## An Optimal Parallel Matching Algorithm for Cographs (1994)

Venue: | Journal of Parallel and Distributed Computing |

Citations: | 4 - 1 self |

### BibTeX

@ARTICLE{Lin94anoptimal,

author = {R. Lin and S. Olariu},

title = {An Optimal Parallel Matching Algorithm for Cographs},

journal = {Journal of Parallel and Distributed Computing},

year = {1994},

volume = {22},

pages = {26--36}

}

### OpenURL

### Abstract

The class of cographs, or complement-reducible graphs, arises naturally in many different areas of applied mathematics and computer science. We show that the problem of finding a maximum matching in a cograph can be solved optimally in parallel by reducing it to parenthesis matching. With an $n$-vertex cograph $G$ represented by its parse tree as input, our algorithm finds a maximum matching in $G$ in O($logn$) time using O($n0$) processors in the EREW-PRAM model. Key Words: list ranking, tree contraction, matching, parenthesis matching, scheduling, operating systems, cographs, parallel algorithms, EREW-PRAM. 1. Introduction A well-known class of graphs arising in a wide spectrum of practical applications [1,2,7] is the class of cographs, or complement-reducible graphs. The cographs are defined recursively as follows: . a single-vertex graph is a cograph; . if $G$ is a cograph, then its complement $G bar$ is also a cograph; . if $G$ and $H$ are cographs, then their union is also a cog...

### Citations

1258 |
Graph Theory with Applications
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- 1976
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Citation Context ...a given graph $G$. The literature on matching is extensive. Matching problems are related to flow problems, covering problems, and scheduling, to name just a few (the interested reader is referred to =-=[4,9,19,23,25]-=- where many other applications are summarized) . We assume the Parallel Random Access Machine model (PRAM, for short) which consists of autonomous processors, each having access to a common memory. At... |

165 | Matching is as easy as matrix inversion
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Citation Context ...observation in mind, we shall adopt the EREW-PRAM as our model of computation. Parallel algorithms to compute a maximum matching in general graphs have been studied but only with moderate success. In =-=[22]-=- a randomized parallel algorithm running in O(${log} sup 2$) expected time determines a maximum matching for general $n$-vertex graphs. The fastest deterministic algorithm [24] for general graphs runs... |

156 |
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Citation Context ...ching, parenthesis matching, scheduling, operating systems, cographs, parallel algorithms, EREW-PRAM. 1. Introduction A well-known class of graphs arising in a wide spectrum of practical applications =-=[1,2,7]-=- is the class of cographs, or complement-reducible graphs. The cographs are defined recursively as follows: . a single-vertex graph is a cograph; . if $G$ is a cograph, then its complement $G bar$ is ... |

120 |
Parallel tree contraction and its applications
- Miller, Reif
- 1985
(Show Context)
Citation Context ...ction is already recognized as one of the fundamental techniques in parallel processing. It has found - 5 - important applications in dynamic expression evaluation [1,6,10,11,15], isomorphism testing =-=[20,21]-=-, among many others. Miller and Reif's tree contraction technique [20] uses two basic operations, RAKE - which removes all leaves in the tree, and COMPRESS - which uses a variant of list ranking to re... |

95 |
An efficient parallel biconnectivity algorithm
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Citation Context ...[5] and Anderson and Miller [3] have showed that list ranking can be done optimally in O($logn$) time using O($n0$) processors in the EREW-PRAM model. The well-known Euler-tour technique developed in =-=[26]-=- allows one to compute a number of tree functions by reducing them to list ranking. To make our presentation selfcontained, we shall present now the details of a variant of this technique. To begin, w... |

93 |
Efficient algorithms for finding maximum matching in graphs
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Citation Context ...a given graph $G$. The literature on matching is extensive. Matching problems are related to flow problems, covering problems, and scheduling, to name just a few (the interested reader is referred to =-=[4,9,19,23,25]-=- where many other applications are summarized) . We assume the Parallel Random Access Machine model (PRAM, for short) which consists of autonomous processors, each having access to a common memory. At... |

87 |
A Simple Parallel Tree Contraction Algorithm
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Citation Context ...ching, parenthesis matching, scheduling, operating systems, cographs, parallel algorithms, EREW-PRAM. 1. Introduction A well-known class of graphs arising in a wide spectrum of practical applications =-=[1,2,7]-=- is the class of cographs, or complement-reducible graphs. The cographs are defined recursively as follows: . a single-vertex graph is a cograph; . if $G$ is a cograph, then its complement $G bar$ is ... |

45 |
Approximate Parallel Scheduling, Part I: The Basic Technique with Applications to Optimal Parallel List Ranking in Logarithmic Time
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Citation Context ... elements following it in the list. List ranking has turned out to be one of the fundamental techniques in parallel processing, playing a crucial role in a vast array of important parallel algorithms =-=[1,3,5,6,17,28]-=-. In particular, Cole and Vishkin [5] and Anderson and Miller [3] have showed that list ranking can be done optimally in O($logn$) time using O($n0$) processors in the EREW-PRAM model. The well-known ... |

44 |
Deterministic Parallel List Ranking
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Citation Context ... elements following it in the list. List ranking has turned out to be one of the fundamental techniques in parallel processing, playing a crucial role in a vast array of important parallel algorithms =-=[1,3,5,6,17,28]-=-. In particular, Cole and Vishkin [5] and Anderson and Miller [3] have showed that list ranking can be done optimally in O($logn$) time using O($n0$) processors in the EREW-PRAM model. The well-known ... |

44 |
A Survey of Parallel Algorithms for Shared Memory Machines. 196
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Citation Context ...is used for writing. In the Exclusive Read Exclusive Write PRAM (EREW-PRAM) model, a memory location cannot be simultaneously accessed by more than one processor. The interested reader is referred to =-=[16,28]-=- for a detailed discussion of the PRAM family. It is easy to see that the more restrictive EREW-PRAM is the weakest member of the PRAM family. Additionally, several - 3 - authors argue that EREW-PRAM ... |

38 | The Accelerated Centroid Decomposition Technique for Optimal Parallel Tree Evaluation in Logarithmic Time
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Citation Context ... elements following it in the list. List ranking has turned out to be one of the fundamental techniques in parallel processing, playing a crucial role in a vast array of important parallel algorithms =-=[1,3,5,6,17,28]-=-. In particular, Cole and Vishkin [5] and Anderson and Miller [3] have showed that list ranking can be done optimally in O($logn$) time using O($n0$) processors in the EREW-PRAM model. The well-known ... |

34 |
Efficient parallel algorithms for graph problems
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Citation Context |

33 | Sublinear-time parallel algorithms for matching and related problems
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Citation Context ...rithm [24] for general graphs runs in O($n logn$) time and uses O($n sup 2$) processors. The first sublinear maximum matching algorithms for bipartite graphs runs in O($n sup {2/3} {log} sup 3$) time =-=[12]-=-. Given the status of the maximum matching problem for general graphs, it makes sense to study parallel algorithms to compute a maximum matching in particular classes of graphs. In [2] a suboptimal pa... |

30 | Parallel tree contraction, part 2: Further applications
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(Show Context)
Citation Context ...ction is already recognized as one of the fundamental techniques in parallel processing. It has found - 5 - important applications in dynamic expression evaluation [1,6,10,11,15], isomorphism testing =-=[20,21]-=-, among many others. Miller and Reif's tree contraction technique [20] uses two basic operations, RAKE - which removes all leaves in the tree, and COMPRESS - which uses a variant of list ranking to re... |

22 |
Optimal parallel algorithm for dynamic expression evaluation and context-free recognition
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Citation Context ...Along with list ranking, tree contraction is already recognized as one of the fundamental techniques in parallel processing. It has found - 5 - important applications in dynamic expression evaluation =-=[1,6,10,11,15]-=-, isomorphism testing [20,21], among many others. Miller and Reif's tree contraction technique [20] uses two basic operations, RAKE - which removes all leaves in the tree, and COMPRESS - which uses a ... |

21 |
Discrete Optimization Algorithms
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- 1983
(Show Context)
Citation Context ...a given graph $G$. The literature on matching is extensive. Matching problems are related to flow problems, covering problems, and scheduling, to name just a few (the interested reader is referred to =-=[4,9,19,23,25]-=- where many other applications are summarized) . We assume the Parallel Random Access Machine model (PRAM, for short) which consists of autonomous processors, each having access to a common memory. At... |

16 | Dynamic parallel tree contraction
- Reif, Tate
- 1994
(Show Context)
Citation Context ...Along with list ranking, tree contraction is already recognized as one of the fundamental techniques in parallel processing. It has found - 5 - important applications in dynamic expression evaluation =-=[1,6,10,11,15]-=-, isomorphism testing [20,21], among many others. Miller and Reif's tree contraction technique [20] uses two basic operations, RAKE - which removes all leaves in the tree, and COMPRESS - which uses a ... |

13 |
Efficient parallel recognition algorithms of cographs and distance hereditary graphs
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- 1995
(Show Context)
Citation Context ...authors have presented an algorithm that solves this task in O($logn$) time using O(${n sup 2 +mn} over {logn}$) processors in the EREW-PRAM model [18]. More recently, the authors as well as Dahlhaus =-=[8]-=- have independently proposed very similar algorithms running in O(${log} sup 2$) time using O($n+m$) processors in the CREW-PRAM model of computation. Yet another such algorithm is contained in [14]. ... |

13 |
Slowing Sequential Algorithms for Obtaining Fast Distributed and Parallel Algorithms: Maximum Matchings
- Shieber, Moran
- 1986
(Show Context)
Citation Context ...th moderate success. In [22] a randomized parallel algorithm running in O(${log} sup 2$) expected time determines a maximum matching for general $n$-vertex graphs. The fastest deterministic algorithm =-=[24]-=- for general graphs runs in O($n logn$) time and uses O($n sup 2$) processors. The first sublinear maximum matching algorithms for bipartite graphs runs in O($n sup {2/3} {log} sup 3$) time [12]. Give... |

11 | Structural parallel algorithmics
- Vishkin
- 1991
(Show Context)
Citation Context ...is used for writing. In the Exclusive Read Exclusive Write PRAM (EREW-PRAM) model, a memory location cannot be simultaneously accessed by more than one processor. The interested reader is referred to =-=[16,28]-=- for a detailed discussion of the PRAM family. It is easy to see that the more restrictive EREW-PRAM is the weakest member of the PRAM family. Additionally, several - 3 - authors argue that EREW-PRAM ... |

10 |
Matching Theory. Annals of Discrete Mathematics
- Lovasz, Plummer
- 1986
(Show Context)
Citation Context |

8 |
Efficient parallel algorithms for solving some tree problems
- He
- 1986
(Show Context)
Citation Context ...If $x$ is regular, than $v$ must be the left child of $x$ and the conclusion follows by (6.3). Therefore, we may assume that $x$ itself is special. By the induction hypothesis applied to $x$, we have =-=(13) Let $y$ and $z$ stand f-=-or the left and right child of $x$, respectively. First, if $v=y$ then, by (11.2), $A(y)��L(y)��-B(y)$, and so $A(v)+B(v)=A(y)+B(y)��L(y)��-B(y)+B(y)=��L(y)��$. Now we may assu... |

8 | Parallel algorithm for cograph recognition with applications
- He
- 1993
(Show Context)
Citation Context ...m matching in cographs is presented: their algorithm runs in O($log sup 2$n) time and using O($n sup 2$) processors in the CRCW model. After this paper was submitted, we learned of an algorithm of Xe =-=[14]-=- that computes a maximum matching in an $n$-vertex cograph in O($logn$) time using O($n$) processors in the CRCW-PRAM. The purpose of this work is to show that the problem of finding a maximum matchin... |

8 |
Binary Tree Algebraic Computation and Parallel Algorithms for Simple Graphs
- He, Yesha
- 1988
(Show Context)
Citation Context ...Along with list ranking, tree contraction is already recognized as one of the fundamental techniques in parallel processing. It has found - 5 - important applications in dynamic expression evaluation =-=[1,6,10,11,15]-=-, isomorphism testing [20,21], among many others. Miller and Reif's tree contraction technique [20] uses two basic operations, RAKE - which removes all leaves in the tree, and COMPRESS - which uses a ... |

8 |
An Nc Recognition Algorithm for Cograph
- Lin, Olariu
- 1991
(Show Context)
Citation Context ...lso creates the cotree representation. Recently, the authors have presented an algorithm that solves this task in O($logn$) time using O(${n sup 2 +mn} over {logn}$) processors in the EREW-PRAM model =-=[18]-=-. More recently, the authors as well as Dahlhaus [8] have independently proposed very similar algorithms running in O(${log} sup 2$) time using O($n+m$) processors in the CREW-PRAM model of computatio... |

5 |
An optimal EREW parallel algorithm for parenthesis matching
- Tsang, Lam, et al.
- 1989
(Show Context)
Citation Context ...nd b-tokens. In fact, if we replace every a-token by a left parenthesis and every b-token by a right parenthesis we get a well-formed sequence of parenthesis. Now applying the parenthesis matching of =-=[27]-=- we identify a set of $m(root)$ matched a-tokens and b-tokens that have been assigned to say, vertices $u$ and $v$ of $G$. The following result is at the basis of the correctness of our algorithm. The... |

1 |
A Parallel Algorithm for Maximum Matching
- Adhar, Peng
- 1990
(Show Context)
Citation Context ...nclusions and open problems We have presented a cost-optimal parallel algorithm to compute a maximum matching in a complement-reducible graph (cograph, for short) thus improving over the algorithm in =-=[2]-=-. However, a number of questions remain open. The most important one is how to construct the cotree representation of a given cograph efficiently. Typically, a cograph recognition algorithm also creat... |