Finding squares and rectangles in sets of points

Abstract

The following problem is studied: Given a set S of n points in the plane, does it contain a subset of four points that form the vertices of a square or rectangle. Both the axis-parallel case and the arbitrarily oriented case are studied. We also investigate extensions to the d-dimensional case. Algorithms are obtained that run in O(n ^1+1/dlog n) time for axis-parallel squares and O(n ^2−1/d) time for axis-parallel rectangles. For arbitrarily oriented squares the time bounds are O(n ²log n), O n ³) and O(n ^d−1/2β(n)) for d=2, d=3 and d≥4, respectively (where β(n) is related to the inverse of Ackermann's function), whereas the algorithm for arbitrarily oriented rectangles takes time O(n ^dlogn). Furthermore, it is shown that recognizing axisparallel rectangles is equivalent to recognizing a K _2,2-subgraph in a bipartite graph, resulting in a O(‖E‖ √‖E‖) time and O(‖V‖ + ‖E‖) space solution to this problem. Also, combinatorial results on the maximal number of squares and rectangles any point set can contain are given.

The work of this author is supported by the Dutch Organisation for Scientific Research (N.W.O.).