The nearest neighbor graph is an important structure in many data mining methods for clustering, advertising, recommender systems, and outlier detection. Constructing the graph requires computing up to n2 similarities for a set of n objects. This high complexity has led researchers to seek approximate methods, which find many but not all of the nearest neighbors. In contrast, we leverage shared memory parallelism and recent advances in similarity joins to solve the problem exactly. Our method considers all pairs of potential neighbors but quickly filters pairs that could not be a part of the nearest neighbor graph, based on similarity upper bound estimates. The filtering is data dependent and not easily predicted, which poses load balance challenges in parallel execution. We evaluated our methods on several real-world datasets and found they work up to two orders of magnitude faster than existing methods, display linear strong scaling characteristics, and incur less than 1% load imbalance during filtering.