Summary
A certain pebble game on graphs has been studied in various contexts as a model for the time and space requirements of computations [1,2,3,8]. In this note it is shown that there exists a family of directed acyclic graphs G n and constants c 1, c 2, c 3 such that
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(1)
G n has n nodes and each node in G n has indegree at most 2.
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(2)
Each graph G n can be pebbled with c 1√n pebbles in n moves.
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(3)
Each graph G n can also be pebbled with c 2√n pebbles, c 2<c1, but every strategy which achieves this has at least 2c 3√n moves.
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References
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Research partially supported by DAAD (German Academic Exchange Service) Grant No. 430/402/653/5
Research partially supported by the National Science Foundation, Grant No. MCS 75-22870 and by the Office of Naval Research, Contract No. N00014-76-C-0688
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Paul, W.J., Tarjan, R.E. Time-space trade-offs in a pebble game. Acta Informatica 10, 111–115 (1978). https://doi.org/10.1007/BF00289150
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DOI: https://doi.org/10.1007/BF00289150