Time-Space Trade-Offs for Computing Euclidean Minimum Spanning Trees

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10807)

Abstract

In the limited-workspace model, we assume that the input of size n lies in a random access read-only memory. The output has to be reported sequentially, and it cannot be accessed or modified. In addition, there is a read-write workspace of O(s) words, where \(s \in \{1, \dots , n\}\) is a given parameter. In a time-space trade-off, we are interested in how the running time of an algorithm improves as s varies from 1 to n.

We present a time-space trade-off for computing the Euclidean minimum spanning tree (\({{\mathrm{EMST}}}\)) of a set V of n sites in the plane. We present an algorithm that computes \({{\mathrm{EMST}}}(V)\) using \(O(n^3\log s /s^2)\) time and O(s) words of workspace. Our algorithm uses the fact that \({{\mathrm{EMST}}}(V)\) is a subgraph of the bounded-degree relative neighborhood graph of V, and applies Kruskal’s MST algorithm on it. To achieve this with limited workspace, we introduce a compact representation of planar graphs, called an s-net which allows us to manipulate its component structure during the execution of the algorithm.

Keywords

Euclidean minimum spanning tree Relative neighborhood graph Time-space trade-off Limited workspace model Kruskal’s algorithm 

Notes

Acknowledgments

This work was initiated at the Fields Workshop on Discrete and Computational Geometry, held July 31–August 04, 2017, at Carleton university. The authors would like to thank them and all the participants of the workshop for inspiring discussions and for providing a great research atmosphere.

References

  1. 1.
    Ahn, H.-K., Baraldo, N., Oh, E., Silvestri, F.: A time-space trade-off for triangulations of points in the plane. In: Cao, Y., Chen, J. (eds.) COCOON 2017. LNCS, vol. 10392, pp. 3–12. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-62389-4_1 Google Scholar
  2. 2.
    Aronov, B., Korman, M., Pratt, S., van Renssen, A., Roeloffzen, M.: Time-space trade-offs for triangulating a simple polygon. In: Proceedings of the 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT), pp. 30:1–30:12 (2016)Google Scholar
  3. 3.
    Arora, S., Barak, B.: Computational Complexity: A Modern Approach. Cambridge University Press, Cambridge (2009)CrossRefMATHGoogle Scholar
  4. 4.
    Asano, T., Buchin, K., Buchin, M., Korman, M., Mulzer, W., Rote, G., Schulz, A.: Memory-constrained algorithms for simple polygons. Comput. Geom. 46(8), 959–969 (2013)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Asano, T., Kirkpatrick, D.: Time-space tradeoffs for all-nearest-larger-neighbors problems. In: Dehne, F., Solis-Oba, R., Sack, J.-R. (eds.) WADS 2013. LNCS, vol. 8037, pp. 61–72. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-40104-6_6 CrossRefGoogle Scholar
  6. 6.
    Asano, T., Mulzer, W., Rote, G., Wang, Y.: Constant-work-space algorithms for geometric problems. J. Comput. Geom. 2(1), 46–68 (2011)MathSciNetMATHGoogle Scholar
  7. 7.
    Bahoo, Y., Banyassady, B., Bose, P., Durocher, S., Mulzer, W.: Time-space trade-off for finding the k-visibility region of a point in a polygon. In: Poon, S.-H., Rahman, M.S., Yen, H.-C. (eds.) WALCOM 2017. LNCS, vol. 10167, pp. 308–319. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-53925-6_24 CrossRefGoogle Scholar
  8. 8.
    Banyassady, B., Korman, M., Mulzer, W., van Renssen, A., Roeloffzen, M., Seiferth, P., Stein, Y.: Improved time-space trade-offs for computing Voronoi diagrams. In: Proceedings of the 34th Symposium on Theoretical Aspects of Computer Science (STACS), pp. 9:1–9:14 (2017)Google Scholar
  9. 9.
    Barba, L., Korman, M., Langerman, S., Sadakane, K., Silveira, R.I.: Space-time trade-offs for stack-based algorithms. Algorithmica 72(4), 1097–1129 (2015)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Barba, L., Korman, M., Langerman, S., Silveira, R.I.: Computing a visibility polygon using few variables. Comput. Geom. 47(9), 918–926 (2014)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    de Berg, M., Cheong, O., van Kreveld, M., Overmars, M.H.: Computational Geometry: Algorithms and Applications, 3rd edn. Springer, Heidelberg (2008).  https://doi.org/10.1007/978-3-540-77974-2 CrossRefMATHGoogle Scholar
  12. 12.
    Chan, T.M., Chen, E.Y.: Multi-pass geometric algorithms. Discrete Comput. Geom. 37(1), 79–102 (2007)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Chan, T.M., Munro, J.I., Raman, V.: Selection and sorting in the “restore" model. In: Proceedings of the 25th Annual ACM-SIAM symposium Discrete Algorithms (SODA), pp. 995–1004 (2014)Google Scholar
  14. 14.
    Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 3rd edn. MIT Press, Cambridge (2009)MATHGoogle Scholar
  15. 15.
    Darwish, O., Elmasry, A.: Optimal time-space tradeoff for the 2D convex-hull problem. In: Schulz, A.S., Wagner, D. (eds.) ESA 2014. LNCS, vol. 8737, pp. 284–295. Springer, Heidelberg (2014).  https://doi.org/10.1007/978-3-662-44777-2_24 Google Scholar
  16. 16.
    Har-Peled, S.: Shortest path in a polygon using sublinear space. J. Comput. Geom. 7(2), 19–45 (2016)MathSciNetMATHGoogle Scholar
  17. 17.
    Jaromczyk, J.W., Toussaint, G.T.: Relative neighborhood graphs and their relatives. Proc. IEEE 80, 1502–1517 (1992)CrossRefGoogle Scholar
  18. 18.
    Korman, M., Mulzer, W., van Renssen, A., Roeloffzen, M., Seiferth, P., Stein, Y.: Time-space trade-offs for triangulations and Voronoi diagrams. Comput. Geom. (2017, to appear)Google Scholar
  19. 19.
    Mitchell, J.S.B., Mulzer, W.: Proximity algorithms. In: Goodman, J.E., O’Rourke, J., Tóth, C.D. (eds.) Handbook of Discrete and Computational Geometry, 3rd edn, pp. 849–874. CRC Press, Boca Raton (2017)Google Scholar
  20. 20.
    Munro, J.I., Paterson, M.S.: Selection and sorting with limited storage. Theoret. Comput. Sci. 12(3), 315–323 (1980)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Munro, J.I., Raman, V.: Selection from read-only memory and sorting with minimum data movement. Theoret. Comput. Sci. 165(2), 311–323 (1996)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Pagter, J., Rauhe, T.: Optimal time-space trade-offs for sorting. In: Proceedings of the 39th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 264–268 (1998)Google Scholar
  23. 23.
    Reingold, O.: Undirected connectivity in log-space. J. ACM 55(4), 17 (2008)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Toussaint, G.T.: The relative neighbourhood graph of a finite planar set. Pattern Recogn. 12(4), 261–268 (1980)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Freie Universität BerlinBerlinGermany
  2. 2.ETH ZurichZurichSwitzerland

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