Computational geometry: an introduction
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Transitions in geometric minimum spanning trees (extended abstract)
SCG '91 Proceedings of the seventh annual symposium on Computational geometry
Data Structures and Algorithms
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Computers and Intractability: A Guide to the Theory of NP-Completeness
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The Design and Analysis of Computer Algorithms
The Design and Analysis of Computer Algorithms
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This paper presents some new methods for ordering spatial entitiesto reflect spatial proximity. The ordering methods work not only for pointsets, but also for a variety of types of 1-D, 2-D, and higher-dimensionalspatial objects. The paper describes some important, less traditionalapplications for spatially sorted data, including list frames for systematicsampling and efficient organization of hierarchical geographicneighborhoods. The new methods derive from methods for ordering vertices oredges in a tree (connected acyclic graph), making novel use of an Euleriantour to assign a cyclic order. The new orderings are canonical in the sensethat they are coordinate system independent, rotation invariant, and do notdepend on prior bucketing of space as do some standard existing methods.They depend instead on the spatial distribution of the input data and on themetric of the underlying space. We call our new orderings tree-orders. Theycan be constructed in linear time from topological graph data structures;and we show that they are fully characterized by a usefulproximity-preserving property called branch-recursion.