Skip to main content
Springer Nature Link
Log in

Finding minimal spanning trees in a euclidean coordinate space

  • Part I Computer Science
  • Published:
BIT Numerical Mathematics Aims and scope Submit manuscript

Abstract

The minimal spanning tree problem of a point set in ak-dimensional Euclidean space is considered and a new version of the multifragmentMST-algorithm of Bentley and Friedman is given. The minimal spanning tree is found by repeatedly joining the minimal subtree with the closest subtree. Ak-d tree is used for choosing the connecting edges. Computation time of the algorithm depends on the configuration of the point set: for normally distributed random points the algorithm is very fast. Two extreme cases demandingO(n logn) andO(n 2) operations,n being the cardinality of the point set, are also given.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alfred Aho, John Hopcroft and Jeffrey Ullman,The Design and Analysis of Computer Algorithms, (Addison-Wesley, 1974).

  2. Jon Louis Bentley,Multidimensional binary search trees used for associative searching, Comm. ACM, Vol. 18, No. 9, September 1975.

  3. Jon Louis Bentley and Jerome H. Friedman,Fast algorithms for constructing minimal spanning trees in coordinate spaces, IEEE Trans. on Computers, Vol. C-27, No. 2, February 1978.

  4. Jon Louis Bentley and Michael Ian Shamos,Divide-and-conquer in multidimensional space, Proc. 8th Ann. ACM Symp. on Theory of Computing, May 1976.

  5. David Cheriton and Robert Endre Tarjan,Finding minimal spanning trees, SIAM J. Computing, Vol. 5, No. 4, December 1976.

  6. Nicos Christofides,Graph Theory; An Algorithmic Approach, (Academic Press, 1975).

  7. A. K. Dewdney,Complexity of nearest neighbour searching in three and higher dimensions, Univ. of Western Ontario, Techn. Rep. No. 28, June 1977.

  8. E. W. Dijkstra,A note on two problems in connection with graphs, Numerische Mathematik, Bd. 1, 269–271, 1959.

    Article  Google Scholar 

  9. Jarmo Ernvall, Jyrki Katajainen and Olli Nevalainen,A minimal spanning tree algorithm for a point set in Euclidean space, Rep. 24, Comp. Sci., Univ. of Turku, Finland, 1980.

    Google Scholar 

  10. D. B. Johnson,Priority queues with update and finding minimal spanning trees, Inf. Proc. Letters, Vol. 4, No. 1, 1975.

  11. Joseph B. Kruskal,On the shortest spanning subtree of a graph and the traveling salesman problem, Proc. Amer. Math. Soc. 7, 48–50, 1956.

    Google Scholar 

  12. Olli Nevalainen and Jarmo Ernvall,A note on a minimal spanning tree algorithm for Euclidean space, Rep. 25, Comp. Sci., Univ. of Turku, Finland, 1980.

    Google Scholar 

  13. R. C. Prim,Shortest connection networks and some generalizations, The Bell Systems Techn. J., November 1957.

  14. Michael Ian Shamos and Don Hoey,Closest-point problems, Proc. 16th Ann. Symp. on Found of Comp. Sci., October 1975.

  15. V. Kevin and M. Whitney,Algorithm 422Minimal spanning tree H, Collected Algorithms of CACM.

  16. Andrew Chi-Chih Yao,An O(|E| log log |V|)algorithm for finding minimum spanning trees, Inf. Proc. Letters, Vol. 4, No. 1, September 1975.

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences Computer Science, University of Turku, SF-20500, Turku 50, Finland

    O. Nevalainen, J. Ernvall & J. Katajainen

Authors
  1. O. Nevalainen
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. J. Ernvall
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. J. Katajainen
    View author publications

    You can also search for this author inPubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Nevalainen, O., Ernvall, J. & Katajainen, J. Finding minimal spanning trees in a euclidean coordinate space. BIT 21, 46–54 (1981). https://doi.org/10.1007/BF01934070

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01934070

Key-words