The theory of quantum algorithms promises unprecedented benefits harnessing the laws mechanics for solving certain computational problems. A prerequisite applying to a wide range real-world problems is loading classical data state. Several circuit-based methods have been proposed encoding as probability amplitudes However, in these methods, either circuit depth or width must grow linearly with ...

We study metric properties of sets which are characterized by their approximation smooth curves at scales positive density and relationships with known problems in complex dynamics analysis. Three direct applications towards removability for bounded analytic functions, optimal bounds Traveling Salesman problem, Hausdorff dimension unfolding attractors discussed the paper. The main technical ing...

In this paper we present a randomized constant factor approximation algorithm for the problem of computing the optimal cost of the metric Minimum Facility Location problem, in the case of uniform costs and uniform demands, and in which every point can open a facility. By exploiting the fact that we are approximating the optimal cost without computing an actual solution, we give the first algori...

This result was originally attributed to Nemhauser, Wolsey, and Fisher [4], but they only claimed this result without a proof. Jenkyns was the first to give a full proof of this result. We will discuss a simplified proof due to Calinescu et al [1]. Proof. If we have (α+1)-approximation for incremental oracle then approximation bound is αk+1. Let β ≤ 1 and β = 1/α. Then we want f(Greedy) ≥ β k +...