## 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.

### Citations

2438 | The Design and Analysis of Computer Algorithms - Aho, Hopcroft, et al. - 1974 |

1259 |
Graph Theory with Applications
- Bondy, Murty
- 1980
(Show Context)
Citation Context ...ounting, and depth-first search, respectively. 2. Basics for planar graphs. A planar digraph is one that can be drawn on a plane such that the edges in the drawing intersect only at common ends [13], =-=[5]-=-. A drawing of a planar digraph can be specified by the clockwise cyclic order of edges incident with each vertex. Such a specification is called a combinatorial embedding and is useful for algorithmi... |

1129 |
Graph Theory
- Harary
- 1995
(Show Context)
Citation Context ...purpose of future study, we address this fundamental issue in the context of minor-closed families; a family of graphs is called minor-closed if it is closed under deletions and contractions of edges =-=[13]-=-. We show that for a minor-closed family, the reachability problem can be transformed via linear-processor NC reductions to the problems of computing strongly connected components and topological orde... |

992 | Depth first search and linear graph algorithms
- Tarjan
- 1972
(Show Context)
Citation Context ...thm for planar directed depth-first search. The depth-first search problem is to construct a forest that corresponds to performing depth-first search in a given graph starting from specified vertices =-=[38]-=-, [3]. For lexicographic depth-first search, Reifshows that the problem is P-complete even for general undirected graphs [35]. For unordered depth-first search, Smith 182gave the first deterministic ... |

801 |
Matrix multiplications via arithmetic progressions
- Coppersmith, Winograd
- 1990
(Show Context)
Citation Context ..., all known polylog-time parallel algorithms for the problem compute the transitive closure of the given graph; the best of them currently uses O(n 2"3~6) processors for an n-vertex graph [21], [12], =-=[7]-=-. Thus, there is a significant gap between the best sequential and parallel complexities of the problem. The reliance on transitive closure techniques and the associated complexity bottleneck are not ... |

388 | Tarjan, A Separator Theorem for Planar Graphs
- Lipton, E
- 1979
(Show Context)
Citation Context ... x and from r to y form an undirected separator of G. Furthermore, such a pair of x and y can be found in O(log n) time using O( n / log n) processors Proof. Lipton and Tarjan give an existence proof =-=[24]-=-. Miller gives a linear-processor algorithm [29]. [:1 3. Directed ear decomposition. In this section, we show that finding a directed ear decomposition is optimally NC-equivalent to finding a CD-pair ... |

272 | Parallel prefix computation
- Ladner, Fischer
- 1980
(Show Context)
Citation Context ...ears under the lexicographic order induced by the double-index (i, j). After the Si,j's and ti's are computed, the double-indexing can be converted into single-indexing by parallel prefix computation =-=[8]-=- in O(log n) time using O(n/log n) processors. To finish the proof, it suffices to describe how to efficiently compute Si,1,...,Si,t~ for R/. For notational brevity, the index i is omitted in the foll... |

120 |
Parallel tree contraction and its applications
- Miller, Reif
- 1985
(Show Context)
Citation Context ...are the leaves of D, the Ai's actually partition D. Thus, the R/'s contain all the vertices of G. As for the complexity, the idea for computing the ai's and Ai's is to use tree contraction techniques =-=[28]-=- to compute for each vertex v the lowest-indexed zi that is a descendant of v in D. This descendant information can then be used to identify ai and At. The bi's and Bi's are processed in the same way.... |

83 |
Finding small simple cycle separators for 2-connected planar graphs
- Miller
- 1986
(Show Context)
Citation Context ...of G. Furthermore, such a pair of x and y can be found in O(log n) time using O( n / log n) processors Proof. Lipton and Tarjan give an existence proof [24]. Miller gives a linear-processor algorithm =-=[29]-=-. [:1 3. Directed ear decomposition. In this section, we show that finding a directed ear decomposition is optimally NC-equivalent to finding a CD-pair of spanning trees as defined below. A convergent... |

52 |
Depth-First Search is Inherently Sequential
- Reif
- 1985
(Show Context)
Citation Context ...rming depth-first search in a given graph starting from specified vertices [38], [3]. For lexicographic depth-first search, Reifshows that the problem is P-complete even for general undirected graphs =-=[35]-=-. For unordered 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 ... |

49 | Optimal parallel evaluation of tree-structured computations by raking - Kosaraju, Delcher - 1988 |

45 | Deterministic parallel list ranking - Anderson, Miller - 1988 |

44 |
A Survey of Parallel Algorithms for Shared Memory Machines. 196
- Karp
(Show Context)
Citation Context ... In contrast, all known polylog-time parallel algorithms for the problem compute the transitive closure of the given graph; the best of them currently uses O(n 2"3~6) processors for an n-vertex graph =-=[21]-=-, [12], [7]. Thus, there is a significant gap between the best sequential and parallel complexities of the problem. The reliance on transitive closure techniques and the associated complexity bottlene... |

41 | Parallel ear decomposition search (EDS) and st-numbering in graphs. Theoret
- Maon, Schieber, et al.
- 1986
(Show Context)
Citation Context ... graph is strongly connected. 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... |

40 |
Flow in planar graphs with multiple sources and sinks
- Miller, Naor
- 1995
(Show Context)
Citation Context ...narity if the sources are not connected. This subtlety has also been a fundamental constraint in many other algorithms for planar graphs including the recent network flow algorithm by Miller and Naor =-=[30]-=-. For the purpose of future study, we address this fundamental issue in the context of minor-closed families; a family of graphs is called minor-closed if it is closed under deletions and contractions... |

29 | Finding triconnected components by local replacements
- Fussell, Ramachandran, et al.
- 1993
(Show Context)
Citation Context ...cted 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 for c... |

29 |
A parallel algorithm for finding a separator in planar graphs
- Gazit, Miller
- 1987
(Show Context)
Citation Context ... respectively. These complexity bounds are obtained using the path algebra algorithms by Pan and Reif [31] in conjunction with the randomized planar undirected separator algorithm by Gazit and Miller =-=[11]-=-. As for smaller classes of planar digraphs, there have been several optimal NC algorithms for the digraph teachability problem all based on properties unique to the class in question; in particular, ... |

26 |
A random NC algorithm for depth first search
- Aggarwal, Anderson
- 1988
(Show Context)
Citation Context ...r 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]. Aggarwal, Anderson, and Kao give the first randomized NC algorithm for general directed graphs [2]. While these result... |

25 | A new graph triconnectivity algorithm and its parallelization, in: STOC ’87
- Miller, Ramachandran
- 1987
(Show Context)
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 ... |

22 | Improved algorithms for graph fourconnectivity
- Kanevsky, Ramachandran
- 1991
(Show Context)
Citation Context ... 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 for computing planar dire... |

21 |
Parallel depth-first search in general directed graphs
- Aggarwal, Anderson, et al.
- 1990
(Show Context)
Citation Context ...r processors, even for planar digraphs. Among the five problems, depth-first search is the most difficult. In fact, depth-first search for general digraphs is not even known to be in deterministic NC =-=[2]-=-, and the planar case has only very recently been shown by Kao to have a deterministic NC algorithm using O(n 4) processors [17]. In addition to devising these linear-processor NC algorithms, we also ... |

19 |
An optimal parallel algorithm for graph planarity
- Ramachandran, Reif
- 1989
(Show Context)
Citation Context ...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 separators [17]... |

14 |
Efficient parallel triconnectivity in logarithmic parallel time
- Ramachandran, Vishkin
(Show Context)
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... |

13 |
Parallel Algorithms for Depth First Searches: I
- Smith
- 1984
(Show Context)
Citation Context ...h, Reifshows that the problem is P-complete even for general undirected graphs [35]. For unordered 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 undi... |

11 | Optimal parallel algorithms ]or expression tree evaluation and list ranking - COLE, VISHKIN - 1988 |

8 |
A nearly optimal parallel algorithm .for constructing depth first spanning trees in planar graphs
- HE, YESHA
- 1988
(Show Context)
Citation Context ...ith 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]. Aggarwal, Anderson, an... |

7 |
Extension of the parallel nested dissection algorithm to the path algebra problems
- Pan, Reif
- 1985
(Show Context)
Citation Context ...e-source and single-source teachability problems require O(n 2) processors and O(n 15) processors, respectively. These complexity bounds are obtained using the path algebra algorithms by Pan and Reif =-=[31]-=- in conjunction with the randomized planar undirected separator algorithm by Gazit and Miller [11]. As for smaller classes of planar digraphs, there have been several optimal NC algorithms for the dig... |

7 |
Optimal Parallel Algorithms for Transitive Closure and
- Tamassia, Vitter
- 1989
(Show Context)
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 ... |

5 |
All graphs have cycle separators and planar directed depth.first search is
- KAO
- 1988
(Show Context)
Citation Context ... search for general digraphs is not even known to be in deterministic NC [2], and the planar case has only very recently been shown by Kao to have a deterministic NC algorithm using O(n 4) processors =-=[17]-=-. In addition to devising these linear-processor NC algorithms, we also take the first step in developing a general theory for understanding the nature of the transitive-closure bottleneck. We discove... |

3 |
Parallel algorithms ]or planar graphs and related problems
- JA'JA, KOSARAJU
- 1988
(Show Context)
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... |

2 |
Separation pair detection
- Fussell, Thurimella
- 1988
(Show Context)
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 ... |

2 | NC algorithms for planar directed graphs I: Strongly connected components - Linear-processor - 1993 |

1 |
reorientation, global order, and planar topology
- Local
- 1989
(Show Context)
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... |

1 |
An efficient parallel algorithm for planarity, FOCS
- KLEIN, IEIF
- 1986
(Show Context)
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 |
Computing ears and branchings
- tsz
- 1985
(Show Context)
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... |

1 |
An O(logn) parallel con. nectivity algorithm
- SHILOACH, VISHIIN
- 1982
(Show Context)
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... |