Efficient Algorithms for the Weighted 2-Center Problem in a Cactus Graph

Ben‐Moshe, Boaz; Bhattacharya, Binay; Shi, Qiaosheng

doi:10.1007/11602613_70

Cited by 20 publications

(12 citation statements)

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“…In particular, if a vertex is irrelevant, then its weight is +∞. 1 An obnoxious center is a point a * ∈ A(G) such that COST(a * ) = max a∈A(G) COST(a).…”

Section: Preliminariesmentioning

confidence: 99%

“…The problem arises naturally when considering the placement of an undesirable facility that will affect the environment, or, in a dual setting, when searching for a place away from existing obnoxious facilities. Algorithmically, obnoxious facilities have received much attention previously; see [1,2,7,11,15,23,25,26,27] and references therein.In this paper, we consider the problem of placing a single obnoxious facility in a graph, either at its vertices or along its edges; this is often referred to as the continuous problem, as opposed to the discrete version, where the facility has to be placed in a vertex of G. A formal definition of the problem is given in Section 2.1. We use n, m for the number of vertices and edges of G, respectively.…”

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confidence: 99%

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confidence: 99%

“…Faster algorithms are known for some special classes of trees [7,25]. For cactus graphs, Zmazek andŽerovnik [27] gave an algorithm using O(cn) time, where c is the number of different weights in the sites, and recently Ben-Moshe, Bhattacharya, and Shi [1] showed an algorithm using O(n log 3 n) time. For general graphs, Tamir [24] showed how to solve the obnoxious center problem in O(nm + n 2 log n) time.…”

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Obnoxious Centers in Graphs

Cabello¹,

Rote²

2010

SIAM J. Discrete Math.

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Abstract. We consider the problem of finding obnoxious centers in graphs. For arbitrary graphs with n vertices and m edges, we give a randomized algorithm with O(n log 2 n + m log n) expected time. For planar graphs, we give algorithms with O(n log n) expected time and O(n log 3 n) worstcase time. For graphs with bounded treewidth, we give an algorithm taking O(n log n) worst-case time. The algorithms make use of parametric search and several results for computing distances on graphs of bounded treewidth and planar graphs.Key words. graph algorithms, facility location, planar graphs, parametric search, bounded treewidth AMS subject classifications. 05C85, 68W05, 90B851. Introduction. A central problem in locational analysis deals with the placement of new facilities that optimize a given objective function. In the obnoxious center problem, there is set of sites in some metric space, each with its own weight, and we want to place a facility that maximizes the minimum of the weighted distances from the given sites. The problem arises naturally when considering the placement of an undesirable facility that will affect the environment, or, in a dual setting, when searching for a place away from existing obnoxious facilities. Algorithmically, obnoxious facilities have received much attention previously; see [1,2,7,11,15,23,25,26,27] and references therein.In this paper, we consider the problem of placing a single obnoxious facility in a graph, either at its vertices or along its edges; this is often referred to as the continuous problem, as opposed to the discrete version, where the facility has to be placed in a vertex of G. A formal definition of the problem is given in Section 2.1. We use n, m for the number of vertices and edges of G, respectively.Previous results. Subquadratic algorithms are known for the obnoxious center problem in trees and cacti. Tamir [25] gave an algorithm with O(n log 2 n) worst-case time for trees. Faster algorithms are known for some special classes of trees [7,25]. For cactus graphs, Zmazek andŽerovnik [27] gave an algorithm using O(cn) time, where c is the number of different weights in the sites, and recently Bhattacharya, and Shi [1] showed an algorithm using O(n log 3 n) time. For general graphs, Tamir [24] showed how to solve the obnoxious center problem in O(nm + n 2 log n) time. We are not aware of other works for special classes of graphs. However, for planar graphs, it is easy to use separators of size O( √ n) [17] to solve the problem in roughly O(n 3/2 ) time. Our results. In general, we follow an approach similar to Tamir [25], using the close connection between the obnoxious center problem and the following covering problem: do a set of disks cover a graph? See Section 2.2 for a formal definition. A summary of our results is as follows:

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“…In particular, if a vertex is irrelevant, then its weight is +∞. 1 An obnoxious center is a point a * ∈ A(G) such that COST(a * ) = max a∈A(G) COST(a).…”

Section: Preliminariesmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Obnoxious Centers in Graphs

Cabello¹,

Rote²

2010

SIAM J. Discrete Math.

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“…Frederickson [6] gives a linear time algorithm for the k-center problems for a tree with unit edge costs. Apart from trees, if we restrict the graph to a cactus, where cycles share at most one vertex, O(n log 2 n) time algorithm is known for the absolute k-center problem [3].…”

Section: Introductionmentioning

confidence: 99%

Efficient Algorithms for the 2-Center Problems

Takaoka

2010

Computational Science and Its Applications – ICCSA 2010

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Abstract. This paper achieves O(n 3 log log n/ log n) time for the 2-center problems on a directed graph with non-negative edge costs under the conventional RAM model where only arithmetic operations, branching operations, and random accessibility with O(log n) bits are allowed. Here n is the number of vertices. This is a slight improvement on the best known complexity of those problems, which is O(n 3 ). We further show that when the graph is with unit edge costs, one of the 2-center problems can be solved in O(n 2.575 ) time.

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The backup 2‐center and backup 2‐median problems on trees

Wang

Chao

2008

Networks

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In this paper, we are concerned with the problem of deploying two servers in a tree network, where each server may fail with a given probability. Once a server fails, the other server will take full responsibility for the services. Here, we assume that the servers do not fail simultaneously. In the backup 2-center problem, we want to deploy two servers at the vertices such that the expected distance from a farthest vertex to the closest functioning server is minimum. In the backup 2-median problem, we want to deploy two servers at the vertices such that the expected sum of distances from all vertices to the set of functioning servers is minimum. We propose an O(n)-time algorithm for the backup 2-center problem and an O(n log n)-time algorithm for the backup 2-median problem, where n is the number of vertices in the given tree network.

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Efficient Algorithms for the Weighted 2-Center Problem in a Cactus Graph

Cited by 20 publications

References 8 publications

Obnoxious Centers in Graphs

Obnoxious Centers in Graphs

Efficient Algorithms for the 2-Center Problems

The backup 2‐center and backup 2‐median problems on trees

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