We give a deterministic O(n log n)-time algorithm to decide if two n-point sets in 4-dimensional Euclidean space are the same up to rotations and translations. It has been conjectured that O(n log n) algorithms should exist for any fixed dimension. The best algorithms in d-space so far are a deterministic algorithm by Brass and Knauer [Int. J. Comput. Geom. Appl., 2000] and a randomized Monte Carlo algorithm by Akutsu [Comp. Geom., 1998]. They take time O(n^2 log n) and O(n^(3/2) log n) respectively in 4-space. Our algorithm exploits many geometric structures and properties of 4-dimensional space.
['Heuna Kim', 'Günter Rote']
Congruence Testing of Point Sets in 4-Space.