The Community for Technology Leaders
RSS Icon
Issue No.05 - September/October (2009 vol.15)
pp: 777-788
Yaniv Frishman , Technion—Israel Institute of Technology, Haifa
Ayellet Tal , Technion—Israel Institute of Technology, Haifa
Many graph layouts include very dense areas, making the layout difficult to understand. In this paper, we propose a technique for modifying an existing layout in order to reduce the clutter in dense areas. A physically inspired evolution process based on a modified heat equation is used to create an improved layout density image, making better use of available screen space. Using results from optimal mass transport problems, a warp to the improved density image is computed. The graph nodes are displaced according to the warp. The warp maintains the overall structure of the graph, thus limiting disturbances to the mental map, while reducing the clutter in dense areas of the layout. The complexity of the algorithm depends mainly on the resolution of the image visualizing the graph and is linear in the size of the graph. This allows scaling the computation according to required running times. It is demonstrated how the algorithm can be significantly accelerated using a graphics processing unit (GPU), resulting in the ability to handle large graphs in a matter of seconds. Results on several layout algorithms and applications are demonstrated.
Graph layout, graph visualization, GPU, anisotropic heat equation, mass transport.
Yaniv Frishman, Ayellet Tal, "Uncluttering Graph Layouts Using Anisotropic Diffusion and Mass Transport", IEEE Transactions on Visualization & Computer Graphics, vol.15, no. 5, pp. 777-788, September/October 2009, doi:10.1109/TVCG.2009.55
[1] I.G. Tollis, G.D. Battista, P. Eades, and R. Tamassia, Graph Drawing: Algorithms for the Visualization of Graphs. Prentice Hall, 1999.
[2] Drawing Graphs: Methods and Models, M. Kaufmann and D.Wagner, eds. Springer-Verlag, 2001.
[3] K.A. Lyons, H. Meijer, and D. Rappaport, “Algorithms for Cluster Busting in Anchored Graph Drawing,” J. Graph Algorithms and Applications, vol. 2, no. 1, pp. 1-24, 1998.
[4] P. Eades, “A Heuristic for Graph Drawing,” Congressus Numerantium, vol. 42, pp. 149-160, 1984.
[5] T. Kamada and S. Kawai, “An Algorithm for Drawing General Undirected Graphs,” Information Processing Letters, vol. 31, no. 1, pp. 7-15, 1989.
[6] T.M.J. Fruchterman and E.M. Reingold, “Graph Drawing by Force-Directed Placement,” Software—Practice and Experience, vol. 21, no. 11, pp. 1129-1164, 1991.
[7] D. Merrick and J. Gudmundsson, “Increasing the Readability of Graph Drawings with Centrality-Based Scaling,” Proc. Asia Pacific Symp. Information Visualisation (APVis '06), vol. 60, pp. 67-76, 2006.
[8] L.V. Kantorovich, “On a Problem of Monge,” Uspekhi Math. Nauk., vol. 3, no. 2, pp. 225-226, 1948.
[9] S. Haker, L. Zhu, A. Tannenbaum, and S. Angenent, “Optimal Mass Transport for Registration and Warping,” Int'l J. Computer Vision, vol. 60, no. 3, pp. 225-240, 2004.
[10] J.-D. Benamou and Y. Brenier, “A Computational Fluid Mechanics Solution to the Monge-Kantorovich Mass Transfer Problem,” Numerische Math., vol. 84, no. 3, pp. 375-393, Oct. 2000.
[11] K. Misue, P. Eades, W. Lai, and K. Sugiyama, “Layout Adjustment and the Mental Map,” J. Visual Languages and Computing, vol. 6, no. 2, pp. 183-210, 1995.
[12] S. Hachul and M. Jünger, “Drawing Large Graphs with a Potential-Field-Based Multilevel Algorithm,” Proc. 12th Int'l Symp. Graph Drawing, pp. 285-295, 2004.
[13] E. Shimizu and R. Inoue, “Time-Distance Mapping: Visualization of Transportation Level of Service,” Proc. Symp. Environmental Issues Related to Infrastructure Development, pp. 221-230, 2003.
[14] J.H. Chuang, C.C. Lin, and H.C. Yen, “Drawing Graphs with Nonuniform Nodes Using Potential Fields,” Proc. 11th Int'l Symp. Graph Drawing (GD '03), pp. 460-465, 2004.
[15] E.R. Gansner and S.C. North, “Improved Force-Directed Layouts,” Proc. Int'l Symp. Graph Drawing, pp. 364-373, 1998.
[16] D. Harel and Y. Koren, “Drawing Graphs with Non-Uniform Vertices,” Proc. Working Conf. Advanced Visual Interfaces (AVI '02), pp. 157-166, 2002.
[17] K. Marriott, P.J. Stuckey, V. Tam, and W. He, “Removing Node Overlapping in Graph Layout Using Constrained Optimization,” Constraints, vol. 8, no. 2, pp. 143-171, 2003.
[18] W. Li, P. Eades, and N. Nikolov, “Using Spring Algorithms to Remove Node Overlapping,” Proc. Asia Pacific Symp. Information Visualisation (APVis '05), vol. 45, pp. 131-140, 2005.
[19] X. Huang, W. Lai, A.S.M. Sajeev, and J. Gao, “A New Algorithm for Removing Node Overlapping in Graph Visualization,” Information Sciences, vol. 177, no. 14, pp. 2821-2844, 2007.
[20] T. Dwyer, K. Marriott, and P.J. Stuckey, “Fast Node Overlap Removal,” Graph Drawing, pp. 153-164, Springer, 2005.
[21] E.R. Gansner and S.C. North, “An Open Graph Visualization System and Its Applications to Software Engineering,” Software—Practice and Experience, vol. 30, no. 11, pp. 1203-1234, 2000.
[22] C. Walshaw, “Graph Collection,” partition/, 2009.
[23] O. Deussen, S. Hiller, C. van Overveld, and T. Strothotte, “Floating Points: A Method for Computing Stipple Drawings,” Computer Graphics Forum, vol. 19, no. 3,ISSN 1067-7055, Aug. 2000.
[24] T.F. Chan, J. Cong, and K. Sze, “Multilevel Generalized Force-Directed Method for Circuit Placement,” Proc. Int'l Symp. Physical Design (ISPD), P. Groeneveld and L. Scheffer, eds., pp. 185-192, 2005.
[25] K. Hayashi, M. Inoue, T. Masuzawa, and H. Fujiwara, “A Layout Adjustment Problem for Disjoint Rectangles Preserving Orthogonal Order,” Proc. Int'l Symp. Graph Drawing, S. Whitesides, ed., pp.183-197, 1998.
[26] M.T. Gastner and M.E.J. Newman, “Diffusion-Based Method for Producing Density-Equalizing Maps,” Proc. Nat'l Academy of Sciences USA, vol. 101, no. 20, pp. 7499-7504, 2004.
[27] W.L. Briggs, V.E. Henson, and S.F. McCormick, A Multigrid Tutorial, second ed. SIAM, 2000.
[28] R. van Liere and W. de Leeuw, “GraphSplatting: Visualizing Graphs as Continuous Fields,” IEEE Trans. Visualization and Computer Graphics, vol. 9, no. 2, pp. 206-212, Apr.-June 2003.
[29] G. Strang, Introduction to Applied Math., Wellesley-Cambridge Press, 1986.
[30] J.W. Demmel, Applied Numerical Linear Algebra. SIAM, 1997.
[31] S.C. Chapra and R.P. Canale, Numerical Methods for Engineers: With Programming and Software Applications, third ed. McGraw Hill, 1998.
[32] Y. Brenier, “Polar Factorization and Monotone Rearrangement of Vector-Valued Functions,” Comm. Pure and Applied Math., vol. 44, no. 4, pp. 375-417, 1991.
[33] M. Knott and C.S. Smith, “On the Optimal Mapping of Distributions,” J. Optimization Theory and Applications, vol. 43, no. 1, pp. 39-49, 1984.
[34] A.T. Adai, S.V. Date, S. Wieland, and E.M. Marcotte, “LGL: Creating a Map of Protein Function with an Algorithm for Visualizing Very Large Biological Networks,” J. Molecular Biology, vol. 340, pp. 179-190, 2004.
[35] “AT&T Graph Library,” http:/, 2009.
[36] D. Archambault, T. Munzner, and D. Auber, “TopoLayout: Multilevel Graph Layout by Topological Features,” IEEE Trans. Visualization and Computer Graphics, vol. 13, no. 2, pp. 305-317, Mar./Apr. 2007.
[37] Y. Frishman and A. Tal, “Multi-Level Graph Layout on the GPU,” IEEE Trans. Visualization and Computer Graphics, vol. 13, no. 6, pp.1310-1317, Nov./Dec. 2007.
[38] “Rocketfuel Maps and Data,” rocketfuel/, 2009.
[39] N. Goodnight, C. Woolley, G. Lewin, D. Luebke, and G. Humphreys, “A Multigrid Solver for Boundary Value Problems Using Programmable Graphics Hardware,” Proc. ACM SIGGRAPH/Eurographics Workshop Graphics Hardware, pp. 102-111, 2003.
[40] T. Rehman, E. Haber, G. Pryor, J. Melonakos, and A. Tannenbaum, “3D Nonrigid Registration via Optimal Mass Transport on the GPU,” Elsevier J. Medical Image Analysis, 2008.
41 ms
(Ver 2.0)

Marketing Automation Platform Marketing Automation Tool