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Minimal Networks: A Review

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Advances in Dynamical Systems and Control

Part of the book series: Studies in Systems, Decision and Control ((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.

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Notes

  1. 1.

    Let the one that did not appreciate this method, solve the following problem: for given three points find the fourth one such that the total lengths of the three segments connecting it with the given three points takes a minimal value.

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  1. Mechanical and Mathematical Faculty, Lomonosov Moscow State University, GSP-1, Leninskie Gory, Moscow, 119991, Russian Federation

    Alexander O. Ivanov & Alexey A. Tuzhilin

  2. Bauman Moscow Technical University, ul. Baumanskay 2-ya, 5, Moscow, 105005, Russia

    Alexander O. Ivanov

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  1. Alexander O. Ivanov
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  1. Lomonosov Moscow State University, Moscow, Russia

    Victor A. Sadovnichiy

  2. Kyiv Polytechnic Institute, National Technical University of Ukraine, Kiev, Ukraine

    Mikhail Z. Zgurovsky

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Ivanov, A.O., Tuzhilin, A.A. (2016). Minimal Networks: A Review. In: Sadovnichiy, V., Zgurovsky, M. (eds) Advances in Dynamical Systems and Control. Studies in Systems, Decision and Control, vol 69. Springer, Cham. https://doi.org/10.1007/978-3-319-40673-2_4

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