Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.Variations of k-d trees represent a fundamental data structure used in Computational Geometry with numerous applications in science. For example particle track tting in the software of the LHC experiments, and in simulations of N-body systems in the study of dynamics of interacting galaxies, particle beam physics, and molecular dynamics in biochemistry. The many-body tree methods devised by Barnes and Hutt in the 1980s and the Fast Multipole Method introduced in 1987 by Greengard and Rokhlin use variants of k-d trees to reduce the computation time upper bounds to O(n log n) and even O(n) from O(n2). We present an algorithm that uses the principle of well-separated pairs decomposition to always produce compressed trees in O(n log n) work. We present and evaluate parallel implementations for the algorithm that can take advantage of multi-core architectures.The Science and Technology Facilities Council, UK
