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Reviewer: Joseph J. O'Rourke

Although I hope most computer scientists have heard of Voronoi diagrams, many would be surprised to learn that research on these diagrams suffices to fill (and overflow) a 500-page monograph. It is a testimony to the lasting richness of these diagrams that we possess such a wealth of research results. The basic Voronoi diagram is a partition of the plane generated by a finite set of n points or “sites.” Each point in the plane is assigned to its nearest site. Points with multiple affiliations under this assignment form the one-dimensional Voronoi diagram, whose roots extend back to the nineteenth century. The diagram and its dual, the Delaunay tessellation, encode proximity to the sites in a form so useful that these concepts have been rediscovered by researchers in a variety of fields a dozen times or more. This compendium of information on the Voronoi diagram gives all researchers easy access to common wisdom, and perhaps will cease the duplication of effort. The book falls into two parts, reflecting the expertise of the authors. The first half focuses on the geometric and computational aspects of Voronoi diagrams, including Sugihara's interests in computer science, and the second half turns more toward applications, incorporating the interests of Boots in geography and Okabe in environmental planning, with special emphasis on the statistical properties of random Voronoi diagrams. Combining these points of view in one monograph makes it a uniquely useful reference. Unfortunately, Wiley has priced it out of reach of most individuals. The book starts with 50 pages of mathematical preliminaries, covering everything from the definition of a derivative to stochastic processes. I found these preliminaries too terse to be useful. The authors could have shortened the book and reduced the price by omitting this chapter. The core of the first half follows: geometric properties of the Voronoi diagram and the Delaunay tessellation; generalization of the diagram in an amazing variety of directions; and algorithms for constructing the diagrams. The list of geometric properties will prove useful to a variety of researchers. The book is surprisingly up to date, including, for example, the characterization theorems of Dillencourt, results on Euclidean distance bounds in Delaunay triangulations, and the ray-shooting monotonicity property of DeFloriani and Nagy. The authors neglect to mention, however, that this last property has been beautifully generalized to arbitrary dimensions by Edelsbrunner [1]. The chapter on generalizations strives too hard to find an umbrella abstraction under which every known Voronoi variation fits, but this effort detracts only slightly from the chapter's usefulness. The figures in this chapter are especially stunning, illustrating, among other things, bisectors in a variety of metrics: Manhattan, supremum, Karlsruhe, and Hausdorff. Some recent work [2–5] is omitted, but the coverage is otherwise excellent. The chapter on algorithms is notable for an exposition of Sugihara and Iri's work on topological robustness. The explanation of Fortune's plane sweep algorithm by analogy with a flowing river is delightful. The description of higher-dimensional algorithms will be unsatisfying to browsers looking for specific information: it takes some effort to extract the fact that a three-dimensional diagram can be constructed in O n 2 time. The recent relevant results of Chazelle [6] are not included here. Fortune's recent survey [7] nicely supplements this chapter. The second half of the book commences with an exhaustive collection of information on random diagrams, especially those whose sites are generated by a Poisson process. Readers learn, for example, that a typical three-dimensional Voronoi cell has constant complexity: approximately 16 faces, 41 edges, and 27 vertices. The tables of data seem endless, but I have no doubt of their utility. The recent related results of Dwyer on point sets uniformly distributed in a bounded region [8] are not included in this chapter. Material on using Delaunay tessellations for interpolation, a major application, and the use of Voronoi diagrams in crystallography, in clustering for pattern recognition, and as a guide to optimization, follows. With more than 600 references, this book is an impressive piece of scholarship and will have enduring relevance.