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 − ε, the first vertex is in this set. We show that for any ε, there exist such algorithms with K independent of the size of the input tree. Moreover, we provid...

We describe a polynomial time O(k log log OPTk(P ))approximation algorithm for the k-guarding problem of finding a minimum number, OPTk(P ), of vertex guards of an n-vertex simple polygon P so that for every point p ∈ P , the number of guards that see p is at least the minimum of k and the number of vertices that see p. Our approach finds O ( k ε log log 1 ε ) size (k, ε)-nets for instances of ...

Metric clustering is a fundamental primitive in machine learning with several applications for mining massive datasets. An important example of metric the k-center problem. While this problem has been extensively studied distributed settings, all previous algorithms use Ω(k) space per and Ω(n k) total work. In paper, we develop first highly scalable approximation algorithm clustering, O~(n^ε) O...

We prove that the sum of the squared Euclidean distances from the n rows of an n×d matrix A to any compact set that is spanned by k vectors in R can be approximated up to (1+ε)-factor, for an arbitrary small ε > 0, using the O(k/ε)-rank approximation of A and a constant. This implies, for example, that the optimal k-means clustering of the rows of A is (1+ε)approximated by an optimal k-means cl...

Aim of this paper is to investigate the existence of periodic solutions of a nonlinear planar autonomous system having a limit cycle x0 of least period T0 > 0 when it is perturbed by a small parameter, T1−periodic, perturbation. In the case when T0/T1 is a rational number l/k, with l, k prime numbers, we provide conditions to guarantee, for the parameter perturbation ε > 0 sufficiently small, t...

We consider variational problems defined on domains ‘weakly’ connected through a separation hyperplane (‘sieve plane’) by an ε-periodically distributed ‘contact zone’. We study the asymptotic behaviour as ε tends to 0 of integral functionals in such domains in the nonlinear and vector-valued case, showing that the asymptotic memory of the sieve is described by a nonlinear ‘capacitarytype’ formu...