On the area requirements of Euclidean minimum spanning trees

In their seminal paper on Euclidean minimum spanning trees, Monma and Suri (1992) proved that any tree of maximum degree 5 admits a planar embedding as a Euclidean minimum spanning tree. Their algorithm constructs embeddings with exponential area; however, the authors conjectured that there exist n-vertex trees of maximum degree 5 that require c^nxc^n area to be embedded as Euclidean minimum spanning trees, for some constant c>1. In this paper, we prove the first exponential lower bound on the area requirements for embedding trees as Euclidean minimum spanning trees.

[1]  Ioannis G. Tollis,et al.  Area Requirement of Gabriel Drawings , 1996, CIAC.

[2]  Sanjeev Arora,et al.  Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems , 1998, JACM.

[3]  Joseph S. B. Mitchell,et al.  Guillotine Subdivisions Approximate Polygonal Subdivisions: A Simple Polynomial-Time Approximation Scheme for Geometric TSP, k-MST, and Related Problems , 1999, SIAM J. Comput..

[4]  Michael Kaufmann,et al.  Polynomial area bounds for MST embeddings of trees , 2011, Comput. Geom..

[5]  Balaji Raghavachari,et al.  Degree-bounded minimum spanning trees , 2004, CCCG.

[6]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[7]  Christos H. Papadimitriou,et al.  On Two Geometric Problems Related to the Traveling Salesman Problem , 1984, J. Algorithms.

[8]  Emilio Di Giacomo,et al.  Drawing a tree as a minimum spanning tree approximation , 2012, J. Comput. Syst. Sci..

[9]  Timothy M. Chan.  Euclidean Bounded-Degree Spanning Tree Ratios , 2003, SCG '03.

[10]  Michael Hoffmann,et al.  The Euclidean degree-4 minimum spanning tree problem is NP-hard , 2009, SCG '09.

[11]  Clyde L. Monma,et al.  Transitions in geometric minimum spanning trees , 1991, SCG '91.

[12]  Giuseppe Liotta,et al.  Proximity Drawings of High-Degree Trees , 2013, Int. J. Comput. Geom. Appl..

[13]  Prosenjit Bose,et al.  Characterizing proximity trees , 1996, Algorithmica.

[14]  Marco Chiesa,et al.  On the Area Requirements of Euclidean Minimum Spanning Trees , 2011, WADS.