Tis a lesson you should heed: Try, try, try again. If at first you don't succeed, Try, try, try again. (William Edward Hickson, 19th century educational writer)
AbstractLet G = (V, E) be an n-vertex graph and M d a d-vertex graph, for some constant d. Is M d a subgraph of G? We consider this problem in a model where all n processes are connected to all other processes, and each message contains up to O(log n) bits. A simple deterministic algorithm that requires O(n (d−2)/d / log n) communication rounds is presented. For the special case that M d is a triangle, we present a probabilistic algorithm that requires an expected O( n 1/3 /(t 2/3 + 1) ) rounds of communication, where t is the number of triangles in the graph, and O(min{n 1/3 log 2/3 n/(t 2/3 + 1), n 1/3 }) with high probability.We also present deterministic algorithms specially suited for sparse graphs. In any graph of maximum degree ∆, we can test for arbitrary subgraphs of diameter D in O( ∆ D+1 /n ) rounds. For triangles, we devise an algorithm featuring a round complexity of O(A 2 /n + log 2+n/A 2 n), where A denotes the arboricity of G.
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