We initiate a thorough study of distributed property testing—producing algorithms for the approximation problems of property testing in the CONGEST model. In particular, for the so-called dense graph testing model we emulate sequential tests for nearly all graph properties having 1-sided tests, while in the general model we obtain faster tests for triangle-freeness and cycle-freeness, and in the sparse model we obtain a faster test for bipartiteness. In addition, we show a logarithmic lower bound for testing bipartiteness and cycle-freeness, which holds even in the stronger LOCAL model. In most cases, aided by parallelism, the distributed algorithms have a much shorter running time than their counterparts from the sequential querying model of traditional property testing. More importantly, the distributed algorithms we develop for testing graph properties are in many cases much faster than what is known for exactly deciding whether the property holds. The simplest property testing algorithms allow a relatively smooth transition to the distributed model. For the more complex tasks we develop new machinery that may be of independent interest.

中文翻译：

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用于测试图属性的快速分布式算法

我们启动了对分布式属性测试的彻底研究——为 CONGEST 模型中的属性测试的近似问题生成算法。特别是，对于所谓的密集图测试模型，我们模拟几乎所有具有 1 边测试的图属性的顺序测试，而在一般模型中，我们获得了三角形自由度和循环自由度的更快测试，而在稀疏模型中我们获得了更快的二分性测试。此外，我们展示了用于测试二分性和无循环性的对数下限，即使在更强的 LOCAL 模型中也是如此。在大多数情况下，在并行性的帮助下，分布式算法的运行时间比传统属性测试的顺序查询模型中的对应算法要短得多。更重要的是，我们开发的用于测试图属性的分布式算法在许多情况下比准确判断属性是否成立的已知算法要快得多。最简单的属性测试算法允许相对平滑地过渡到分布式模型。对于更复杂的任务，我们开发可能具有独立兴趣的新机器。