## A Linear Time Algorithm for the Bottleneck Biconnected Spanning Subgraph Problem (1996)

Venue: | Information Processing Letters |

Citations: | 4 - 0 self |

### BibTeX

@ARTICLE{Manku96alinear,

author = {Gurmeet Singh Manku},

title = {A Linear Time Algorithm for the Bottleneck Biconnected Spanning Subgraph Problem},

journal = {Information Processing Letters},

year = {1996},

volume = {59},

pages = {1--7}

}

### OpenURL

### Abstract

A linear time algorithm for the Bottleneck Biconnected Spanning Subgraph problem is presented. This improves the hitherto best-known solution, which has a running time of O(m+ n log n), where m and n are the number of edges and vertices of the graph. Keywords Algorithms - Combinatorial Problems - Biconnectivity 1 Introduction Given a biconnected graph G = (V; E) with weight function w : E ! R, let F be the family of biconnected subgraphs of G that span all the vertices in V . The bottleneck weight is defined as wB (G) = min G 0 2F max e2G 0 w(e) and GB = (V; EB ) is a Bottleneck Biconnected Spanning Subgraph of G if 8e 2 EB : w(e) wB (G). BBSS finds applications in solving the bottleneck traveling salesman problem [4][8], in communication networks [7] and in approximation algorithms of some `hard' bottleneck problems [5][9]. BBSS with vertices in the Euclidean plane has been studied by Chang, Tang and Lee [3], who propose an O(n 2 ) algorithm. For general graphs, Punnen an...

### Citations

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Citation Context ... in H, as shown in Section 2.6. Step (D) computes fl, the median of weights of dotted edges and constructs E dl ` E d , containing all edges weighing less than fl. Median can be computed in O(d) time =-=[1]-=-, where jE d j = d. Step (E) computes the mbcc's of L, which can be done in O(d) time, as shown in Section 2.4. Step (F) shrinks L, which again takes O(d) time. Since jE dl j ! jE d j=2, the new graph... |

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Citation Context ...in G 0 2F max e2G 0 w(e) and GB = (V; EB ) is a Bottleneck Biconnected Spanning Subgraph of G if 8e 2 EB : w(e)swB (G). BBSS finds applications in solving the bottleneck traveling salesman problem [4]=-=[8]-=-, in communication networks [7] and in approximation algorithms of some `hard' bottleneck problems [5][9]. BBSS with vertices in the Euclidean plane has been studied by Chang, Tang and Lee [3], who pr... |

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Citation Context ...= min G 0 2F max e2G 0 w(e) and GB = (V; EB ) is a Bottleneck Biconnected Spanning Subgraph of G if 8e 2 EB : w(e)swB (G). BBSS finds applications in solving the bottleneck traveling salesman problem =-=[4]-=-[8], in communication networks [7] and in approximation algorithms of some `hard' bottleneck problems [5][9]. BBSS with vertices in the Euclidean plane has been studied by Chang, Tang and Lee [3], who... |

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Citation Context ...eneck Spanning Tree (BST ) T of G, which is defined as follows. Let F T be the family of spanning trees of G. Then, ff = min T 0 2FT max e2T 0 w(e) and 8e 2 T; w(e)sff. T can be computed in O(m) time =-=[2]-=-. It is easily seen that wB (G)sff. Step (B) modifies the weights of some edges in E. Any non-tree edge with weight at most ff is increased in weight to ff. All tree-edges are assigned a weight of \Ga... |

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Citation Context ...blem [4][8], in communication networks [7] and in approximation algorithms of some `hard' bottleneck problems [5][9]. BBSS with vertices in the Euclidean plane has been studied by Chang, Tang and Lee =-=[3]-=-, who propose an O(n 2 ) algorithm. For general graphs, Punnen and Nair [10] present an O(m + n log n) algorithm. This paper presents a linear time algorithm for the general case, which first maps the... |

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Citation Context ...EB : w(e)swB (G). BBSS finds applications in solving the bottleneck traveling salesman problem [4][8], in communication networks [7] and in approximation algorithms of some `hard' bottleneck problems =-=[5]-=-[9]. BBSS with vertices in the Euclidean plane has been studied by Chang, Tang and Lee [3], who propose an O(n 2 ) algorithm. For general graphs, Punnen and Nair [10] present an O(m + n log n) algorit... |

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Citation Context ...: w(e)swB (G). BBSS finds applications in solving the bottleneck traveling salesman problem [4][8], in communication networks [7] and in approximation algorithms of some `hard' bottleneck problems [5]=-=[9]-=-. BBSS with vertices in the Euclidean plane has been studied by Chang, Tang and Lee [3], who propose an O(n 2 ) algorithm. For general graphs, Punnen and Nair [10] present an O(m + n log n) algorithm.... |

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Citation Context ...of some `hard' bottleneck problems [5][9]. BBSS with vertices in the Euclidean plane has been studied by Chang, Tang and Lee [3], who propose an O(n 2 ) algorithm. For general graphs, Punnen and Nair =-=[10]-=- present an O(m + n log n) algorithm. This paper presents a linear time algorithm for the general case, which first maps the problem onto a special kind of graph which we call B Graphs. Linear time is... |