In this paper we consider the classical Connected Dominating Set (CDS) problem. Twenty years ago, Guha and Khuller developed two algorithms for this problem - a centralized greedy approach with an approximation guarantee of H(D) +2, and a local greedy approach with an approximation guarantee of 2(H(D)+1) (where H() is the harmonic function, and D is the maximum degree in the graph). A local greedy algorithm uses significantly less information about the graph, and can be useful in a variety of contexts. However, a fundamental question remained - can we get a local greedy algorithm with the same performance guarantee as the global greedy algorithm without the penalty of the multiplicative factor of "2" in the approximation factor? In this paper, we answer that question in the affirmative.