## Optimal Algorithms for the Vertex Updating Problem of a Minimum Spanning Tree (1992)

Venue: | In Proc. of the 6th Intl Parallel Proccessing Symposium (IPPS '92 |

Citations: | 3 - 2 self |

### BibTeX

@INPROCEEDINGS{Johnson92optimalalgorithms,

author = {Donald B. Johnson and Panagiotis Metaxas},

title = {Optimal Algorithms for the Vertex Updating Problem of a Minimum Spanning Tree},

booktitle = {In Proc. of the 6th Intl Parallel Proccessing Symposium (IPPS '92},

year = {1992},

pages = {306--314},

publisher = {IEEE Press}

}

### OpenURL

### Abstract

The vertex updating problem for a minimum spanning tree (MST) is defined as follows: Given a graph G = (V; EG ) and its MST T , update T when a new vertex z is introduced along with weighted edges that connect z with the vertices of G. We present a set of rules that, together with a valid tree-contraction schedule, are used to produce simple optimal parallel algorithms that run in O(lgn) parallel time using n= lg n EREW PRAMs where n = jV j. These rules can also be used to derive simple linear-time sequential algorithms for the same problem. The previously best known parallel result was a rather complicated algorithm that used n processors of the more powerful CREW PRAM model. Furthermore, we show how our solution can be used to solve the multiple vertex updating problem: Update a given MST when k new vertices are introduced simultaneously. This problem is solved in O(lg k \Delta lg n) parallel time using k \Deltan lg k \Deltalg n EREW PRAM processors. 1 Introduction Definition. ...

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Citation Context ... MST problem can be solved using kn=(lg n lg k) EREW PRAM processors in O(lg n lg k) parallel time. Proof. The algorithm that we use is a well-known algorithm whose main idea is attributed to Boruvka =-=[17]-=- and was described in its parallel form in [3]. The analysis however and the time-processors bounds for the bipartite-MST problem are new. First, let us give some definitions. A pseudotree is a direct... |

581 |
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Citation Context ...s useful to observe that the edge with minimum weight incident to some node will always be included into the MST. Actually, many sequential and parallel algorithms are based on this observation (i.e. =-=[15, 16, 3]-=-). Edge inclusion makes use of this observation. Another useful observation is that whenever some edge is found to correspond to the MaxWE of some cycle it can be removed from the tree without affecti... |

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Citation Context ...rformance by a factor of lg n. As indicated above, the model of parallel computation we will use throughout this paper is the EREW PRAM (exclusive-read-exclusive-write parallel random access machine) =-=[10]-=-. Representation. Upon introducing the new vertexsz along with n weighted 2 edges, \Gamma n 2 \Delta cycles are created. If we break these cycles by deleting the maximum weight edge (MaxWE) that appea... |

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Citation Context ...when p = n= log n is the best possible. Proof. a. the optimal parallel algorithms. There are, actually, several valid tree contraction schedules that produce optimal behavior in our algorithm. First, =-=[12]-=- proposed such a schedule which was constructed on the fly by an optimal randomized algorithm. The problem and its applications drew the attention of researchers, and soon several optimal deterministi... |

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Citation Context ...o-right. Here, the input is supposed to be a regular binary tree, i.e. a binary tree in which every internal node has exactly two children. The numbering can be done using the eulerian tour technique =-=[18]-=- within the desired bounds. 2. Prune the odd-numbered leaves that are the left children of their parent. Then, shortcut their parent. This is the shunt operation. 3. Shunt the odd-numbered leaves that... |

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Citation Context ... two new weighted edges connecting z to each of the two leaves, the updating MST problem is equivalent to the problem of computing the maximum of n + 1 weights on the cycle created, for which problem =-=[6]-=- have proven a lower bound of \Omega\Gamma/13 n). 2 3.1 Binarization We now discuss how a general tree can be transformed into a binary tree using the procedure binarize: Each node v having k children... |

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Citation Context ...ich was constructed on the fly by an optimal randomized algorithm. The problem and its applications drew the attention of researchers, and soon several optimal deterministic algorithms were presented =-=[1, 11, 5, 8, 7]-=-. We will briefly describe here the simplest of these schedules (called Shunting) which was proposed independently by [1] and [11]. The algorithm is composed of a number of phases, each containing the... |

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Citation Context ...s useful to observe that the edge with minimum weight incident to some node will always be included into the MST. Actually, many sequential and parallel algorithms are based on this observation (i.e. =-=[15, 16, 3]-=-). Edge inclusion makes use of this observation. Another useful observation is that whenever some edge is found to correspond to the MaxWE of some cycle it can be removed from the tree without affecti... |

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Citation Context ...ich was constructed on the fly by an optimal randomized algorithm. The problem and its applications drew the attention of researchers, and soon several optimal deterministic algorithms were presented =-=[1, 11, 5, 8, 7]-=-. We will briefly describe here the simplest of these schedules (called Shunting) which was proposed independently by [1] and [11]. The algorithm is composed of a number of phases, each containing the... |

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Citation Context ...s (using tree-contraction) as well as sequential ones (using, for example, depth-first-search). History. The vertex updating problem of a minimum spanning tree was first addressed by Spira and Pan in =-=[16]-=-, where an O(n) sequential algorithm was presented. Another solution using depth-first-search and having the same time complexity was later given by Chin and Houck in [2], while Pawagi and Ramakrishna... |

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Citation Context ... and the linked lists are stored in an array. This representation of the input is not crucial, since it can be derived in O(lg n) time using n= lg n processors from any reasonable representation (see =-=[5]-=- for a discussion on the representation). We should mention at the outset that in the course of our description we treat every case where read or write conflicts might be expected to occur and we show... |

22 |
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Citation Context ...ich was constructed on the fly by an optimal randomized algorithm. The problem and its applications drew the attention of researchers, and soon several optimal deterministic algorithms were presented =-=[1, 11, 5, 8, 7]-=-. We will briefly describe here the simplest of these schedules (called Shunting) which was proposed independently by [1] and [11]. The algorithm is composed of a number of phases, each containing the... |

18 | Optimal tree contraction in an EREW model
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Citation Context |

16 |
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Citation Context ...ddressed by Spira and Pan in [16], where an O(n) sequential algorithm was presented. Another solution using depth-first-search and having the same time complexity was later given by Chin and Houck in =-=[2]-=-, while Pawagi and Ramakrishnan [14] gave a parallel solution to the problem. Their algorithm, which runs in O(lg n) time 1 using n 2 CREW PRAMs, precomputes all maximum weight edges on paths between ... |

11 |
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(Show Context)
Citation Context ...G 's vertices by kn = jE k j new weighted edges, but they are not connected among themselves. We are asked to compute the new MST T 0 . The problem of multiple vertex updates was considered by Pawagi =-=[13]-=- and a parallel algorithm was presented running in O(lgn lg k) time using nk CREW PRAM processors. We will show how our solution for the (single) vertex update problem can be used to achieve a better ... |

9 |
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(Show Context)
Citation Context ...rks in the same parallel time, but uses n CREW PRAM processors. Even though their idea is rather simple, the implementation details make the algorithm rather complex. More recently, Jung and Mehlhorn =-=[9]-=- have given an optimal solution for the more powerful CRCW PRAM model by reduction to an expression evaluation problem. They use an optimal tree contraction algorithm as a subroutine, as do we. Howeve... |

9 |
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(Show Context)
Citation Context ...where an O(n) sequential algorithm was presented. Another solution using depth-first-search and having the same time complexity was later given by Chin and Houck in [2], while Pawagi and Ramakrishnan =-=[14]-=- gave a parallel solution to the problem. Their algorithm, which runs in O(lg n) time 1 using n 2 CREW PRAMs, precomputes all maximum weight edges on paths between any two nodes in the tree, and then ... |

8 |
A parallel vertex insertion algorithm for minimum spanning trees
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(Show Context)
Citation Context ...s in O(lg n) time 1 using n 2 CREW PRAMs, precomputes all maximum weight edges on paths between any two nodes in the tree, and then breaks the cycles simultaneously in constant time. Varman and Doshi =-=[19]-=- presented an efficient solution that works in the same parallel time, but uses n CREW PRAM processors. Even though their idea is rather simple, the implementation details make the algorithm rather co... |