Abstract
We improve upon the running time of several graph and network algorithms when applied to dense graphs. In particular, we show how to compute on a machine with word size λ=Ω (logn) a maximal matching in ann-vertex bipartite graph in timeO(n 2+n 2.5/λ)=O(n 2.5/logn), how to compute the transitive closure of a digraph withn vertices andm edges in timeO(n 2+nm/λ), how to solve the uncapacitated transportation problem with integer costs in the range [O.C] and integer demands in the range [−U.U] in timeO ((n 3 (log log/logn)1/2+n2 logU) lognC), and how to solve the assignment problem with integer costs in the range [O.C] in timeO(n 2.5 lognC/(logn/loglogn)1/4).
Assuming a suitably compressed input, we also show how to do depth-first and breadth-first search and how to compute strongly connected components and biconnected components in timeO(nλ+n 2/λ), and how to solve the single source shortest-path problem with integer costs in the range [O.C] in time0 (n 2(logC)/logn). For the transitive closure algorithm we also report on the experiences with an implementation.
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References
H. Alt, N. Blum, K. Mehlhorn, and M. Paul. Computing a maximum cardinality matching in a bipartite graph in timeO(n 1.5√m/logn).Inform. Process. Lett., 37:237–240, 1991.
R. K. Ahuja, A. V. Goldberg, J. B. Orlin, and R. E. Tarjan. Finding minimum-cost flows by double scaling.Math. Programming, 53:243–266, 1992.
R. K. Ahuja, R. L. Magnanti, and J. B. Orlin. Network flows.Handbook in Oper. Res. Management Sci., 1:211–360, 1991.
D. Angluin and L. G. Valiant. Fast probabilistic algorithms for hamiltonian circuits and matching.J. Comput. Systems Sci., 18:155–193, 1979.
D. P. Bertsekas. A new algorithm for the assignment problem.Math. Programming, 21:152–171, 1981.
J. Cheriyan, T. Hagerup, and K. Mehlhorn. Can a maximum flow be computed ino(nm) time.Proc. 17thICALP Conference, pp. 235–248. Lecture Notes in Computer Science, vol. 443. Springer-Verlag, Berlin, 1990. The full version is available as Technical Report MPI-I-91-120, Max-Planck-Institut für Informatik, Saarbrücken.
J. Cheriyan, M. Y. Kao, and R. Thurimella. Scan-first search and sparse certificates: an improved parallel algorithm fork-vertex connectivity.SIAM J. Comput., 22:157–174, 1993.
T. H. Cormen, C. E. Leiserson, and R. L. Rivest.Introduction to Algorithms. McGraw-Hill/The MIT Press, New York/Cambridge, MA, 1990.
E. W. Dijkstra. A note on two problems in connexion with graphs.Numer. Math., 1:269–271, 1959.
E. W. Dijkstra.Selected Writings in Computing: A Personal Perspective. Springer-Verlag, Berlin, 1982.
T. Feder and R. Motwani. Clique partitions, graph compression and speeding-up algorithms.Proc. 23rd ACM STOC, pp. 123–133, 1991.
M. L. Fredman and D. E. Willard. Blasting through the information theoretic barrier with fusion trees.Proc. 22nd ACM STOC, pp. 1–7, 1990.
A. Goralcikova and V. Koubek. A reduct and closure algorithm for graphs.Proc. Mathematical Foundations of Computer Science, pp. 301–307. Lecture Notes in Computer Science, vol. 74. Springer-Verlag, Berlin, 1979.
H. N. Gabow and R. E. Tarjan. Faster scaling algorithms for network problems.SIAMJ. Comput., 18:1013–1036, 1989.
J. E. Hopcroft and R. M. Karp. Ann 5/2 algorithm for maximum matchings in bipartite graphs.SIAM J. Comput., 2:225–231, 1973.
D. Kirkpatrick and S. Reisch.Upper Bounds for Sorting Integers on Random Access Machines. EATCS Monographs on Theoretical Computer Science, vol. 28, pp. 263–276. Springer-Verlag, Berlin, 1984.
K. Mehlhorn.Data Structures and Efficient Algorithms, vol. I–III. Springer-Verlag, Berlin, 1984.
K. Mehlhorn and St. Näher. LEDA: A platform for combinatorial and geometric computing.Comm. ACM, 38(1):96–102, 1995.
St. Näher. LEDA manual. Technical Report MPI-I-93-109, Max-Planck-Institut für Informatik, 1993.
I. B. Orlin and R. K. Ahuja. New scaling algorithms for the assignment and minimum mean cycle problems.Math. Programming, 54:41–56, 1992.
M. Sharir. A strong-connectivity algorithm and its application in data flow analysis.Comput. Math. Appl., 7(1):67–72, 1981.
K. Simon. An improved algorithm for transitive closure on acyclic digraphs.Proc. 13th ICALP Conference, pp. 376–386. Lecture Notes in Computer Science, vol. 226. Springer-Verlag, Berlin, 1986.
R. E. Tarjan. Depth-first search and linear graph algorithms.SIAM J. Comput., 1:146–160, 1972.
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Communicated by R. Sedgewick.
Most of this research was carried out while both authors worked at the Fachbereich Informatik, Universität des Saarlandes, Saarbrücken, Germany. The research was supported in part by ESPRIT Project No. 3075 ALCOM. The first author acknowledges support also from NSERC Grant No. OGPIN007.
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Cheriyan, J., Mehlhorn, K. Algorithms for dense graphs and networks on the random access computer. Algorithmica 15, 521–549 (1996). https://doi.org/10.1007/BF01940880
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DOI: https://doi.org/10.1007/BF01940880