Abstract
We introduce and study (1 + ε)-EMST drawings, i.e. planar straight-line drawings of trees such that, for any fixed ε> 0, the distance between any two vertices is at least \(\frac{1}{1 + \varepsilon}\) the length of the longest edge in the path connecting them. (1 + ε)-EMST drawings are good approximations of Euclidean minimum spanning trees. While it is known that only trees with bounded degree have a Euclidean minimum spanning tree realization, we show that every tree T has a (1 + ε)-EMST drawing for any given ε> 0. We also present drawing algorithms that compute (1 + ε)-EMST drawings of trees with bounded degree in polynomial area. As a byproduct of one of our techniques, we improve the best known area upper bound for Euclidean minimum spanning tree realizations of complete binary trees.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
Bose, P., Di Battista, G., Lenhart, W., Liotta, G.: Proximity constraints and representable trees. In: Tamassia, R., Tollis, I.G. (eds.) GD 1994. LNCS, vol. 894, pp. 340–351. Springer, Heidelberg (1995)
Chan, T.M.: A near-linear area bound for drawing binary trees. Algorithmica 34(1), 1–13 (2002)
Di Battista, G., Eades, P., Tamassia, R., Tollis, I.G.: Graph Drawing. Prentice Hall, Upper Saddle River (1999)
Eades, P., Whitesides, S.: The realization problem for euclidean minimum spanning trees in NP-hard. Algorithmica 16(1), 60–82 (1996)
Eades, P., Whitesides, S.: The realization problem for Euclidean minimum spanning trees is NP-hard. Algorithmica 16, 60–82 (1996)
Frati, F., Kaufmann, M.: Polynomial area bounds for MST embeddings of trees. RT-DIA-122-2008, Dept. of Comput. Sc. Univ. Roma Tre (2008)
Jaromczyk, J.W., Toussaint, G.T.: Relative neighborhood graphs and their relatives. Proc. IEEE 80(9), 1502–1517 (1992)
Kaufmann, M.: Polynomial area bounds for MST embeddings of trees. In: Hong, S.-H., Nishizeki, T., Quan, W. (eds.) GD 2007. LNCS, vol. 4875, pp. 88–100. Springer, Heidelberg (2008)
Kaufmann, M., Wagner, D. (eds.): Drawing Graphs. LNCS, vol. 2025. Springer, Heidelberg (2001)
King, J.: Realization of degree 10 minimum spanning trees in 3-space. In: Canadian Conference on Computational Geometry (CCCG 2006) (2006)
Liotta, G.: Proximity drawings. In: Tamassia, R. (ed.) Handbook of Graph Drawing and Visualization. CRC Press, Boca Raton (to appear)
Liotta, G., Di Battista, G.: Computing proximity drawings of trees in the 3-dimemsional space. In: Sack, J.-R., Akl, S.G., Dehne, F., Santoro, N. (eds.) WADS 1995. LNCS, vol. 955, pp. 239–250. Springer, Heidelberg (1995)
Monma, C., Suri, S.: Transitions in geometric minimum spanning trees. Discrete Comput. Geom. 8, 265–293 (1992)
Preparata, F.P., Shamos, M.I.: Computational Geometry: An Introduction, 3rd edn. Springer, Heidelberg (October 1990)
Valiant, L.G.: Universality considerations in VLSI circuits. IEEE Trans. Computers 30(2), 135–140 (1981)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Di Giacomo, E., Didimo, W., Liotta, G., Meijer, H. (2010). Drawing a Tree as a Minimum Spanning Tree Approximation. In: Cheong, O., Chwa, KY., Park, K. (eds) Algorithms and Computation. ISAAC 2010. Lecture Notes in Computer Science, vol 6507. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-17514-5_6
Download citation
DOI: https://doi.org/10.1007/978-3-642-17514-5_6
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-17513-8
Online ISBN: 978-3-642-17514-5
eBook Packages: Computer ScienceComputer Science (R0)