Estimating the number of triangles in a graph is one of the most fundamental problems in sublinear algorithms. In this work, we provide an algorithm that approximately counts the number of triangles in a graph using only polylogarithmic queries when the number of triangles on any edge in the graph is polylogarithmically bounded. Our query oracle Tripartite Independent Set (TIS) takes three disjoint sets of vertices A, B and C as inputs, and answers whether there exists a triangle having one endpoint in each of these three sets. Our query model generally belongs to the class of group queries (Ron and Tsur ACM Trans. Comput. Theory 8(4), 15, 2016; Dell and Lapinskas 2018) and in particular is inspired by the Bipartite Independent Set (BIS) query oracle of Beame et al. (2018). We extend the algorithmic framework of Beame et al., with TIS replacing BIS, for approximately counting triangles in graphs.

估计图中三角形的数量是次线性算法中最基本的问题之一。在这项工作中，我们提供了一种算法，当图中任何边上的三角形数量是多对数有界时，该算法仅使用多对数查询来近似计算图中三角形的数量。我们的查询预言机Tripartite Independent Set (TIS) 将三个不相交的顶点A、B和C集合作为输入，并回答在这三个集合中的每一个中是否存在具有一个端点的三角形。我们的查询模型通常属于组查询的类别（Ron和Tsur ACM Trans。Comput。Theory 8（4），2016年15月； Dell和Lapinskas 2018），并且特别受到了Beame 等人的二部独立集(BIS) 查询预言机。(2018)。我们扩展了 Beame 等人的算法框架，用TIS代替BIS来近似计算图中的三角形。