在顺序环境中构建测试器具有挑战性，但在分布式环境中则容易得多。以$k$-边连通性为例，我们在CONGEST模型中设计了超快速的测试算法。随机距离的引入为分布式环境提供了更自然的适配，为新兴计算模型的未来研究开辟了有前景的途径。