## Towards overcoming the transitive-closure bottleneck: efficient parallel algorithms for planar digraphs (1993)

Venue: | J. Comput. System Sci |

Citations: | 11 - 1 self |

### BibTeX

@ARTICLE{Klein93towardsovercoming,

author = {Philip N. Klein},

title = {Towards overcoming the transitive-closure bottleneck: efficient parallel algorithms for planar digraphs},

journal = {J. Comput. System Sci},

year = {1993}

}

### OpenURL

### Abstract

Abstract. Currently, there is a significant gap between the best sequential and parallel complexities of many fundamental problems related to digraph reachability. This complexity bottleneck essentially reflects a seemingly unavoidable reliance on transitive closure techniques in parallel algorithms for digraph reachability. To pinpoint the nature of the bottleneck, we de* velop a collection of polylog-time reductions among reachability problems. These reductions use only linear processors and work for general graphs. Furthermore, for planar digraphs, we give polylog-time algorithms for the following problems: (1) directed ear decomposition, (2) topological ordering, (3) digraph reachability, (4) descendent counting, and (5) depth-first search. These algorithms use only linear processors and therefore reduce the complexity to within a polylog factor of optimal.

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Citation Context ...nected. The undirected version of ear decomposition has proved tremendously useful in parallel algorithms. For instance, it is the basis of efficient algorithms for st-numbering [26], triconnectivity =-=[27]-=-, [34], [10], [9], 4connectivity [16], and planarity [22], [33]. We expect © 1990 ACM 089791-361-2/90/0005/0181 $1.50 181that directed ear decomposition will also be very useful. Indeed, it has been ... |

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Citation Context .... The undirected version of ear decomposition has proved tremendously useful in parallel algorithms. For instance, it is the basis of efficient algorithms for st-numbering [26], triconnectivity [27], =-=[34]-=-, [10], [9], 4connectivity [16], and planarity [22], [33]. We expect © 1990 ACM 089791-361-2/90/0005/0181 $1.50 181that directed ear decomposition will also be very useful. Indeed, it has been used b... |

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Citation Context ...perties unique to the class in question; in particular, for planar st-digraphs, Vitter and Tamassia have given optimal algorithms that solve the digraph teachability problem as well as other problems =-=[39]-=-. The descendent counting problem is to compute for each vertex the number of vertices that can be reached from the vertex through directed paths. More generally, the problem is to sum the weights of ... |

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Citation Context ...ed depth-first search, Smith 182gave the first deterministic NC algorithm for planar undirected graphs [37]; the processor complexity for this case was later shown to be linear by Ja'Ja and Kosaraju =-=[15]-=- and by He and Yesha [14]. Aggarwal and Anderson give the first randomized NC algorithm for general undirected graphs [1]. Kao gives the first deterministic NC algorithm for planar directed graphs [17... |

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Citation Context ...undirected version of ear decomposition has proved tremendously useful in parallel algorithms. For instance, it is the basis of efficient algorithms for st-numbering [26], triconnectivity [27], [34], =-=[10]-=-, [9], 4connectivity [16], and planarity [22], [33]. We expect © 1990 ACM 089791-361-2/90/0005/0181 $1.50 181that directed ear decomposition will also be very useful. Indeed, it has been used by Kao ... |

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Citation Context ... edge points from child to parent (respectively, from parent to child). A CD-pair of spanning trees consists of a convergent spanning tree and a divergent spanning tree both rooted at the same vertex =-=[20]-=-, [19]. Observe that a digraph has a CD-pair of spanning trees if and only if it is strongly connected. Also recall that a digraph has an ear decomposition if and only if it is strongly connected. Con... |

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Citation Context ...roved tremendously useful in parallel algorithms. For instance, it is the basis of efficient algorithms for st-numbering [26], triconnectivity [27], [34], [10], [9], 4connectivity [16], and planarity =-=[22]-=-, [33]. We expect © 1990 ACM 089791-361-2/90/0005/0181 $1.50 181that directed ear decomposition will also be very useful. Indeed, it has been used by Kao for computing planar directed cycle separator... |

1 |
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Citation Context ...ing a topological ordering in tile present paper. Lovasz has previously given a directed ear decomposition algorithm for general digraphs that relies on breadth-first search and has a high complexity =-=[25]-=-. We give a parallel reduction for obtaining a directed ear decomposition from an arbitrary pair of directed spanning trees, one convergent, one divergent, and both rooted at the same vertex. This red... |

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Citation Context ...ide Ri; let Gi be the subgraph of G induced by Vi. Furthermore, let I]i be the set of ears/~j' inside each P~. These P~'s, Gi's, and l]i's can be found using undirected connected component algorithms =-=[36]-=-. By induction, the t~'s can be arranged into a topological segmentation T' for G. To refine T' into a topological ordering of G, it suffices to first compute a topological ordering Ti for.each Gi and... |