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A Simple Parallel Algorithm to Draw Cubic Graphs
October 2000 (vol. 11 no. 10)
pp. 1009-1018

Abstract—The main contribution of this work is to offer a simple and cost-efficient parallel algorithm that, given an arbitrary $n$-vertex cubic graph $G$ as input, produces an orthogonal grid drawing of $G$ in ${\rm O}(\log n)$ time, using $n$ processors on an EREW PRAM. Our algorithm matches the time and cost performance of the best previously-known algorithm while at the same time improving the constant factors involved in two important metrics: layout area and number of bends. More importantly, however, our algorithm stands out by its conceptual simplicity and ease of implementation.

[1] K. Abrahamson, N. Dadoun, D.G. Kirkpatrick, and T. Przytycka, "A Simple Parallel Tree Contraction Algorithm," J. Algorithms, vol. 10, no. 2, pp. 287-302, 1989.
[2] A. Aho, J.K. Hopcroft, and J.D. Ullman, The Design and Analysis of Computer Algorithms. Reading, Mass.: Addison Wesley, 1973.
[3] R.J. Anderson and G.L. Miller, “Deterministic Parallel List Ranking,” Algorithmica, vol. 6, pp. 859–868 1991.
[4] T. Biedl and G. Kant, “A Better Heuristic for Orthogonal Graph Drawings,” Computational Geometry: Theory and Applications, vol. 9, pp. 159–180, 1998.
[5] T. Calamoneri, “Does Cubicity Help to Solve Problems?” PhD thesis, IX-2-97, Univ. of Rome (La Sapienza), 1997.
[6] T. Calamoneri, S. Jannelli, and R. Petreschi, “Experimental Comparison of Graph Drawing Algorithms for Cubic Graphs,” J. Graph Algorithms and Applications, vol. 3, no. 2, pp. 1–23, 1999.
[7] T. Calamoneri and R. Petreschi, “An Efficient Orthogonal Grid Drawing Algorithm for Cubic Graphs” Proc. COCOON '95, pp. 31–40, 1995.
[8] T. Calamoneri and R. Petreschi, “A Parallel Algorithm for Orthogonal Grid Drawings of Cubic Graphs,” J. Parallel and Distributed Computing, vol. 55, pp. 94-108, 1998.
[9] R. Cole, "Parallel Merge Sort," SIAM J. Computing, vol. 17, pp. 770-785, 1988.
[10] R. Cole and U. Vishkin, "Approximate Parallel Scheduling. Part 1: The Basic Technique with Applications to Optimal Parallel List Ranking in Logarithmic Time," SIAM J. Computing, vol. 18, pp. 128-142, 1988.
[11] G. DiBattista, P. Eades, R. Tamassia, and I.G. Tollis, "Algorithms for Drawing Graphs: An Annotated Bibliography," Computational Geometry: Theory and Applications, vol. 4, no 5, pp. 235-282, 1994. Also available via anonymous ftp from,, andin.
[12] G. Di Battista, G. Liotta, and F. Vargiu, “Spirality and Optimal Orthogonal Drawings,” SIAM J. Computers, vol. 27, no. 6, pp. 1,764-1,811, 1998.
[13] A. Gibbons and W. Rytter, Efficient Parallel Algorithms. Cambridge Univ. Press, 1988.
[14] R. Greenlaw and R. Petreschi, “Cubic Graphs,” ACM Computing Surveys, vol. 27, no. 4, pp. 471–495, 1995.
[15] J. J'aJ'a, An Introduction to Parallel Algorithms.New York: Addison-Wesley, 1992.
[16] J. Jàjà and J. Simon, “Parallel Algorithms in Graph Theory: Planarity Testing,” SIAM J. Computing, vol. 11, pp. 314–328, 1982.
[17] G. Kant, “Drawing Planar Graphs Using the Canonical Ordering,” Proc. 33th Ann. IEEE Symp. Foundations of Computer Science (FOCS '92), pp. 101–110, 1992.
[18] Y. Liu, P. Marchioro, and R. Petreschi, “At Most Single Bend Embedding of Cubic Graphs,” Applied Mathematics, vol. 9, no. 2, pp. 127–142, 1994.
[19] G.L. Miller and J.H. Reif, “Parallel Tree Contraction and Its Applications,” Proc. 26th Ann. IEEE Symp. Foundations of Computer Science (FOCS '85), pp. 478–489, 1985.
[20] A. Papakostas and I. G. Tollis, "Improved Algorithms and Bounds for Orthogonal Drawings," Proc. 1994 Symp. Graph Drawing (GD '94), pp. 40-51, 1994.
[21] F.P. Preparata and M.I. Shamos, Computational Geometry. Springer-Verlag, 1985.
[22] B.T. Preas and M.J. Lorenzetti, eds., Physical Design Automation of VLSI Systems. Menlo Park, Calif.: Benjamin/Cummings, 1988.
[23] M.J. Quinn, Parallel Computing: Theory and Practice.New York: McGraw-Hill, 1994.
[24] M.S. Rahman, S. Nakano, and T. Nishizeki, “A Linear-Time Algorithm for Optimal Orthogonal Drawings of Triconnected Cubic Plane Graphs,” Proc. Graph Drawing `97, pp. 99–110, 1997.
[25] R. Tamassia, I.G. Tollis, and J.S. Vitter, “Lower Bounds and Parallel Algorithms for Planar Orthogonal Grid Drawings,” Proc. IEEE Symp. Parallel and Distributed Processing, pp. 1–8, 1991.
[26] R. Tamassia, J.S. Vitter, “Parallel Transitive Closure and Point Location in Planar Structures,” SIAM J. Computing, vol. 20, no. 4, pp. 708–725, 1991.
[27] J.C. Wyllie, “The Complexity of Parallel Computations,” PhD thesis, Dept. of Computer Science, Cornell Univ., 1979.

Index Terms:
Cubic graphs, orthogonal drawing, computer graphics. visualization, layout, parallel algorithms.
Citation:
Tiziana Calamoneri, Stephan Olariu, Rossella Petreschi, "A Simple Parallel Algorithm to Draw Cubic Graphs," IEEE Transactions on Parallel and Distributed Systems, vol. 11, no. 10, pp. 1009-1018, Oct. 2000, doi:10.1109/71.888641
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