Finding Adam in random growing trees

Abstract: We investigate algorithms to find the first vertex in large trees generated by either the uniform attachment or preferential attachment model. We require the algorithm to output a set of $K$ vertices, such that, with probability at least $1-\epsilon$, the first vertex is in this set. We show that for any $\epsilon$, there exist such algorithms with $K$ independent of the size of the input tree. Moreover, we provide almost tight bounds for the best value of $K$ as a function of $\epsilon$. In the uniform attachment case we show that the optimal $K$ is subpolynomial in $1/\epsilon$, and that it has to be at least superpolylogarithmic. On the other hand, the preferential attachment case is exponentially harder, as we prove that the best $K$ is polynomial in $1/\epsilon$. We conclude the paper with several open problems.