Abstract
Let N be a single-source single-sink flow network with n nodes, m arcs, and positive arc costs. We present a pseudo-polynomial algorithm that computes a maximum flow of minimum cost for N in time O(χ 3/4 m√log n), where χ is the cost of the flow. This improves upon previously known methods for networks where the minimum cost of the flow is small. We also show an application of our flow algorithm to a well-known graph drawing problem. Namely, we show how to compute a planar orthogonal drawing with the minimum number of bends for an n- vertex embedded planar graph in time O(n 7/4√log n). This is the first subquadratic algorithm for bend minimization. The previous best bound for this problem was O(n 2 log n) [19].
Research supported in part by the National Science Foundation under grant CCR-9423847.
Chapter PDF
Similar content being viewed by others
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
R.K. Ahuja, T.L. Magnanti, and J.B. Orlin. Network Flows: Theory, Algorithms, and Applications. Prentice Hall, Englewood Cliffs, NJ, 1993.
T. Biedl and G. Kant. A better heuristic for orthogonal graph drawings. In Proc. 2nd Annu. European Sympos. Algorithms (ESA '94), volume 855 of Lecture Notes in Computer Science, pages 24–35. Springer-Verlag, 1994.
G. Di Battista, P. Eades, R. Tamassia, and I. G. Tollis. Algorithms for drawing graphs: an annotated bibliography. Comput. Geom. Theory Appl., 4:235–282, 1994.
G. Di Battista, A. Garg, G. Liotta, R. Tamassia, E. Tassinari, and F. Vargiu. An experimental comparison of three graph drawing algorithms. In Proc. 11th Annu. ACM Sympos. Comput. Geom., pages 306–315, 1995.
G. Di Battista, A. Giammarco, G. Santucci, and R. Tamassia. The architecture of Diagram Server. In Proc. IEEE Workshop on Visual Languages (VL '90), pages 60–65, 1990.
G. Di Battista, G. Liotta, and F. Vargiu. Spirality of orthogonal representations and optimal drawings of series-parallel graphs and 3-planar graphs. In Proc. Workshop Algorithms Data Struct., volume 709 of Lecture Notes in Computer Science, pages 151–162. Springer-Verlag, 1993.
G. Di Battista, G. Liotta, and F. Vargiu. Diagram Server. J. Visual Lang. Comput., 6(3):275–298, 1995. (special issue on Graph Visualization, edited by I. F. Cruz and P. Eades).
S. Even and G. Granot. Grid layouts of block diagrams — bounding the number of bends in each connection. In R. Tamassia and I. G. Tollis, editors, Graph Drawing (Proc. GD '94), volume 894 of Lecture Notes in Computer Science, pages 64–75. Springer-Verlag, 1995.
L.R. Ford and D.R. Fulkerson. A primal-dual algorithm for the capacitated hitchcock problem. Naval Research Logistics Quarterly, 4:47–54, 1957.
L.R. Ford and D.R. Fulkerson. Flows in Networks. Princeton University Press, Princeton, NJ, 1962.
U. Fößmeier and M. Kaufmann. On bend-minimum orthogonal upward drawing of directed planar graphs. In R. Tamassia and I. G. Tollis, editors, Graph Drawing (Proc. GD '94), volume 894 of Lecture Notes in Computer Science, pages 52–63. Springer-Verlag, 1995.
A. Garg and R. Tamassia. On the computational complexity of upward and rectilinear planarity testing. In R. Tamassia and I. G. Tollis, editors, Graph Drawing (Proc. GD '94), volume 894 of Lecture Notes in Computer Science, pages 286–297. Springer-Verlag, 1995.
G. Kant. Drawing planar graphs using the canonical ordering. Algorithmica, 16:4–32, 1996. (special issue on Graph Drawing, edited by G. Di Battista and R. Tamassia).
Y. Liu, P. Marchioro, R. Petreschi, and B. Simeone. Theoretical results on at most 1-bend embeddability of graphs. Technical report, Dipartimento di Statistica, Univ. di Roma “La Sapienza”, 1990.
K. Mehlhorn. Graph Algorithms and NP-Completeness, volume 2 of Data Structures and Algorithms. Springer-Verlag, Heidelberg, West Germany, 1984.
A. Papakostas and I. G. Tollis. Improved algorithms and bounds for orthogonal drawings. In R. Tamassia and I. G. Tollis, editors, Graph Drawing (Proc. GD '94), volume 894 of Lecture Notes in Computer Science, pages 40–51. Springer-Verlag, 1995.
D.D. Sleator. An O(nm log n) Algorithm for Maximum Network Flow. PhD thesis, Dept. Comput. Sci., Stanford Univ., Palo Alto, California, 1980.
J. A. Storer. On minimal node-cost planar embeddings. Networks, 14:181–212, 1984.
R. Tamassia. On embedding a graph in the grid with the minimum number of bends. SIAM J. Comput, 16(3):421–444, 1987.
R. Tamassia and I. G. Tollis. Planar grid embedding in linear time. IEEE Trans. Circuits Syst., CAS-36(9):1230–1234, 1989.
R. E. Tarjan. Data Structures and Network Algorithms, volume 44 of CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial Applied Mathematics, 1983.
L. Valiant. Universality considerations in VLSI circuits. IEEE Trans. Comput., C-30(2):135–140, 1981.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1997 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Garg, A., Tamassia, R. (1997). A new minimum cost flow algorithm with applications to graph drawing. In: North, S. (eds) Graph Drawing. GD 1996. Lecture Notes in Computer Science, vol 1190. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-62495-3_49
Download citation
DOI: https://doi.org/10.1007/3-540-62495-3_49
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-62495-0
Online ISBN: 978-3-540-68048-2
eBook Packages: Springer Book Archive