Properties of the k‐centra in a tree network

Abstract

The k-centrum of a graph G = ( V , E ) is the set of vertices each minimizing the sum of the distances from it to the corresponding k farthest vertices of G. In this article properties of k-centrum of trees, where edges have positive real lengths, are investigated.

Finding an Euclidean anti-k-centrum location of a set of points

“…Also, the extension of the results to distances derived from norms different than the Euclidean is a task that will require specific, complementary tools, so will be discussed elsewhere. 1 ord . Therefore, the processing of the half-bisector is finished.…”

“…, k, in total time O(kn 3 + n 3 log n) when the number of nonempty forbidden regions is fixed. 1 In this case x is a triple point and therefore, by Lemma 5, x is an orderk-Voronoi vertex for k = |C < x | + 1 and also k = |C < x | + 2. 2 In this case, x is an inner point of an arc e of bis(pi, pj) defined by ×C ij ∪ and, by Lemma 3, x is an inner point of an edge of the order-k-Voronoi diagram for k = |C < e | + 1(= |CP| + 1).…”

“…The attractive version of this problem, minimizing the largest sum of distances, was introduced in the context of a tree graph by Slater [18], who proved structural properties and relations between the k-centrum vertices for consecutive values of k. Later Andreatta and Mason [1] extended these results to networks with arbitrary edge-lengths, Tamir [19] discussed efficient algorithms for solving kcentrum problems for several types of graph and network structures. Finally, in continuous space Ogryczak and Tamir [13] have showed how to formulate the 1 (or, similarly, ∞ ) version of the problem as a linear program, solvable in O(n) time using the results of Zemel [20].…”

Parking slot assignment for urban distribution: Models and formulations

“…Thus, to compute the extra time we assume that exactly c requests are assigned just at the end of the interval (one at each place), and that these assignments correspond to the c largest durations of the requests indexed in set Q α;β . That is, the extra time coincides with the ccentrum [22][23][24][25] …”

Worst‐case incremental analysis for a class of p‐facility location problems

“…Single-facility k-centrum problems were first discussed in Slater [15] and Andreatta and Mason [1,2]. The reader is referred to Tamir [16] for structural and algorithmic results concerning the multifacility problem.…”