On determining the congruence of point sets in d dimensions

AbstractThis paper considers the following problem: given two point sets A and B (|A| = |B| = n) in d dimensional Euclidean space, determine whether or not A is congruent to B. This paper presents an O(n(d−1)2 log n) time randomized algorithm. The birthday paradox, which is well-known in combinatorics, is used effectively in this algorithm. Although this algorithm is Monte-Carlo type (i.e., it may give a wrong result), this improves a previous O(nd−2 log n) time deterministic algorithm considerably. This paper also shows that if d is not bounded, the problem is at least as hard as the graph isomorphism problem in the sense of the polynomiality. Several related results are described too