Skip to main content
SpringerLink
Log in

Fast parallel algorithms and the complexity of parallelism (basic issues and recent advances)

  • Invited Papers
  • Conference paper
  • First Online:
Book cover Parcella '88 (Parcella 1988)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 342))

  • 137 Accesses

Abstract

The paper classifies the problems solvable very fast in parallel with a feasible number of processors and contrasts them with problems hard to parallelize. The notions of NC1-reducibilities, RNC1-reducibilities, log-space reducibilities are presented and problems complete (or randomly complete) for P are defined and discussed. The notion of optimal parallel time-number of processors tradeoff is highlighted and several techniques to achieve optimal tradeoff are presented. Several results about fast parallel approximations to P-complete problems are described. Finally, certain milestones in the design of efficient parallel algorithms are noted and their impact is discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • [Aleliunas, Karp, Lipton, Lovasz, Rackoff, 1979] "Random Walks, Universal Traversal Sequences and the Complexity of Maze problems", 20th IEEE FOCS Symposium, Oct. 1979, pp. 218–223.

    Google Scholar 

  • [Anderson, Mayr, 1986] "Approximating P-complete problems" by Anderson R. and Mayr E., Tech. Report, Stanford University.

    Google Scholar 

  • [Adleman, L., 1978] "Two theorems on random polynomial time", in Proc. 19th IEEE Found. Comput. Sci., pp. 75–83.

    Google Scholar 

  • [Aho, Horcroft, Ullman, 1974] "The Design and Analysis of Computer Algorithms", Addison-Wesley, Reading, Mass.

    Google Scholar 

  • [Awerbuch, Shiloach, 1983] "New connectivity and MSF algorithm for ultra-computer and PRAM", preprint, IBM-Israel Scientific Center, technion, Haifa.

    Google Scholar 

  • [Aggarwal et al, 1985] "Parallel computational geometry", A. Aggarwal, B. Chacelle, L. Guibas, C. O'Dunlaing, C. Yap, IEEE Foundations of Computes Science, pp. 468–477, 1985.

    Google Scholar 

  • [Ajtai, Kamlcs, Szemeredi, 1983] "Sorting in clogn parallel steps", Canbinatorica 3:1–19, 1983.

    Google Scholar 

  • [Borodin, Von zur Gathen, Hopcroft, 1982] "Fast parallel matrix and GCD computations" by Borodin A., Von zur Gathen J. and Hopcroft J., Inform. and Control 52, 241–256.

    Google Scholar 

  • [Berkowitz, 1984] "On computing the determinant in small parallel time using a small number of processors, Inform. Process, Lett. 18, 147–150.

    Google Scholar 

  • [Borodin, 1977] "On relating time and space to size and depth." SIAM J. Comput. 6, 733–744.

    Google Scholar 

  • [Borodin 1982] Structured vs. general models in computational complexity, in "Logic and Algorithmic", Enseign. Math. (no. 30) pp. 47–65, Univ. Geneva, Geneva.

    Google Scholar 

  • [Beame, Cook, Hoover, 1984] "Log depth circuits for division and related problems", in Proc. 17th IEEE Found. Comput. Sci.

    Google Scholar 

  • [Borodin, Cook, Pippenger, 1983] "Parallel computation for well-endowed rings and space-bounded probabilistic machines", Inform. and Control 58, 113–136.

    Google Scholar 

  • [Borodin, Von zur Gathen, Hopcroft, 1982] "Fast parallel matrix and GCD computations", Inform. and Control 52, 241–256.

    Google Scholar 

  • [Cole, 87] "Parallel Merge Sort" R. Cole, 27th IEEE FOCS.

    Google Scholar 

  • [Cole, Vishkin, 86] "Approximate and Exact Parallel Scheduling with Applications to List, Tree and Graph problems", R. Cole, U. Vishkin, 27th IEEE Symposium on Foundations of Computer Science, 1986.

    Google Scholar 

  • [Chandra, Kozen, Stockmeyer, 1981] Alternion, J. Assoc. Comput. Mach. 28, No. 1, 114–113.

    Google Scholar 

  • [Chandra, Stockmeyer, Vishkin, 1982] "Complexity theory for unbounded fan-in parallelism", in Proc. 23rd IEEE Found. Comput. Sci., pp. 1–13.

    Google Scholar 

  • [Cook, 1981] "Towards a complexity theory of synchronous parallel computation, Enseign. Math. 27, 99–124.

    Google Scholar 

  • [Cook, 1983] "The classification of problems which have fast parallel algorithms", in Proc. 1983 International FCT Conference, Lecture notes in Computer Science Vol. 158, pp. 78–93, Springer-Verlag, Berlin/New York.

    Google Scholar 

  • [Cook, Ruzzo, 1983] unpublished theorem.

    Google Scholar 

  • [Csanky, 1976 Fast parallel matrix inversion algorithms, SIAM J. Comput. 5, 618–623.

    Google Scholar 

  • [Dymond, Cook, 1980] "Hardware complexity and parallel computation", in Proc. 21st IEEE Found. Comput. Sci., 360–372.

    Google Scholar 

  • [Doakin, Lipton, Reiss, 1979] "Linear programming is log-space hard for P, Inform. Process. Lett. 8, 96–97.

    Google Scholar 

  • [Dekel, Nassimi, Sahni, 1981] "Parallel matrix and graph algorithms, SIAM J. Comput. 10, 657–675.

    Google Scholar 

  • [Feather, 1984] M.Sc. thesis, Department of Computer Science, University of Toronto.

    Google Scholar 

  • [Furst, Saxe, Sipser, 1981] "Parity, circuits, and the polynomial-time hierarchy", in Proc. 22nd IEEE Found. Comput. Sci., pp. 260–270.

    Google Scholar 

  • [Garey, Johnson, 1979] "Computers and Intractability: A Guide to the Theory of NP-Completeness", Freeman, San Francisco.

    Google Scholar 

  • [Von zur Gathen, 1983] "Parallel algorithms for algebraic problems", in Proc. 15th ACM Sympos. Theory of Comput., pp. 17–23.

    Google Scholar 

  • [Gill, 1977] "Computational Complexity of probabilistic Turing Machine", SIAM J. Comput. 6, 675–695.

    Google Scholar 

  • [Goldschlager, 1977;1978;1982] "Synchronous Parallel Computation", Ph.D. thesis, University of Toronto; in Proc. ACM Sympos. Theory of Comput., pp. 89–94, Assoc. Comput. Mach. 29, No. 4, 1073–1086.

    Google Scholar 

  • [Goldschlager, 1977] "The monotone and planar circuit value problems are log space complete for P. SIGACT News 9, No. 2, 25–29.

    Google Scholar 

  • [Greibach, 1973] "The hardest context-free language", SIAM J. Comput. 2, 304–310.

    Google Scholar 

  • [Goldschlager, Shaw, Staples, 1982] "The maximum flow problem is log space complete for P. Theoret. Comput. Sci. 21, 105–111.

    Google Scholar 

  • [Gupta, 1985] M.Sc. thesis, Dept. of Computer Science, University of Toronto.

    Google Scholar 

  • [Hoover, Ruzzo, 1984] "A compendium of problems complete for P", by H. Hoover and W.L. Ruzzo, unpublished manuscript, 1984.

    Google Scholar 

  • [Hong, 1980] "On some space complexity problems about the set of assignments satisfying a boolean cormula", in Proc. 12th ACM Sympos. Theory of Comput., pp. 310–317.

    Google Scholar 

  • [Hoover, 1979] "Some Topics in Circuit Complexity", M.Sc. thesis, University of Toronto, Department of Computer Science, Department of Computer Science Technical Report, 139/80.

    Google Scholar 

  • [Ibarra, Mcran, Rosier, 1980] "A note on the parallel compexity of computing the rank of order n matrices", Inform. Process. Lett. 11, 162.

    Google Scholar 

  • [Ja'Ja'J, Simon, 1982] "Parallel algorithms in graph theory Planarity testing", SIAM J. Comput. 11, 314–328.

    Article  Google Scholar 

  • [Jones, Laaser, 1977] "Complete problems for deterministic polynomial time, Theoretical Computer Science 3, 105–117.

    Article  Google Scholar 

  • [Jones, Lien, Laaser, 1976] "New problems complete for nondeterministic log space", Math. Systems Theory 10, 1–17.

    Google Scholar 

  • [Kyrousis, Spirakis, 1988] "Probabilistic log-space reductions and problems probabilistically hard for P", L. Kyrousis and P. Spirakis, "Scandinaviocn Workshop on Algorithm Theory (SWAT 88)", July 5–8, 1988, Proceedings.

    Google Scholar 

  • [Kozen, Yap, 1985] "Algebraic cell decomposition in NC", D. Kozen & C. Yap, IEEE Foundations of Computer Science Proceedings, pp. 515–521, 1985.

    Google Scholar 

  • [Karp, Upfal, Wigderson, 1985] "Constructing a perfect matching is in Random NC", 1985, ACM STOC, pp. 22–33.

    Google Scholar 

  • [Khachlan, 1979] "A polynomial time algorithm for linear programming", Dokl. Akad. Nauk. SSSR 144, No. 5, 1093–96; transl. in Soviet Math. Dokl. 20, 191–194.

    Google Scholar 

  • [Karp, Upfal, Wigderson, 1985] "Constructing a perfect matching is in random NC", in Proc. 17th ACM Sympos. Theory of Comput., pp. 22–32.

    Google Scholar 

  • [Karp, Wigderson, 1984] "A fast parallel algorithm for the maximal independent set problem", in Proc. 16th ACM Sympos. Theory of Comput., pp. 266–272.

    Google Scholar 

  • [Ladner, 1975] "The circuit value problem is log space complete for p", SIGACT News 7, No. 1, 18–20.

    Article  Google Scholar 

  • [Lev., 1978] "Size-Bounds and Parallel Algorithms for Network", Doctoral thesis, Report CST-8-80. Dept. of Computer Science, University of Edimburgh.

    Google Scholar 

  • [Lynch, 1977] "Log space recognition and translation of parenthesis languages, J. Assoc. Comput. Math. 24, No. 4, 583–590.

    Google Scholar 

  • [Lipton, Snyder, Zalcstein, 1976] "The complexity of the Word and Isomorphism Problems for Finite Groups", Tech. Rep. 91/76, Yale University.

    Google Scholar 

  • [McKenzie, 1984] "Parallel Complexity and Permutation Groups", Ph.D. thesis, University of Toronto, Department of Computer Science.

    Google Scholar 

  • [Miller, 1978] "On the nlogn isomorphism technique", in Proc. 10th Sympos. Theory of Comput., pp. 51–58.

    Google Scholar 

  • [McKenzie, Cook, 1983] "The parallel complexity of the Abelian permutation group membership problem", in Proc. 24th IEEE Found of Comput. Sci., pp. 154–161.

    Google Scholar 

  • [McKenzie, Cook, 1985] "The parallel complexity of Abelian Permutation group problems, University of Toronto, Dept. of Computer Science, Technical Report No. 181/85.

    Google Scholar 

  • [Muller, Preparata, 1975] "Bounds to complexities of networks for sorting and switching", J. Assoc. Comput. Mach. 22, No. 2, 195–201.

    Google Scholar 

  • [Pippenger, 1979] "On simultaneous resource bounds (preliminary version)", in Proc. 20th IEEE Found. of Comput. Sci., pp. 307–311.

    Google Scholar 

  • [Papadimitriou, Steiglitz, 1982] "Combinatorial Optimization: Algorithms and Complexity", Prentice-Hall, Englewood Cliffs, N.J.

    Google Scholar 

  • [Pratt, Stockmeyer, 1976] "A characterization of the power of vector machines", J. Comput. System Sci. 12, 198–221.

    Google Scholar 

  • [Reif, 1982] "Symmetric complementation", in Proc. 14th ACM Sympos. Theory of Comput. pp. 201–214.

    Google Scholar 

  • [Reif, 1983] "Logarithmic depth circuits for algebraic functions", in 24th IEEE Found. of Comput. Sci., pp. 138–145, revised version, Preprint, 1984.

    Google Scholar 

  • [Ruzzo, 1980a] unpublished list of problems in NC2.

    Google Scholar 

  • Ruzzo, 1980b "Tree-size bounded alternation", J. Comput. System Sci. 21, No. 2, 218–235.

    Google Scholar 

  • [Ruzzo, 1981] "On uniform circuit complexity", J. Comput. System Sci. 22, No. 3, 365–383.

    Google Scholar 

  • [Ruzzo, 1984] private communication

    Google Scholar 

  • [Ruzzo, Tompa, 1982] unpublished result. See Stockmeyer, L. and Vishkin L., "Simulation of Parallel Random Access Machines by Circuits", Report RC-9362. IBM Research. Yorktown Heights. N.Y.

    Google Scholar 

  • [Savage, 1976] "The Complexity of Computing", Wiley, New York.

    Google Scholar 

  • [Skyum, Vallant, 1981] "A complexity theory based on Boolean algebra", in Proc. 22nd IEEE Found. of Comput. Sci., pp. 244–253.

    Google Scholar 

  • [Schwartz, 1980a] "Ultracomputers", ACM Trans. Program, Lang. Systems 2, No. 4, 484–521.

    Google Scholar 

  • [Schwartz, 1980b] "Probabilistic algorithms for verification of polynomial identities", J. Assoc. Comput. Mach. 27, No. 4, 701–717.

    Google Scholar 

  • [Shannon, 1949] "The synthesis of two terminal switching circuits", BSTJ 28, 55–98.

    Google Scholar 

  • [Subsgrough, 1978] "On the tape complexity of deterministic context-free languages", J. Assoc. Comput. Mach. 25, No. 3, 405–414.

    Google Scholar 

  • [Spirakis, 1987] "The parallel complexity of deadlock detection", P. Spirakis, Theoretical Computer Science, 52, 1987, 155–163.

    Google Scholar 

  • [Spirakis, 1988] "Optimal Parallel Randomized Algorithms for Sparse Addition and Identification", P. Spirakis, Information and Computation Journal, Academic Press, New York, Vol. 76, No. 1, Jan. 1988, pp. 1–12.

    Google Scholar 

  • [Schwartz, 1980] "Fast probabilistic algorithms for verification of polynomial identities", JACM 27, 4, 1980, pp. 701–717.

    Google Scholar 

  • Tompa, 1984 private communication.

    Google Scholar 

  • [Vallant, 1979] "Completeness classes in algebra", in Proc. 11th ACM Sympos. Theory of Comput., pp. 249–261.

    Google Scholar 

  • [Vallant, Skyum, Berkowitz, Rackoff, 1983] "Fast parallel computation of polynomials using few processors", SIAM J. Comput. 12, No. 4, 641–644.

    Google Scholar 

  • [Vishkin, 1983] "Synchronous parallel computation — A survey, preprint Courant Institute, New York University.

    Google Scholar 

  • [Wilson, 1983] "Relativised circuit complexity", in Proc. 24th IEEE Found. of Comput. Sci., pp. 329–342.

    Google Scholar 

  • [Warmuth, 1987] "Parallel Approximation Algorithms for One-Dimensional Bin Packing", by M. Warmuth, tech. report, Univ. of California Santa Cruz, 1987.

    Google Scholar 

  • [Yap, 1987] "What can be parallelized in computational geometry?" C.K. Yap, Parallel Algorithms and Architectures, Academic-Verlag, Berlin, May 25–30, 1987.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Computer Technology Institute, PO Box 1122, 26110, Patras, Greece

    Paul G. Spirakis

  2. Courant Institute of Math. Science, NYU, USA

    Paul G. Spirakis

Authors
  1. Paul G. Spirakis
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Gottfried Wolf Tamáas Legendi Udo Schendel

Rights and permissions

Reprints and permissions

Copyright information

© 1989 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Spirakis, P.G. (1989). Fast parallel algorithms and the complexity of parallelism (basic issues and recent advances). In: Wolf, G., Legendi, T., Schendel, U. (eds) Parcella '88. Parcella 1988. Lecture Notes in Computer Science, vol 342. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-50647-0_110

Download citation

  • DOI: https://doi.org/10.1007/3-540-50647-0_110

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-50647-8

  • Online ISBN: 978-3-540-46062-6

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation