Approximately Counting Triangles in Sublinear Time
Approximately Counting Triangles in Sublinear Time
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DOI：
10.1109/focs.2015.44
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发表时间：
2015-04
期刊：
影响因子：
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通讯作者：
T. Eden;Amit Levi;D. Ron;C. Seshadhri
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文献类型：
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作者：
T. Eden;Amit Levi;D. Ron;C. Seshadhri
我们考虑了估计图中三角形的数量。三角形中的三角形，其中算法的查询访问允许的查询是度量查询，顶点对查询和邻居查询。估计帽子{t}使得具有高恒定概率，（1- epsilon）t <; hat {t}κ（1+epsilon）t，其中t是图G中的三角形的数量。该算法为O（n/t̂ {1/3} + min {m，m̂ {3/2}/t}）poly（log n，1/epsilon），其中n是图和m中的顶点数量是边缘的数量，预期的运行时间为（n/t̂ {1/3} + m̂ {3/2}/t）poly（log n，1/epsilon）。 {1/3} + min {m，m̂ {3/2}/t}）的查询是必要的，从而确定该算法的查询复杂性是最佳的，可以最佳地进行n（以及对1/epsilon的依赖性的polygarithmic因子） 。
We consider the problem of estimating the number of triangles in a graph. This problem has been extensively studied in both theory and practice, but all existing algorithms read the entire graph. In this work we design a sublinear-time algorithm for approximating the number of triangles in a graph, where the algorithm is given query access to the graph. The allowed queries are degree queries, vertex-pair queries and neighbor queries. We show that for any given approximation parameter 0<;epsilon<;1, the algorithm provides an estimate hat{t} such that with high constant probability, (1-epsilon) t<;hat{t}κ(1+epsilon)t, where t is the number of triangles in the graph G. The expected query complexity of the algorithm is O(n/t̂{1/3} + min {m, m̂{3/2}/t}) poly(log n, 1/epsilon), where n is the number of vertices in the graph and m is the number of edges, and the expected running time is (n/t̂{1/3} + m̂{3/2}/t) poly(log n, 1/epsilon). We also prove that Omega(n/t̂{1/3} + min {m, m̂{3/2}/t}) queries are necessary, thus establishing that the query complexity of this algorithm is optimal up to polylogarithmic factors in n (and the dependence on 1/epsilon).