Balancing minimum spanning trees and shortest-path trees
Rent the article at a discountRent now
* Final gross prices may vary according to local VAT.Get Access
We give a simple algorithm to find a spanning tree that simultaneously approximates a shortest-path tree and a minimum spanning tree. The algorithm provides a continuous tradeoff: given the two trees and aγ>0, the algorithm returns a spanning tree in which the distance between any vertex and the root of the shortest-path tree is at most 1+√2γ times the shortest-path distance, and yet the total weight of the tree is at most 1+√2/γ times the weight of a minimum spanning tree. Our algorithm runs in linear time and obtains the best-possible tradeoff. It can be implemented on a CREW PRAM to run a logarithmic time using one processor per vertex.
- I. Althöfer, G. Das, D. Dobkin, D. Joseph, and J. Soares, On sparse spanners of weighted graphs,Discrete and Computational Geometry,9(1) (1993), 81–100.
- B. Awerbuch, A. Baratz, and D. Peleg, Cost-sensitive anlaysis of communication protocols,Proc. 9th Symp. on Principles of Distributed Computing, 1990, pp. 177–187.
- B. Awerbuch, A. Baratz, and D. Peleg, Efficient broadcast and light-weight spanners, Manuscript (1991).
- K. Bharath-Kumar and J. M. Jaffe, Routing to multiple destinations in computer networks,IEEE Transactions on Communications,31(3) (1983), 343–351.
- B. Chandra, G. Das, G. Narasimhan, and J. Soares, New sparseness results on graph spanners,Proc. 8th Symp. on Conputational Geometry, 1992, pp. 192–201.
- L. P. Chew, There are planar graphs almost as good as the complete graph,Journal of Computer and System Sciences,39(2) (1989), 205–219.
- J. Cong, A. B. Kahng, G. Robins, M. Sarrafzadeh, and C. K. Wong, Performance-driven global routing for cell based IC's,Proc. IEEE Internat. Conf. on Computer Design, 1991, pp. 170–173.
- J. Cong, A. B. Kahng, G. Robins, M. Sarrafzadeh, and C. K. Wong, Provably good performance-driven global routing,IEEE Transactions on CAD, (1992), 739–752.
- J. Cong, A. B. Kahng, G. Robins, M. Sarrafzadeh, and C. K. Wong, Provably good algorithms for performance-driven global routing,Proc. IEEE Internat. Symp. on Circuits and Systems, San Diego, 1992, pp. 2240–2243.
- T. H. Cormen, C. E. Leiserson, and R. L. Rivest,Introduction to Algorithms, MIT Press, Cambridge, MA, 1989.
- E. W. Dijkstra, A note on two problems in connexion with graphs,Numerische Mathematik,1 (1959), 269–271.
- M. L. Freeman and R. E. Tarjan, Fibonacci heaps and their uses in improved network optimization algorithms,Journal of the ACM,34(3) (1987), 596–615.
- H. N. Gabow, Z. Galil, T. Spencer, and R. E. Tarjan, Efficient algorithms for finding minimum spanning trees in undirected and directed graphs,Combinatorica,6(2) (1986), 109–122.
- J. JáJáIntroduction to Parallel Algorithms, Addison-Wesley, Reading, MA, 1991.
- J. B. Kruskal, On the shortest spanning subtree of a graph and the traveling salesman problem,Proceedings of the American Mathematical Society,7, (1956), pp. 48–50.
- C. Levcopoulos and A. Lingas, There are planar graphs almost as good as the complete graphs and almost as cheap as minimum spanning trees,Algorithmica,8(3) (1992), 251–256.
- D. Peleg and J. D. Ullman, An optimal synchronizer for the hypercube,Proc. 6th Symp. on Principles of Distributed Computing, 1987, pp. 77–85.
- F. P. Preparata and M. I. Shamos,Computational Geometry, Springer-Verlag, New York, 1985.
- R. C. Prim, Shortest connection networks and some generalizations,Bell System Technical Journal,36 (1957), 1389–1401.
- P. M. Vaidya, A sparse graph almost as good as the complete graph on points inK dimensions,Discrete and Computational Geometry,6 (1991), 369–381.
- Balancing minimum spanning trees and shortest-path trees
Volume 14, Issue 4 , pp 305-321
- Cover Date
- Print ISSN
- Online ISSN
- Additional Links
- Minimum spanning trees
- Graph algorithms
- Parallel algorithms
- Shortest paths
- Industry Sectors
- Author Affiliations
- 1. Department of Computer Science and Institute for Advanced Computer Studies, University of Maryland, 20742, College Park, MD, USA
- 2. Department of Computer Science, University of Texas at Dallas, Box 830688, 75083-0688, Richardson, TX, USA
- 3. Department of Computer Science, Princeton University, 08544, Princeton, NJ, USA