Title: A Truly Constant-time Distribution-aware Negative Sampling. Abstract: Softmax classifiers with a very large number of classes naturally occur in many applications such as natural language processing and information retrieval. The calculation of full-softmax is very expensive from the computational and energy perspective. There have been a variety of sampling approaches to overcome this challenge, popularly known as negative sampling (NS). Ideally, NS should sample negative classes from a distribution that is dependent on the input data, the current parameters, and the correct positive class. Unfortunately, due to the dynamically updated parameters and data samples, there does not exist any sampling scheme that is truly adaptive and also samples the negative classes in constant time every iteration. Therefore, alternative heuristics like random sampling,  static frequency-based sampling, or learning-based biased sampling; which primarily trade either the sampling cost or the adaptivity of samples per iteration, are adopted. In this paper, we show a class of distribution where the sampling scheme is truly adaptive and provably generates negative samples in constant time. We demonstrate a negative sampling implementation that is significantly faster, in terms of wall clock time, compared to the most optimized TensorFlow implementations of standard softmax or other sampling approaches on the best available GPUs (V100s).