Fault-Tolerant Placement of Additional Mesh Nodes in Rural Wireless Mesh Networks: A Minimum Steiner Tree Based Centre of Mass With Bounded Edge Length

Wireless mesh networks are presented as an attractive solution to reduce the digital divide between rural and developed areas. In a multi-hop fashion, they can cover larger spaces. However, their planning is subject to many constraints including robustness. In fact, the failure of a node may result in the partitioning of the network. The robustness of the network is therefore achieved by carefully placing additional nodes. This work tackles the problem of additional nodes minimization when planning bi and tri-connectivity from a given network. We propose a vertex augmentation approach inspired by the placement of Steiner points. The idea is to incrementally determine cut vertices and bridges in the network and to carefully place additional nodes to ensure connectivity, bi and tri-connectivity. The approach relies on an algorithm using the centre of mass of the blocks derived after the partitioning of the network. The proposed approach has been compared to a modified version of a former approach based on the Minimum Steiner Tree. The different experiments carried out show the competitiveness of the proposed approach to connect, bi-connect, and tri-connect the wireless mesh networks.

Generalized Regularity and the Symmetry of Branches of ''Botanological'' Networks

We derive the generalized regularity of convex quadrilaterals in R^2, which gives a new evolutionary class of convex quadrilaterals that we call generalized regular quadrilaterals in R^2. The property of generalized regularity states that the Simpson line defined by the two Steiner points passes through the corresponding Fermat-Torricelli point of the same convex quadrilateral. We prove that a class of generalized regular convex quadrilaterals consists of convex quadrilaterals, such that their two opposite sides are parallel. We solve the problem of vertical evolution of a ''botanological'' thumb (a two way communication weighted network) w.r to a boundary rectangle in R^2 having two roots,two branches and without having a main branch, by applying the property of generalized regularity of weighted rectangles. We show that the two branches have equal weights and the two roots have equal weights, if the thumb inherits a symmetry w.r to the midperpendicular line of the two opposite sides of the rectangle, which is perpendicular to the ground (equal branches and equal roots). The geometric, rotational and dynamic plasticity of weighted networks for boundary generalized regular tetrahedra and weighted regular tetrahedra lead to the creation of ''botanological'' thumbs and ''botanological'' networks (with a main branch) having symmetrical branches

Shortest Directed Networks in the Plane

Given a set of sources and a set of sinks as points in the Euclidean plane, a directed network is a directed graph drawn in the plane with a directed path from each source to each sink. Such a network may contain nodes other than the given sources and sinks, called Steiner points. We characterize the local structure of the Steiner points in all shortest-length directed networks in the Euclidean plane. This characterization implies that these networks are constructible by straightedge and compass. Our results build on unpublished work of Alfaro, Campbell, Sher, and Soto from 1989 and 1990. Part of the proof is based on a new method that uses other norms in the plane. This approach gives more conceptual proofs of some of their results, and as a consequence, we also obtain results on shortest directed networks for these norms.

Reorganizing topologies of Steiner trees to accelerate their eliminations

We describe a technique to reorganize topologies of Steiner trees by exchanging neighbors of adjacent Steiner points. We explain how to use the systematic way of building trees, and therefore topologies, to find the correct topology after nodes have been exchanged. Topology reorganizations can be inserted into the enumeration scheme commonly used by exact algorithms for the Euclidean Steiner tree problem in [Formula: see text]-space, providing a method of improvement different than the usual approaches. As an example, we show how topology reorganizations can be used to dynamically change the exploration of the usual branch-and-bound tree when two Steiner points collide during the optimization process. We also turn our attention to the erroneous use of a pre-optimization lower bound in the original algorithm and give an example to confirm its usage is incorrect. In order to provide numerical results on correct solutions, we use planar equilateral points to quickly compute this lower bound, even in dimensions higher than two. Finally, we describe planar twin trees, identical trees yielded by different topologies, whose generalization to higher dimensions could open a new way of building Steiner trees.

Hybrid greedy genetic algorithm for the Euclidean Steiner tree problem

This paper presents a genetic algorithm for the Euclidean Steiner tree problem. This is an optimization problem whose objective is to obtain a minimum length tree to interconnect a set of fixed points, and for this purpose to be achieved, new auxiliary points, called Steiner points, can be added. The proposed heuristic uses a genetic algorithm to manipulate spanning trees, which are then transformed into Steiner trees by inserting and repositioning the Steiner points. Greedy genetic operators and evolutionary strategies are tested. Results of numerical experiments for benchmark library problem (OR-Library) are presented and discussed.This paper presents a genetic algorithm for the Euclidean Steiner tree problem. This is an optimization problem whose objective is to obtain a minimum length tree to interconnect a set of fixed points, and for this purpose to be achieved, new auxiliary points, called Steiner points, can be added. The proposed heuristic uses a genetic algorithm to manipulate spanning trees, which are then transformed into Steiner trees by inserting and repositioning the Steiner points. Greedy genetic operators and evolutionary strategies are tested. Results of numerical experiments for benchmark library problem (OR-Library) are presented and discussed.