Move rules and trade-offs in the pebble game

Abstract

The pebble game on directed acyclic graphs is commonly encountered as an abstract model for register allocation problems. The traditional move rule of the game asserts that one may "put a pebble on node x once all its immediate predecessors have a pebble", leaving it open whether the pebble to be placed on x should be taken from some predecessor of x or from the free pool (the strict interpretation). We show that allowing pebbles to slide along an edge as a legal move enables one to save precisely one pebble over the strict interpretation. However, in the worst case the saving may be obtained only at the cost of squaring the time needed to pebble the dag. It shows that one has to be very careful in describing properties of pebblings; the interpretation of the rules can seriously affect the results. As a main result we prove a linear to exponential time trade-off for any fixed interpretation of the rules when a single pebble is saved. There exist families of dags with indegrees ≤2, with the property that they can be pebbled in linear time when one more pebble than the minimum needed is available but which require exponential time when the extra pebble is dropped.