Time-space trade-offs in a pebble game

  • W. J. Paul
  • R. E. Tarjan
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 52)


A certain pebble game on graphs has been studied in various contexts as a model for time and space requirements of computations [1,2,3,7]. In this note it is shown that there exists a family of directed acyclic graphs Gn and constants c1,c2,c3 such that
  1. 1)

    Gn has n nodes and each node in Gn has indegree at most 2.

  2. 2)

    Each graph Gn can be pebbled with \(c_1 \sqrt n\) pebbles in n moves.

  3. 3)

    Each graph Gn can also be pebbled with \(c_2 \sqrt n\) pebbles, c2 < c1,


but every strategy which achieves this has at least \(2^{c_3 \sqrt n }\) moves.


Bipartite Graph Directed Acyclic Graph Output Node Storage Location Input Node 


  1. 1.
    S.A. Cook: An observation on time-storage trade off Proceedings 5th ACM-STOC 1973. 29–33Google Scholar
  2. 2.
    J. Hopcroft, W. Paul and L. Valiant: On time versus space and related problems 16th IEEE-FOCS 1975, 57–64.Google Scholar
  3. 3.
    M.S. Paterson and C.E. Hewitt: Comparative schematology Record of Project MAC Conf. on Concurrent Systems and Parallel Computation 1970, 119–128Google Scholar
  4. 4.
    W.Paul, R.E. Tarjan and J.R. Celoni: Space bounds for a game on graphs 8th ACM-STOC 1976, 149–160Google Scholar
  5. 5.
    M.S. Pinsker: On the complexity of a concentrator 7th International Teletraffic Congress, Stockholm 1973Google Scholar
  6. 6.
    N. Pippenger: Superconcentrators Technical Report IBM Yorktown Heights 1976Google Scholar
  7. 7.
    R. Sethi: Complete register allocation problems Proceedings 5th ACM-STOC 1973, 182–195Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1977

Authors and Affiliations

  • W. J. Paul
    • 1
  • R. E. Tarjan
    • 2
  1. 1.Fakultät für Mathematik der Universität BielefeldBielefeld 1Germany
  2. 2.Computer Science DepartmentStanford UniversityStanfordUSA

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