Convertibility among Grid Filling Curves

Abstract

Consider a square grid graph Gn on n2 grid points 1,..., n x (1,..., n) in the plane, where every unit-distance pair of points are connected by an edge segment. A grid filling curve, or GFC for short, is a hamilton path of Gn between two corner vertices (1, 1) and (n, n). Since a parity argument shows that such a hamilton path exists only if n is odd, we will assume that n is odd throughout the paper. It is easy to construct a GFC under this assumption: a standard snakeorder traversal su?ces. We are, however, interested in generating random GFCs (see below for motivations). To this end, we de?ne a simple local operation called a flip that converts a GFC into another and apply a sequence of random ?ips to the standard GFC hoping that the result will be a random GFC in some useful sense. The ?rst thing we would want to establish in such a random generation method is that every GFC has a positive probability of being generated. We show in this paper that it is indeed the case: our result states that given any two GFCs, there is a sequence of ?ips that converts one into the other.

Grid filling curves are not only interesting by themselves but also applied to various problems, such as e?cient disk seek methods3 and many others. One typical application among them is found in digital halftoning which is a technique to convert an image with several bits for brightness levels into a binary image consisting of black and white dots. One well-known method is the one called “error di?usion” 5 which propagates quantization errors to neighboring pixels following the order of raster scan.