Abstract
Graphs can be represented symbolically by the Ordered Binary Decision Diagram (OBDD) of their characteristic function. To solve problems in such implicitly given graphs, specialized symbolic algorithms are needed which are restricted to the use of functional operations offered by the OBDD data structure. In this paper, two symbolic algorithms for the single-source shortest-path problem with nonnegative positive integral edge weights are presented which represent symbolic versions of Dijkstra’s algorithm and the Bellman-Ford algorithm. They execute \(\mathcal{O}(N\cdot{\rm log}(NB))\) resp. \(\mathcal{O}(NM\cdot{\rm log}(NB))\) OBDD-operations to obtain the shortest paths in a graph with N nodes, M edges, and maximum edge weight B. Despite the larger worst-case bound, the symbolic Bellman-Ford-approach is expected to behave much better on structured graphs because it is able to handle updates of node distances effectively in parallel. Hence, both algorithms have been studied in experiments on random, grid, and threshold graphs with different weight functions. These studies support the assumption that the Dijkstra-approach behaves efficient w. r. t. space usage, while the Bellman-Ford-approach is dominant w. r. t. runtime.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bahar, R.I., Frohm, E.A., Gaona, C.M., Hachtel, G.D., Macii, E., Pardo, A., Somenzi, F.: Algebraic decision diagrams and their applications. In: ICCAD 1993, pp. 188–191. IEEE Press, Los Alamitos (1993)
Bellman, R.: On a routing problem. Quarterly of Applied Mathematics 16, 87–90 (1958)
Bloem, R., Gabow, H.N., Somenzi, F.: An algorithm for strongly connected component analysis in n log n symbolic steps. In: Johnson, S.D., Hunt Jr., W.A. (eds.) FMCAD 2000. LNCS, vol. 1954, pp. 37–54. Springer, Heidelberg (2000)
Bryant, R.E.: Symbolic manipulation of Boolean functions using a graphical representation. In: DAC 1985, pp. 688–694. ACM Press, New York (1985)
Bryant, R.E.: Graph-based algorithms for Boolean function manipulation. IEEE Transactions on Computers 35, 677–691 (1986)
Dijkstra, E.W.: A note on two problems in connexion with graphs. Numerische Mathematik 1, 269–271 (1959)
Gentilini, R., Piazza, C., Policriti, A.: Computing strongly connected components in a linear number of symbolic steps. In: SODA 2003, pp. 573–582. ACM Press, New York (2003)
Gentilini, R., Policriti, A.: Biconnectivity on symbolically represented graphs: A linear solution. In: Ibaraki, T., Katoh, N., Ono, H. (eds.) ISAAC 2003. LNCS, vol. 2906, pp. 554–564. Springer, Heidelberg (2003)
Hachtel, G.D., Somenzi, F.: Logic Synthesis and Verification Algorithms. Kluwer Academic Publishers, Boston (1996)
Hachtel, G.D., Somenzi, F.: A symbolic algorithm for maximum flow in 0–1 networks. Formal Methods in System Design 10, 207–219 (1997)
Hojati, R., Touati, H., Kurshan, R.P., Brayton, R.K.: Efficient ω-regular language containment. In: Probst, D.K., von Bochmann, G. (eds.) CAV 1992. LNCS, vol. 663, pp. 396–409. Springer, Heidelberg (1993)
Mahadev, N.V.R., Peled, U.N.: Threshold Graphs and Related Topics. Elsevier Science, Amsterdam (1995)
Moon, J.H., Kukula, K.: Ravi, and F. Somenzi. To split or to conjoin: The question in image computation. In: DAC 2000, pp. 23–28. ACM Press, New York (2000)
Ravi, K., Bloem, R., Somenzi, F.: A comparative study of symbolic algorithms for the computation of fair cycles. In: Johnson, S.D., Hunt Jr., W.A. (eds.) FMCAD 2000. LNCS, vol. 1954, pp. 143–160. Springer, Heidelberg (2000)
Sawitzki, D.: Implicit flow maximization by iterative squaring. In: Van Emde Boas, P., Pokorný, J., Bieliková, M., Štuller, J. (eds.) SOFSEM 2004. LNCS, vol. 2932, pp. 301–313. Springer, Heidelberg (2004)
Sawitzki, D.: Implicit flow maximization on grid networks. Technical report, Universität Dortmund (2004)
Sawitzki, D.: On graphs with characteristic bounded-width functions. Technical report, Universität Dortmund (2004)
Sawitzki, D.: A symbolic approach to the all-pairs shortest-paths problem (2004) (submitted)
Wegener, I.: Branching Programs and Binary Decision Diagrams. SIAM, Philadelphia (2000)
Woelfel, P.: Symbolic topological sorting with OBDDs. In: Rovan, B., Vojtáš, P. (eds.) MFCS 2003. LNCS, vol. 2747, pp. 671–680. Springer, Heidelberg (2003)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Sawitzki, D. (2004). Experimental Studies of Symbolic Shortest-Path Algorithms. In: Ribeiro, C.C., Martins, S.L. (eds) Experimental and Efficient Algorithms. WEA 2004. Lecture Notes in Computer Science, vol 3059. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24838-5_36
Download citation
DOI: https://doi.org/10.1007/978-3-540-24838-5_36
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-22067-1
Online ISBN: 978-3-540-24838-5
eBook Packages: Springer Book Archive