SSDE: Fast Graph Drawing Using Sampled Spectral Distance Embedding

Civril, Ali and Magdon-Ismail, Malik and Bocek-Rivele, Eli (2007) SSDE: Fast Graph Drawing Using Sampled Spectral Distance Embedding. In: Graph Drawing 14th International Symposium, GD 2006, September 18-20, 2006, Karlsruhe, Germany , pp. 30-41 (Official URL:

Full text not available from this repository.


We present a fast spectral graph drawing algorithm for drawing undirected connected graphs. Classical Multi-Dimensional Scaling yields a quadratic-time spectral algorithm, which approximates the real distances of the nodes in the final drawing with their graph theoretical distances. We build from this idea to develop the linear-time spectral graph drawing algorithm SSDE. We reduce the space and time complexity of the spectral decomposition by approximating the distance matrix with the product of three smaller matrices, which are formed by sampling rows and columns of the distance matrix. The main advantages of our algorithm are that it is very fast and it gives aesthetically pleasing results, when compared to other spectral graph drawing algorithms. The runtime for typical 10^5 node graphs is about one second and for 10^6 node graphs about ten seconds.

Item Type:Conference Paper
Additional Information:10.1007/978-3-540-70904-6_5
Classifications:M Methods > M.100 Algebraic
P Styles > P.720 Straight-line
ID Code:759

Repository Staff Only: item control page


I. Borg and P. Groenen. Modern Multidimensional Scaling. Springer-Verlag, 1997.

A. Civril, M. Magdon-Ismail, and E. Bocek-Rivele. SDE: Graph drawing using spectral distance embedding. In GD'05, pages 512-513, 2005.

T. M. J. Fruchterman and E. M. Reingold. Graph drawing by force-directed placement. Software - Practice And Experience, 21(11):1129-1164, 1991.

G. H. Golub and C.H. Van Loan. Matrix Computations. Johns Hopkins U. Press, 1996.

K. M. Hall. An r-dimensional quadratic placement algorithm. Management Science, 17:219-229, 1970.

D. Harel and Y. Koren. A fast multi-scale method for drawing large graphs. In GD'00, volume 1984, pages 183-196, 2000.

D. Harel and Y. Koren. Graph drawing by high-dimensional embedding. In GD'02, 2002.

W. Johnson and J. Lindenstrauss. Extensions of lipschitz maps into a hilbert space. Contemp. Math., 26:189-206, 1984.

T. Kamada and S. Kawai. An algorithm for drawing general undirected graphs. Information Processing Letters, 31(1):7-15, 1989.

M. Kaufmann and D. Wagner, editors. Drawing Graphs: Methods and Models. Number 2025 in LNCS. Springer-Verlag, 2001.

Y. Koren. On spectral graph drawing. In COCOON 03, volume 2697, pages 496-508, 2003.

Y. Koren. One dimensional layout optimization, with applications to graph drawing by axis separation. Computational Geometry: Theory and Applications, 32:115-138, 2005.

Y. Koren, D. Harel, and L. Carmel. Drawing huge graphs by algebraic multigrid optimization. Multiscale Modeling and Simulation, 1(4):645-673, 2003. SIAM.

J. B. Kruskal and J. B. Seery. Designing network diagrams. In Proc. First General Conference on Social Graphics, 1980.

J. Maeda and K. Murata. Restoration of band-limited images by an iterative regularized pseudoinverse method. Journal of Optical Society of America, 1(1):28-34, 1984.

J. Matousek. Open problems on embeddings of nite metric spaces. Discr. Comput. Geom., to appear.

P.Drineas, R. Kannan, and M. W. Mahoney. Fast Monte Carlo algorithms for matrices III: Computing a compressed approximate matrix decomposition. SIAM Journal on Computing, 36(1):184-206, 2006.

J. C. Platt. FastMap, MetricMap, and landmarkMDS are all Nystrom algorithms. In Proc. 10th Int. Workshop on Artificial Intelligence and Statistics, pages 261-268, 2005.

I. G. Tollis, G. Di Battista, P. Eades, and R. Tamassia. Graph Drawing: Algorithms for the Visualization of Graphs. Prentice Hall, 1999.

V. Vazirani. Approximation Algorithms. Springer-Verlag, 2001.

C. Walshaw. A multilevel algorithm for force-directed graph drawing. In GD'00, volume 1984, 2000.