Given a graphG, a subgraphG' is at-spanner ofG if, for everyu,v ɛV, the distance fromu tov inG' is at mostt times longer than the distance inG. In this paper we give a simple algorithm for constructing sparse spanners for arbitrary weighted graphs. We then apply this algorithm to obtain specific results for planar graphs and Euclidean graphs. We discuss the optimality of our results and present several nearly matching lower bounds.
I. Althöfer, On Optimal Realizations of Finite Metric Spaces by Graphs,Discrete Comput. Geom. 3 (1988), 103–122.
B. Awerbuch, Complexity of Network Synchronization,J. Assoc. Comput. Mach. (1985), 804–823.
B. Awerbuch, D. Bar-Noy, N. Linial, D. Peleg, Compact Distributed Data Structures for Adaptive Routing,Proc. STOC, 1989, pp. 479–489.
B. Awerbuch, D. Peleg, Routing with Polynomial Communication-Space Tradeoff,SIAM J. Discrete Math., to appear.
H. J. Bandelt, A. W. M. Dress, Reconstructing the Shape of a Tree from Observed Dissimilarity Data,Adv. in Appl. Math. 7 (1986), 309–343.
M. Bern, Private Communication to David Dobkin, 1989.
B. Bollobás,Extremal Graph Theory, Academic Press, New York, 1978.
L. P. Chew, There is a Planar Graph Almost as Good as the Complete Graph,Proc. ACM Symp. on Computational Geometry, 1986, pp. 169–177.
J. H. Conway, N. J. A. Sloane,Sphere Packing, Lattices, and Groups, Springer-Verlag, New York, 1988.
A. W. M. Dress, Trees, Tight Extensions of Metric Spaces,Adv. in Math. 53 (1984), 321–402.
D. P. Dobkin, S. J. Friedman, K. J. Supowit, Delaunay Graphs Are Almost as Good as Complete Graphs,Proc. FOCS, 1987, pp. 20–26.
G. Das, D. A. Joseph, Which Triangulations Approximate the Complete Graph?,Proc. Internat. Symp. on Optimal Algorithms, 1989, pp. 168–192, LNCS, Vol. 401, Springer-Verlag, Berlin.
P. Erdös, H. Sachs, Reguläre Graphen gegebener Taillenweite mit minimaler Knotenzahl,Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg Math.-Natur. Reihe 12 (1963), 251–257.
J. M. Keil, Approximating the Complete Euclidean Graph,Proc. SWAT, 1988, pp. 208–213, LNCS, Vol. 318, Springer-Verlag, Berlin.
J. M. Keil, C. A. Gutwin, The Delaunay Triangulation Closely Approximates the Complete Euclidean Graph,Proc. WADS, 1989, pp. 47–56, LNCS, Vol. 382, Springer-Verlag, Berlin.
C. Levcopoulos, A. Lingas, There Are Planar Graphs Almost as Good as the Complete Graphs and as Short as Minimum Spanning Trees,Proc. Internat. Symp. on Optimal Algorithms, 1989, pp. 9–13, LNCS, Vol. 401, Springer-Verlag, Berlin.
J. Longyear, Regulard-valent Graphs of Girth 6 and 2(d*d−d+1) Vertices,J. Combin. Theory 9 (1970), 420–422.
A. Lubotzky, R. Phillips, P. Sarnak, Ramanujan Graphs,Combinatorica 8(3) (1988), 261–277.
G. A. Margulis, Explicit Group-Theoretical Constructions of Combinatorial Schemes and Their Application to the Design of Expanders and Concentrators,Problems Inform. Transmission 24(1) (1988), 39–46 (translated fromProblemy Peredachi Informatsii).
D. Peleg, A. Schäffer, Graph Spanners,J. Graph Theory 13(1) (1989), 99–116.
D. Peleg, J. Ullman, An Optimal Synchronizer for the Hypercube,SIAM J. Comput. 18 (1989), 740–747.
P. Peleg, E. Upfal, A Tradeoff Between Space and Efficiency for Routing Tables,Proc. STOC, 1988, pp. 43–52.
P. Raghaven, M. Snir, Memory Versus Randomness in Online Algorithms,Proc. ICALP, 1989, pp. 687–703, LNCS, Vol. 372, Springer-Verlag, Berlin.
R. Sedgewick, J. S. Vitter, Shortest Paths in Euclidean Graphs,Algorithmica 1 (1986), 31–48.
J. M. S. Simões-Pereira, A Note on the Tree Realizability of a Distance Matrix,J. Combin. Theory 6 (1969), 303–310.
R. E. Tarjan,Data Structures and Network Algorithms, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1983.
The work of G. Das and D. Joseph was supported by NSF PYI Grant DCR-8402375. The work of D. Dobkin was supported by NSF Grant CCR-8700917. The work of J. Soares was supported by CNPq proc 203039/87.4 (Brazil) and NSF Grant CCR-9014562. This research was accomplished while G. Das was a student at the University of Wisconsin-Madison. A preliminary version was presented at the Second Scandinavian Workshop on Algorithm Theory, Bergen, Norway, 1990, under the title “Generating Sparse Spanners for Weighted Graphs,” and proceedings appear in the series Lecture Notes in Computer Science, Springer-Verlag. The preliminary version also appears as Princeton University Technical Report CS-TR-261-90, and as University of Wisconsin-Madison Computer Sciences Technical Report 882.
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Althöfer, I., Das, G., Dobkin, D. et al. On sparse spanners of weighted graphs. Discrete Comput Geom 9, 81–100 (1993). https://doi.org/10.1007/BF02189308