The problem of private publication of graph data has attracted a lot of attention recently. The prevalence of differential privacy makes the problem more promising. However, a large body of existing works on differentially private release of graphs have not answered the question about the upper bounds of privacy budgets. In this paper, for the first time, such a bound is provided. We prove that with a privacy budget of O(log n), there exists an algorithm capable of releasing a noisy output graph with edge edit distance of O(1) against the true graph. At the same time, the complexity of our algorithm Top-m Filter is linear in the number of edges m. This lifts the limits of the state-of-the-art, which incur a complexity of O(n^2) where n is the number of nodes and runnable only on graphs having n of tens of thousands.