## A new graph triconnectivity algorithm and its parallelization (1987)

Venue: | Combinatorica |

Citations: | 26 - 3 self |

### BibTeX

@INPROCEEDINGS{Miller87anew,

author = {Gary L. Miller and Vijaya Ramachandran},

title = {A new graph triconnectivity algorithm and its parallelization},

booktitle = {Combinatorica},

year = {1987},

pages = {254--263}

}

### Years of Citing Articles

### OpenURL

### Abstract

We present a new algorithm for finding the triconnected components of an undirected graph. The algorithm is based on a method of searching graphs called ‘open ear decomposition’. A parallel implementation of the algorithm on a CRCW PRAM runs in O(log 2 n) parallel time using O(n + m) processors, where n is the number of vertices and m is the number of edges in the graph.

### Citations

288 | Parallel Merge Sort
- Cole
- 1988
(Show Context)
Citation Context ...ar embedding, the Euler order gives the sequence in which the stars first appear on the Euler tour. We first sort the stars in G(P) with respect to the Euler order in O(log q) time using q processors =-=[Co]-=- (as a preprocessing step we coalesce all stars with the same span, -- note that any such pair must interlace). By the following observation it is relatively easy to maintain the stars in order even i... |

185 |
Dividing a graph into triconnected components
- Hopcroft, Tarjan
- 1973
(Show Context)
Citation Context ...g O(n + m) processors on a CRCW PRAM, where n is the number of vertices in the graph and m is the number of edges. A sequential linear-time algorithm for the problem is available in Hopcroft & Tarjan =-=[HoTa]-=-, but it is based on depth first search, and is not known to be efficiently parallelizable. Finding triconnected components of a graph is important in determining the connectivity structure of the gra... |

120 |
Parallel Tree Contraction and its Application
- Miller, Reif
- 1985
(Show Context)
Citation Context ...for determining if two planar graphs are isomorphic. Parallel NC algorithms for testing triconnectivity and for finding triconnected components are reported in Ja’Ja’ & Simon [JaSi] and Miller & Reif =-=[MiRe]-=- but neither match our processor bound for general graphs. Ja’Ja’ & Simon [JaSi] give an NC algorithm for finding triconnected components using M(n) processors, where M(n) is the number of processors ... |

101 |
Parallel algorithms for shared memory machines
- Karp, Ramachandran
- 1990
(Show Context)
Citation Context ...TRARY CRCW PRAM that runs in parallel time T using P processors can be simulated by an EREW PRAM (and hence by a CREW PRAM) in parallel time T log P using the same number of processors, P (see, e.g., =-=[KarRa]-=-). Let S be a problem which, on an input of size n, can be solved on a PRAM by a parallel algorithm in parallel time t(n) with p(n) processors. The quantity w(n) = t(n) ⋅ p(n) represents the work done... |

85 |
Graph Algorithms, Computer Science
- Even
- 1979
(Show Context)
Citation Context ...rating pair. We define triconnected components at the end of this section. Let G = (V , E) be a biconnected graph, and let Q be a subgraph of G. We define the bridges of Q in G as follows (see, e.g., =-=[Ev]-=-): Let V ′ be the vertices in G − Q, and consider the partition of V ′ into classes such that two vertices are in the same class if and only if there is a path connecting them which does not use any v... |

59 |
Approximate and exact parallel scheduling with applications to list, tree, and graph problems
- Cole, Vishkin
- 1986
(Show Context)
Citation Context ...l edges between adjacent separating pairs in each G j. All of this computation can be performed in logarithmic time with a linear number of processors using parallel algorithms for graph connectivity =-=[CV]-=- and sorting [Co]. Step 1b can be performed with similar bounds in the same manner. In order to fully parallelize Algorithm 3 we need to perform the above computation in parallel for several nontrivia... |

44 | Parallel Ear Decomposition Search (EDS) and st-Numbering
- Maon, Schieber, et al.
(Show Context)
Citation Context ...ion of an ‘efficient parallel algorithm’). Our algorithm uses an efficient parallel algorithm for finding an open ear decomposition that we developed earlier in Miller & Ramachandran [MiRa] (see also =-=[MaScVi]-=-). More recently, building on the results we present, Ramachandran & Vishkin [RaVi] have obtained an efficient parallel triconnectivity algorithm that runs in logarithmic time. Also, Kanevsky & Ramach... |

43 | Tutte, Connectivity in Graphs - T - 1966 |

42 |
Finding biconnected components and computing tree functions in logarithmic parallel time
- TARJAN, VISHKIN
- 1984
(Show Context)
Citation Context ...nt of the other, or the two stars must be siblings. Thus this step can be done in O(log p) time with O(p) processors using either parallel tree contraction [MiRe] or the Euler tour technique on trees =-=[TaVi]-=-. At this point at most one star in T ′ can interlace with a given star in T and at most one star in T can interlace with a star in T ′. Let us call a star that currently interlaces with a star in the... |

23 | Improved algorithms for graph fourconnectivity
- Kanevsky, Ramachandran
(Show Context)
Citation Context ... recently, building on the results we present, Ramachandran & Vishkin [RaVi] have obtained an efficient parallel triconnectivity algorithm that runs in logarithmic time. Also, Kanevsky & Ramachandran =-=[KanRa]-=- have used open ear decomposition to obtain better sequential and parallel algorithms for graph four connectivity. The rest of the paper is organized as follows. Section 2 provides a brief overview of... |

21 |
Efficient parallel ear decomposition with applications
- Miller, Ramachandran
- 1986
(Show Context)
Citation Context ...2 for the definition of an ‘efficient parallel algorithm’). Our algorithm uses an efficient parallel algorithm for finding an open ear decomposition that we developed earlier in Miller & Ramachandran =-=[MiRa]-=- (see also [MaScVi]). More recently, building on the results we present, Ramachandran & Vishkin [RaVi] have obtained an efficient parallel triconnectivity algorithm that runs in logarithmic time. Also... |

15 |
Parallel algorithms in graph theory: Planarity testing
- Ja'Ja', Simon
- 1980
(Show Context)
Citation Context ...etermining planarity and for determining if two planar graphs are isomorphic. Parallel NC algorithms for testing triconnectivity and for finding triconnected components are reported in Ja’Ja’ & Simon =-=[JaSi]-=- and Miller & Reif [MiRe] but neither match our processor bound for general graphs. Ja’Ja’ & Simon [JaSi] give an NC algorithm for finding triconnected components using M(n) processors, where M(n) is ... |

14 |
Efficient parallel triconnectivity in logarithmic parallel time
- Ramachandran, Vishkin
(Show Context)
Citation Context ...lgorithm for finding an open ear decomposition that we developed earlier in Miller & Ramachandran [MiRa] (see also [MaScVi]). More recently, building on the results we present, Ramachandran & Vishkin =-=[RaVi]-=- have obtained an efficient parallel triconnectivity algorithm that runs in logarithmic time. Also, Kanevsky & Ramachandran [KanRa] have used open ear decomposition to obtain better sequential and par... |

9 |
Finding the triconnected components of a graph
- Tarjan, Hopcroft
- 1973
(Show Context)
Citation Context ... m) time algorithm with O(m) processors to find the triconnected components.s-28We also note that Algorithm 3 can be easily modified to obtain the tree of 3-connected components (or ‘auxiliary graph’ =-=[HoTa2]-=-) with the same time and processor bounds. For this we construct the tree of triconnected components by introducing a vertex for each copy of edge (a, b, i) added at a split and we place an edge betwe... |

5 |
Non-separable and planar graphs," Tr ans
- Whitney
- 1932
(Show Context)
Citation Context ... York, NY, May 1987. † Supported by NSF Grant DCR 8514961. ‡ Supported by NSF Grant ECS 8404866 and the Semiconductor Research Corporation Grant 86-12-109.s1. Introduction -2An open ear decomposition =-=[Wh, Lo]-=- of an undirected graph is a partition of its edge set into an ordered collection of paths called ears that satisfy certain properties. In this paper we present an efficient parallel algorithm based o... |

4 | Lovasz,"Computing ears and branchings in parallel - unknown authors - 1985 |