Abstract
The opinion has sometimes been voiced that parallel randomized computations have only a pure theoretical interest. We instead believe that randomized algorithms have and will have several applications ranging from very theoretical to quite practical. The aim of this chapter has been to provide strong evidence of our convinction.
In our view the covered material represents only the starting point of a deeper study in the enourmous potentiality of probabilistic methods in the design and analysis of parallel algorithms. By the examples we have illustrated, readers from different research areas may find interesting ideas in their own topics.
This work was partially supported by the Human Capital and Mobility project SCOOP — Solving Combinatorial Optimization Problems in Parallel — of the European Union, and it was done when the first author had a Post-Doc Fellowship at the Centre Universitaire d'Informatique of University of Geneva.
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References
N. Alon and J.H. Spencer: The Probabilistic Method. Wiley-Interscience Publication, 1992.
A. Aggarwal, R.J. Anderson: A Random NC Algorithm for Depth First Search. 19th Annual ACM-STOC, 325–334, 1987.
R.J. Anderson: A parallel algorithm for the maximal path problem. Combinatorica 7(3), 400–415, 1987.
L. Babai, Fortnow L. Levin L and M. Szegedy: Checking computation in polylogarithmic time. 23th Annual ACM-STOC 21–28, 1991.
B. Bollobas: Random Graphs. Academic Press, 1985.
D.P. Bovet and P. Crescenzi: Introduction to the Theory of Complexity. Prentice Hall, 1994.
A. Clementi, L. Kucera, J. Rolim: A Note on Parallel Randomized Algorithms for Searching Problems”, DIMACS Series in Discrete Mathematics and Theoretical Computer Sciences, American Mathematical Society, 22, 33–44, 1995.
E. Cohen: Polylog-time and near-linear work approximation scheme for undirected shortest paths. 26th Annual ACM-STOC, 16–26, 1994.
D. Coppersmith, P. Raghavan, M. Tompa: Parallel Graph Algorithms that are Efficient on Average. 28th Annual IEEE-FOCS, 260–269, 1987.
L. Csanky: Fast Parallel Matrix inversion Algorithms. SIAM J. of Computing 5, 618–623, 1976.
P. Erdos and A. Renyi: On Random Graphs I. Publ. Math. Debrecen 6, 290–297, 1959.
L.R. Ford, D.R. Fulkerson: Flows in Networks. Princeton Univ. Press, Princeton, NJ, 1962.
Z. Galil, V. Pan: Improved Processor Bounds for Algebraic and Combinatorial Problems in RNC. 26th Annual IEEE-FOCS, 490–495, 1985.
M.R. Garey, D.S. Johnson: Computers and intractability: a guide to the theory of NP-completeness. Freeman, San Francisco, 1979.
R. K. Ghosh, G. P. Bhattacharjee: A parallel search algorithm for directed acyclic graphs. BIT 24, 134–150, 1984.
R. Greenlaw: Polynomial Completeness and Parallel Computation. Synthesis of Parallel Algorithms, Ed. J. Reif, Morgan-Kaufmann, 1993.
J. Gil, Y. Matias, U. Vishkin: Towards a theory of nearly constant time parallel algorithms. 32th Annual IEEE-FOCS, 698–710, 1991.
J. JáJá: Parallel Algorithms. Addison-Wesley, 1992.
R. Karp: An introduction to randomized algorithms. Discr Appl Math, 34, 165–201, 1991.
R.M. Karp, A. Wigderson: A fast parallel algorithm for the maximal independent set problem. J. of ACM, 32, 762–773, 1985.
R.M. Karp, E. Upfal, A. Wigderson: Constructing a Maximum Matching is in Random NC. Combinatorica 6(1), 35–48, 1986. A preliminary version also appeared in 17th Annual ACM-STOC, 1985.
R. Karp and V. Ramachandran: Parallel Algorithms for Shared-Memory Machines. Handbook of T.C.S., Ed. J. van Leeuwen, Elsevier Science, Vol. A, Chapter 17, 1990.
P.N. Klein, S. Sairam: A Parallel Randomized Approximation Scheme for Shortest Paths. 24th Annual ACM-STOC, 750–758, 1992.
P.N. Klein, S. Sairam: A linear-processor polylog-time algorithm for shortest paths in planar graphs. Proc. of the 34th Annual IEEE FOCS 259–270, 1994.
D. Kavvadias, G.E. Pantziou, P.G. Spirakis, C. D. Zaroliagis: Hammock-on-Ears Decomposition: A Technique for the Efficient Parallel Solution of Shortest Paths and Other Problems. 19th MFCS, LNCS, 841, 462–472, 1994.
L. Kučera: Expected behaviour of graph coloring algorithms. Fundamentals in Computation Theory, LNCS, 56, 447–451, 1984.
L. Lovasz: On Determinants, Matchings and Random Algorithm. Fundamentals of Computing Theory, ed. L. Budach, Akademia-Verlag, Berlin, 1979.
M. Luby: A simple parallel algorithm for the maximal independent set problem. SIAM J. on Computing, 15, 1036–1053, 1986. (also in 17th Annual ACM-STOC).
S. Micali V.V. Vazirani: An O(√¦V∥E¦) algorithm for finding maximum matching in general graphs. the 21st Annual Symp. on IEEE-FOCS. 17–27, 1980.
K. Mulmuley, U.V. Vazirani, V.V Vazirani: Matching is as easy as matrix inversion. Combinatorica 7, 105–113, 1987 (also in 19th Annual ACM-STOC, 345–354, 1987.
S. Nikoletseas, K. Palem, P. Spirakis, M. Yung: Short Vertex Disjoint paths and Multiconnectivity in Random Graphs: Reliable Networks Computing. 21st ICALP, LNCS, 508–519, 1994.
V. Pan: Fast and Efficient Algorithms for the Exact Inversion of Integer Matrices. Fifth Annual Foundations of Software Technology and Theoretical Computer Science Conference, pp. 504–521, 1985.
G. Pantziou, P. Spirakis and C. Zaroliagis: Coloring Random Graphs Efficiently in Parallel, through Adaptive Techniques. CTI TR-90.10.25, Comp. Techn. Institute, Patras. Also presented in the ALCOM Workshop on Graphs Algorithms, Data Structures and Computational Geometry, Berlin, October, 1990.
P. Raghavan, Motwani: Randomized Algorithms. Cambridge University Press, 1995.
P. Raghavan, C.D. Thompson: Provably good routing in graphs: regular arrays. 17th Annual ACM-STOC, 79–87, 1985.
E. Reghbati, D. Corniel: Parallel computations in graph theory. SIAM J. on Computing 7, 230–237, 1978.
J. H. Reif: depth first search is inherently sequential. IPL 20, 229–234, 1985.
J.T. Schwartz: Fast Probabilistic Algorithms for Verification of Polynomial Identities. J. of ACM 27(4), 701–717, 1980.
M. Serna, P.G. Spirakis: Tight RNC Approximations to Max Flow. 8th Annual STACS, LNCS 480, 118–126, 1991.
J. R. Smith: Parallel algorithms for depth first searches I: planar graphs. SIAM J. on Computing 15(3), 814–830, 1986.
W.T. Tutte: The Factorization of Linear Graphs. J. London Math. Soc. 22, pp. 107–111, 1947.
J. Ullmann, M. Yannakakis: High Probability parallel transitive closure algorithms. SIAM J. of Computing 20, 100–125, 1991.
E. Urland: Experimental tests of efficient shortest paths heuristics for random graphs on the CM-2. Techn. Rep. 71, University of Geneva, August, 1994.
J. Van Leeuwen: Graph Algorithms, in Handbook of T.C.S. Ed. J. van Leeuwen, Elsevier Science, Vol. A, 10, 1990.
J.S. Vitter, P. Flajolet: Average-Case Analysis of Algorithms and Data Structures, in Handbook of T.C.S. Ed. J. van Leeuwen, Elsevier Science, Vol. A, 9, 1990.
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Clementi, A., Rolim, J.D.P., Urland, E. (1996). Randomized parallel algorithms. In: Ferreira, A., Pardalos, P. (eds) Solving Combinatorial Optimization Problems in Parallel. Lecture Notes in Computer Science, vol 1054. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0027117
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