## A Probabilistic Minimum Spanning Tree Algorithm (1978)

Venue: | Information Processing Letters |

Citations: | 5 - 0 self |

### BibTeX

@ARTICLE{Rohlf78aprobabilistic,

author = {F. James Rohlf},

title = {A Probabilistic Minimum Spanning Tree Algorithm},

journal = {Information Processing Letters},

year = {1978},

volume = {7},

pages = {44--48}

}

### OpenURL

### Abstract

This paper is concerned with the problem of computing spanning tree (MST) for n points in a p-dimensional space where the "distance" between each pair of points i and j satisfies the relationship' dq max {Ixti - xtql} , where xki is the coordinate of object i along the ktti dimension. This relationship is clearly satisfied by all Minkowski metrics dq = [ Ixki - xnjl r] x/r, r > 1

### Citations

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357 |
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Citation Context ...g points not belonging to V ordered according to the magnitude of their distance to the closest point in the component of the MST currently under consideration (but only if this distance issas a heap =-=[7]-=- since only the smallest distance is needed each time (step K) and a heap requires little effort to update as new elements are added or as the position of an element is revised (O(log m) where m = Ial... |

333 |
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Citation Context ...he computation of the MST can also be directly applied to a number of clustering algorithms. Single linkage clusters can be obtained from a MST with only O(n) effort [10]. A centroid cluster analysis =-=[12]-=- can be carried out by making only modest changes to the above procedure. Since most implementations of cluster analyses have effort of at least O(n 2) there is clearly the potential for dramatic gain... |

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Citation Context ...fort required by the algorithm is reduced considerably by taking advantage of the geometrical properties of the space. The most efficient previously published algorithm is that of Bently and Friedman =-=[1] (which us-=-es a "k-d tree structure"). It has an effort of O(n log n) for normally distributed data sets. This has been given as the lower bound to the MST problem [2,11]. The increased efficiency of t... |

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Citation Context ...suggests that more complex procedures on the heap could reduce the computational effort even more. The merging of the components of the MST in step M was implemented using a method analogous to Yao's =-=[14]-=- quick-merge algorithm. 4. Discussion Fig. 2 gives a plot of t (CPU time, IBM 370/168) versus n (the number of data points) for p = 2, 3, and 4 dimensions for the 8-MST method for random samVolume 7, ... |

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Citation Context ...t of data to give the shortest expected running time. At least two methods can be used in step D to partition R e so as to reduce the number of distances that have to be computer. The method of Yuval =-=[15, 16]-=- has the advantage that maximally only p + 1 grid systems are needed. However, it has the disadvantage that the mesh of each grid is quite coarse (8(p + 1)) for larger values ofp. The method of Rabin ... |

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ACM Algorithm 422: Minimal spanning tree [H
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Citation Context ... be computed for large numbers of points. For an arbitrary measure of "distance" between all pairs of points it is not difficult to construct algorithms which achieve the optimal O(n 2) effo=-=rt (e.g., [4,8,10,18]-=-). If not * On leave from the Department of Ecology and Evolution, State University of New York, Stony Brook, N.Y. 11794. all of the distances are defined then the algorithm of Yao [13] may be more ef... |

3 |
Finding near neighbors in k-dimensional space
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Citation Context ...t of data to give the shortest expected running time. At least two methods can be used in step D to partition R e so as to reduce the number of distances that have to be computer. The method of Yuval =-=[15, 16]-=- has the advantage that maximally only p + 1 grid systems are needed. However, it has the disadvantage that the mesh of each grid is quite coarse (8(p + 1)) for larger values ofp. The method of Rabin ... |

1 |
Algorithm 76: Hierarchial clustering using the minimum spanning tree
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Citation Context ...RS January 1978 This approach to the computation of the MST can also be directly applied to a number of clustering algorithms. Single linkage clusters can be obtained from a MST with only O(n) effort =-=[10]-=-. A centroid cluster analysis [12] can be carried out by making only modest changes to the above procedure. Since most implementations of cluster analyses have effort of at least O(n 2) there is clear... |

1 |
An O(IEI log log IVI) algorithm for finding minimum spanning trees
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Citation Context ... (e.g., [4,8,10,18]). If not * On leave from the Department of Ecology and Evolution, State University of New York, Stony Brook, N.Y. 11794. all of the distances are defined then the algorithm of Yao =-=[13]-=- may be more efficient since the effort for it is O(e log log n) where e is the number of pairwise distances (edges). The algorithm presented below is restricted to the case where the input consists o... |