A Note on the Pebble Game

A wmhmatorial ‘pebble’ game on graphs has been uwd to establish tradeoffs between time and space required for arithmetic expression evaluation [Si and for Turing machine simulation f I]. One places pebbles on the vertices of a directed acyclic graph G in steps according to the following rules: (1) A letup is either a placement of a pebble on an empty vertex or a remowl of a pebble from a vertex. (ii! A pebble may be placed on a vertex only if there are pebbles on all immediate predecessors of the vertex. (Thus, a vertex with no predecessors can always be pebbled.) (iii) A pebble may always be removed from a vertex. A pebhli~g strategy is a sequence of steps in the pebble game. The goal is to find a pebbling strategy that places a pebble on every vertex of G at least once when the supply of pebbles is limited. This pebble game has been studied extensively; Lengauer and Tarjan [ 21 provide an exhaustive list of references. This note develops an explicit strategy that uses O(n/log n) pebbles to pebble every directed acyclic graph G with n vertices and bounded indegree. Furthermore. for every S > O(n/log n), a variation of this strate 8 y uses S pebbles to pebble G in at most SZto n’Sr steps. The proofs of these upper bounds, which employ an overlap argument 131, seem more natural than the original proofs [ 1,2,4]. Fix a directed acyclic graph G (V, E) with vertices V and edges E. Let n = IVI and d be the maxi-