Fast Deterministic Selection.

Abstract

The selection problem, in forms such as finding the median or choosing the k top ranked items in a dataset, is a core task in computing with numerous applications in fields as diverse as statistics, databases, Machine Learning, finance, biology, and graphics. The selection algorithm Median of Medians, although a landmark theoretical achievement, is seldom used in practice because it is slower than simple approaches based on sampling. The main contribution of this paper is a fast linear-time deterministic selection algorithm MedianOfNinthers based on a refined definition of MedianOfMedians. A complementary algorithm MedianOfExtrema is also proposed. These algorithms work together to solve the selection problem in guaranteed linear time, faster than state-of-the-art baselines, and without resorting to randomization, heuristics, or fallback approaches for pathological cases. We demonstrate results on uniformly distributed random numbers, typical low-entropy artificial datasets, and real-world data. Measurements are open-sourced alongside the implementation at https://github.com/andralex/MedianOfNinthers.