This paper addresses combinatorial optimization schemes for solving the multicriteria Steiner tree problem for communication network topology design (e.g., wireless mesh network). The solving scheme is based on several models: multicriteria ranking, clustering, minimum spanning tree, and minimum Steiner tree problem. An illustrative numerical example corresponds to designing a covering long-dis...

Given a set of i (i = 1; 2; : : :; k) orientations (angles) in the plane, one can deene a distance function which induces a metric in the plane, called the orientation metric 3]. In the special case where all the angles are equal, we call the metric a uniform orientation metric 2]. Speciically, if there are orientations, forming angles ii ; 0 i ?1, with the x-axis, where 2 is an integer, we cal...

Given a directed graph G = (V,E) and an integer k ≥ 1, a Steiner k-transitive-closure-spanner (Steiner k-TC-spanner) of G is a directed graph H = (VH , EH) such that (1) V ⊆ VH and (2) for all vertices v, u ∈ V , the distance from v to u in H is at most k if u is reachable from v in G, and ∞ otherwise. Motivated by applications to property reconstruction and access control hierarchies, we conce...

In this papel; we present eficient parallel algorithms forjinding a minimum weighted connected dominating set, a minimum weighted Steiner tree for a distance-hereditary graph which take O(1og n) time using O(n+m) processors on a CRCW PRAM, where n and m are the number of vertices and edges of a given graph, respectively. We also find a maximum weighted clique of a distance-hereditary graph in O...

In this note we present a way to approximate the Steiner problem by a family of elliptic energies of Modica-Mortola type, with an additional term relying on a weighted geodesic distance which takes care of the connectedness constraint. Résumé. Dans cette note nous présentons une méthode d’approximation du problème de Steiner par une famille de fonctionnelles de type Modica-Mortola, avec un term...

We give an overview of some of the mathematical results of Dominique de Caen. These include a short proof of König’s theorem, results on Turán numbers, biclique partitions, clique coverings, hypergraphical Steiner systems, p-ranks, stellar permutations, the construction of new distance-regular graphs, large sets of equiangular lines, a bound on the probability of a union, and more.

Given a family of graphs F , and graph G ∈ F with weights on the edges, the vertices of G are partitioned into terminals T and Steiner nodes S. The shortest paths (according to edge weights) define a metric on the set of vertices. We wish to embed the set T in a weighted graph G′ ∈ F such that the distance between any two vertices x, y ∈ T in the graph G′ is “close” to their distance in G. More...

This paper gives the average distance analysis for the Euclidean tree constructed by a simple greedy but efficient algorithm of the on-line Steiner tree problem. The algorithm accepts the data one by one following the order of input sequence. When a point arrives, the algorithm adds the shortest edge, between the new point and the points arriving already, to the previously constructed tree to f...

Geometric spanner is a fundamental structure in computational geometry and plays an important role in many geometric networks design applications. In this paper, we consider a generalization of the classical geometric spanner problem (called segment spanner): Given a set S of disjoint 2-D segments, find a spanning network G with minimum size so that for any pair of points in S, there exists a p...

The concept of $n$-distance was recently introduced to generalize the classical definition distance functions $n$ arguments. In this paper we investigate through a number examples based on certain geometrical constructions. particular, our study shows which extent computation best constant associated with an may sometimes be difficult and tricky. It also reveals that two important graph theoret...