Matching Points with Squares

Abstract

Given a class C of geometric objects and a point set P , a C-matching of P is a set M = {C 1 , . . . , C k } ⊆ C of elements of C such that each C i contains exactly two elements of P and each element of P lies in at most one C i . If all of the elements of P belong to some C i , M is called a perfect matching. If, in addition, all of the elements of M are pairwise disjoint, we say that this matching M is strong. In this paper we study the existence and characteristics of C-matchings for point sets in the plane when C is the set of isothetic squares in the plane. A consequence of our results is a proof that the Delaunay triangulations for the L ∞ metric and the L 1 metric always admit a Hamiltonian path.

Matching points with disks with a common intersection

“…Observe that our result goes in the direction opposite to that of known results on matching red and blue points: Our goal is that all matching objects have a common intersection, whereas in previous work (e.g., [2,3,4,5,6,7,8], and the references in [8]) it is required that all matching objects are pairwise disjoint.…”

“…To show the lemma it suffices to prove that (x, y) = ( −λβ+α 1+λ 2 , λα+β 1+λ 2 ) satisfies equation (5).…”

“…This class of problems is well studied in discrete and computational geometry, starting from the classic result that n red points and n blue points can always be perfectly matched with n pairwise non-crossing segments, where each segment connects a red point with a blue point [1]. The study has been continued in plenty of directions, for both the monochromatic and bichromatic versions, by using pairwise disjoint segments [2,3], rectangles and squares [4,5,6,7], and more general geometric objects [8].…”

Fixed-orientation equilateral triangle matching of point sets

“…The class of down-triangles (and up-triangles) admits a shrinkability property [5]: each triangle object in this class that contains two points p and q, can be shrunk such that p and q lie on its boundary. It is also clear that we can continue the shrinking process-from the edge that does not contain neither p or q-until at least one of the points, p or q, becomes a triangle vertex and the other point lies on the edge opposite to this vertex.…”

“…Given a point set P and a class C of geometric objects, the maximum C -matching problem is to compute a subclass C ′ of C of maximum cardinality such that no point from P belongs to more than one element of C ′ and for each C ∈ C ′ , there are exactly two points from P which lie inside C. Dillencourt [4] proved that every point set admits a perfect circle-matching.Ábrego et al [5] studied the isothetic square matching problem. Bereg et al concentrated on matching points using axis-aligned squares and rectangles [6].…”