Browse Books

Reviewer: Harvey Cohn

Spanners may be used in computational geometry for solving some proximity problems. They have also found applications in other areas, such as in motion planning in telecommunication networks: network reliability and optimization of roaming in mobile networks. A typical problem involves creating a system of roads (edges) to connect a given set of cities (vertices) in optimal manner in terms of mileage (thus the edges are weighted). Obviously, a complete graph would connect each pair of edges, giving maximal total weight (mileage), but minimal weight between two cities. A sparse graph, on the other hand, would do the opposite. Say, for example, the obvious parameter of t, where t>1 is the maximal ratio of the distance between two cities along the minimal network path to the direct distance between the cities. Such a graph is called a "t-spanner network," and, in practice, it can only be found approximately. An obvious method is to compare network roads with the direct roads between pairs of cities, which requires heavy terminology and patient use of geometry. The introduction of new points (junction cities) might improve the optimization. For the given t, what emerges is a nondeterministic polynomial-time hard (NP-hard) problem that can only be approximated. Each chapter of the book begins with a summary, often in nontechnical terms, and a sometimes-relevant historical quotation. Illustrations are plentiful, including maps of US road networks for different values of t approaching 1. As the book progresses, the definitions, methods, and algorithms become increasingly arcane, so that only the first few chapters are likely to achieve a universal response, despite the patient explanations of the plethoric neologisms. In any case, this book belongs in departmental libraries as a landmark compendium. In this book, Narasimhan and Smid celebrate the completion of a revolution. Galileo is quoted to the effect that his new conception of empirical science requires the use of mathematics, which was then, largely, geometry. After Newton, mathematics expanded progressively in scope, but graph theory arose a century ago with a naivety that seemingly rejected classical mathematics. Yet, here, famous "pure" names not ordinarily linked to networks are on display, such as Delaunay, Voronoi, Steiner, and Millnor. So graphs seem to "come home" and find their place as classical Euclidean objects, and not just abstract drawings of vertices and edges. Online Computing Reviews Service