Time-space trade-offs in a pebble game

Abstract

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 G_n and constants c₁,c₂,c₃ such that

• 1) G_n has n nodes and each node in G_n has indegree at most 2.

• 2) Each graph G_n can be pebbled with $c_1 \sqrt n$ pebbles in n moves.

• 3) Each graph G_n 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.

Research partially supperted 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. N 0014-76-C-0688.

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