Years and Authors of Summarized Original Work
1999; Krznaric, Levcopoulos, Nilsson
Problem Definition
Let S be a set of n points in d-dimensional real space where d ≥ 1 is an integer constant. A minimum spanning tree (MST) of S is a connected acyclic graph with vertex set S of minimum total edge length. The length of an edge equals the distance between its endpoints under some metric. Under the so-called L p metric, the distance between two points x and y with coordinates (x 1, x 2, …, x d ) and (y 1, y 2, …, y d ), respectively, is defined as the pth root of the sum \(\sum \limits _{i=1}^{d}\left \vert x_{i} - y_{i}\right \vert ^{p}\).
Key Results
Since there is a very large number of papers concerned with geometric MSTs, only a few of them will be mentioned here.
In the common Euclidean L 2 metric, which simply measures straight-line distances, the MST problem in two dimensions can be solved optimally in time O(nlog n), by using the fact that the MST is a subgraph of the Delaunay...
Recommended Reading
Agarwal PK, Edelsbrunner H, Schwarzkopf O, Welzl E (1991) Euclidean minimum spanning trees and bichromatic closest pairs. Discret Comput Geom 6:407–422
Bespamyatnikh S (1997) On constructing minimum spanning trees in R 1 k. Algorithmica 18(4):524–529
Bespamyatnikh S (1998) An optimal algorithm for closest-pair maintenance. Discret Comput Geom 19(2):175–195
Callahan PB, Kosaraju SR (1993) Faster algorithms for some geometric graph problems in higher dimensions. In: SODA, Austin, pp 291–300
Chatterjee S, Connor M, Kumar P (2010) Geometric minimum spanning trees with GeoFilterKruskal. In: Experimental Algorithms: proceedings of SEA, Ischia Island. LNCS, vol 6049, pp 486–500
Cheriton D, Tarjan RE (1976) Finding minimum spanning trees. SIAM J Comput 5(4):724–742
Clarkson KL (1984) Fast expected-time and approximation algorithms for geometric minimum spanning trees. In: Proceedings of STOC, Washington, DC, pp 342–348
Czumaj A, Ergün F, Fortnow L, Magen A, Newman I, Rubinfeld R, Sohler C (2005) Approximating the weight of the Euclidean minimum spanning tree in sublinear time. SIAM J Comput 35(1):91–109
Eppstein D (1995) Dynamic Euclidean minimum spanning trees and extrema of binary functions. Discret Comput Geom 13:111–122
Gabow HN, Bentley JL, Tarjan RE (1984) Scaling and related techniques for geometry problems. In: STOC, Washington, DC, pp 135–143
Krznaric D, Levcopoulos C, Nilsson BJ (1999) Minimum spanning trees in d dimensions. Nord J Comput 6(4):446–461
Narasimhan G, Zachariasen M (2001) Geometric minimum spanning trees via well-separated pair decompositions. ACM J Exp Algorithms 6:6
Salowe JS (1991) Constructing multidimensional spanner graphs. Int J Comput Geom Appl 1(2):99–107
Vaidya PM (1988) Minimum spanning trees in k-Dimensional space. SIAM J Comput 17(3):572–582
Yao AC (1982) On constructing minimum spanning trees in k-Dimensional spaces and related problems. SIAM J Comput 11(4):721–736
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer Science+Business Media New York
About this entry
Cite this entry
Levcopoulos, C. (2014). Minimum Geometric Spanning Trees. In: Kao, MY. (eds) Encyclopedia of Algorithms. Springer, Boston, MA. https://doi.org/10.1007/978-3-642-27848-8_236-2
Download citation
DOI: https://doi.org/10.1007/978-3-642-27848-8_236-2
Received:
Accepted:
Published:
Publisher Name: Springer, Boston, MA
Online ISBN: 978-3-642-27848-8
eBook Packages: Springer Reference Computer SciencesReference Module Computer Science and Engineering