Sorting Spatial Data for Sampling and Other Geographic Applications

Abstract

This paper presents some new methods for ordering spatial entities to reflect spatial proximity. The ordering methods work not only for point sets, but also for a variety of types of 1-D, 2-D, and higher-dimensional spatial objects. The paper describes some important, less traditional applications for spatially sorted data, including list frames for systematic sampling and efficient organization of hierarchical geographic neighborhoods. The new methods derive from methods for ordering vertices or edges in a tree (connected acyclic graph), making novel use of an Eulerian tour to assign a cyclic order. The new orderings are canonical in the sense that they are coordinate system independent, rotation invariant, and do not depend on prior bucketing of space as do some standard existing methods. They depend instead on the spatial distribution of the input data and on the metric of the underlying space. We call our new orderings tree-orders. They can be constructed in linear time from topological graph data structures; and we show that they are fully characterized by a useful proximity-preserving property called branch-recursion.