For the list object, introduced in Chapter 5, it was shown that each data element contains at most one predecessor element and one successor element. Therefore, for any given data element or node in the list structure, we can talk in terms of a next element and a previous element.

Graphs and Digraphs

Computational geometry.

Voronoi diagrams partition space according to the influence certain sites exert on their environment. Since the 17th century, such structures play an important role in many areas like Astronomy, Physics, Chemistry, Biology, Ecology, Economics, Mathematics and Computer Science. They help to describe zones of political influence, to determine the hospital nearest to an accident site, to compute collision-free paths for mobile robots, to reconstruct curves and surfaces from sample points, to refine triangular meshes, and to design location strategies for competing markets. This unique book offers a state-of-the-art view of Voronoi diagrams and their structure, and it provides efficient algorithms towards their computation. Readers with an entry-level background in algorithms can enjoy a guided tour of gently increasing difficulty through a fascinating area. Lecturers might find this volume a welcome source for their courses on computational geometry. Experts are offered a broader view, including many alternative solutions, and up-to-date references to the existing literature; they might benefit in their own research or application development. Readership: Students of mathematics and computer science, scientists and engineers working in mathematics, natural sciences and economics.

Voronoi Diagrams and Delaunay Triangulations

Handbook of Approximation: Algorithms and Metaheuristics

We improve previous lower bounds on the number of simple polygonizations, and other kinds of crossing-free subgraphs, of a set of N points in the plane by analyzing a suitable configuration. We also prove that the number of crossing-free perfect matchings and spanning trees is minimum when the points are in convex position.

Lower bounds on the number of crossing-free subgraphs of K N

Let G be a connected plane geometric graph with n vertices. In this paper, we study bounds on the number of edges required to be added to G to obtain 2-vertex or 2-edge connected plane geometric graphs. In particular, we show that for G to become 2-edge connected, 2n3 additional edges are required in some cases and that 6n7 additional edges are always sufficient. For the special case of plane geometric trees, these bounds decrease to n2 and 2n3, respectively.

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Augmenting the connectivity of geometric graphs

Given a finite set of N nodes and the time required for traveling among nodes, in the traveling repairman problem, we seek a route that minimizes the sums of the delays for reaching each node. In this note, we present a linear algorithm for solving the traveling repairman problem when the underlying graph is a path, improving the Θ(N2) time and space complexity of the previously best algorithm for this problem. We also provide a linear algorithm for solving the walk problem with deadlines (WPD) on paths. © 2002 Wiley Periodicals, Inc.

A note on the traveling repairman problem

We consider combinatorial and computational issues that are related to the problem of moving coins from one configuration to another. Coins are defined as non-overlapping discs, and moves are defined as collision free translations, all in the Euclidean plane. We obtain combinatorial bounds on the number of moves that are necessary and/or sufficient to move coins from one configuration to another. We also consider several decision problems related to coin moving, and obtain some results regarding their computational complexity.

Moving coins

In this study, we provide methods for drawing a tree with n vertices on a convex polygon, without crossings and using the minimum number of edges of the polygon. We apply the results to obtain planar packings of two trees in some specific cases. © 2002 Wiley Periodicals, Inc. J Graph Theory 40: 172–181, 2002

Javier Tejel

Papers

Lower bounds on the number of crossing-free subgraphs of K N

Augmenting the connectivity of geometric graphs

A note on the traveling repairman problem

Moving coins

Packing trees into planar graphs