On Reducing the Space Requirements of a Straight-Line Algorithm

Abstract

The simultaneous time and space requirements of a straight-line algorithm can be determined by playing a well-known “pebble game” on a directed acyclic graph whose vertices and edges represent operations and argument assignments of the algorithm, respectively. In the game, pebble placements are made on vertices only when all predecessors have pebbles, time is the number of such placements made, and space is the number of pebbles used to reach all outputs of the graph. When space is restricted, extra time may be required to repebble some vertices, causing a tradeoff between time and space. Previous research has identified computations exhibiting different tradeoff characteristics, such as an extreme time-space tradeoff, in which a reduction in space causes time to rise from polynomial to superpolynomial in the size of the graph, and a favorable time-space tradeoff, in which a significant decrease in space can be achieved at the expense of a small (e.g. constant factor) increase in time. In this paper, we show that a computation is not limited to having only one tradeoff characteristic, but may exhibit both an extreme and a favorable time-space tradeoff. For three families of graphs, we derive upper and lower bounds on pebbling time for certain values of space that provide evidence of each family possessing both types of tradeoffs. We also provide upper and lower bounds on the maximum amount of time that can result from pebbling a graph when S pebbles are used.