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Minimal Networks: A Review

  • Alexander O. Ivanov
  • Alexey A. Tuzhilin
Chapter
Part of the Studies in Systems, Decision and Control book series (SSDC, volume 69)

Abstract

Minimal Networks Theory is a branch of mathematics that goes back to 17th century and unites ideas and methods of metric, differential, and combinatorial geometry and optimization theory. It is still studied intensively, due to many important applications such as transportation problem, chip design, evolution theory, molecular biology, etc. In this review we point out several significant directions of the Theory. We also state some open problems which solution seems to be crucial for the further development of the Theory. Minimal Networks can be considered as one-dimensional minimal surfaces. The simplest example of such a network is a shortest curve or, more generally, a geodesic. The first ones are global minima of the length functional considered on the curves connecting fixed boundary points. The second ones are the curves such that each sufficiently small part of them is a shortest curve. A natural generalization of the problem appears, if the boundary consists of three and more points, and additional branching points are permitted. Steiner minimal trees are analogues of the shortest curves, and locally minimal networks are generalizations of geodesics. We also include some results concerning so-called minimal fillings and minimal networks in the spaces of compacts.

Keywords

Minimal Span Tree Ambient Space Euclidean Plane Minimal Tree Boundary Vertex 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Mechanical and Mathematical FacultyLomonosov Moscow State UniversityMoscowRussian Federation
  2. 2.Bauman Moscow Technical UniversityMoscowRussia

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