On determining the congruity of point sets in higher dimensions

Abstract

This 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. First, this paper presents a randomized algorithm which works in O(n(d-1)/2(log n)2) time. This improves the previous result (an O(nd-2 log n) time deterministic algorithm). The birthday paradox, which is a well-known property in combinatorics, is used effectively in our algorithm. Next, this paper 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.