An improved algorithm for the p-center problem on interval graphs with unit lengths

Abstract

The p-center problem is to locate p facilities in a network of n demand points so as to minimize the longest distance between a demand point and its nearest facility. We consider this problem by modelling the network as an interval graph whose edges all have unit lengths. We present an O(n) time algorithm for the problem under the assumption that the endpoints of the intervals are sorted, which improves on the existing best algorithm for the problem that has a run time of O(pn).

Modelling gateway placement in wireless networks: Geometric k-centres of unit disc graphs

“…Therefore, by (14), (15), and since g(q) ∈ f (P ), ∀p ∈ f (P ), ∃q ∈ F such that dist UDG(P ) p, g(q)…”

“…The algorithm of Cheng et al [15] finds a vertex k-centre in O (n) time by checking whether there exists a solution of radius r v for each of the two possible integer values for r v bounded by (17). By Observation 16 and Lemma 17, the geometric k-radius of a connected interval graph is at most…”

“…Bespamyatnikh et al [4] present an O (nk)-time algorithm for finding a vertex k-centre of a circular-arc graph (and therefore also for any interval graph). Cheng et al [15] improve the running time with an O (n)-time algorithm for finding a vertex k-centre of an interval graph. Each of these algorithms requires that the input list of intervals be sorted (e.g., by left endpoints).…”

Solving the constrained p-center problem using heuristic algorithms

“…In addition to exact algorithms, many approximation and heuristic algorithms have been suggested to solve the p-center problem [4,[7][8][9]. Caruso et al [10] and Cheng et al [11] have discussed the p-center on the graphs and trees with the aim of to solve a variety of problems, such as the location of client/server problems or the location on the street line of a city. On the other hand, some papers discuss continuous p-center problems.…”

Rapidly converging solution for p-centers in nonconvex regions

“…Over time it has helped to solve all variations of the location problem and eventually it was accepted as an efficient and effective choice for solving spatial problems. In [12,13] the p−center was considered for demand planes depicted by graphs and trees, such as client/server problems. This paper attempts to solve the polygonal p-center problem with the help of various computational geometric structures.…”