Upper and lower bounds on time-space tradeoffs in a pebble game

Abstract : We derive asymptotically tight time-space tradeoffs for pebbling three different classes of directed acyclic graphs. Let N be the size of the graph, S the number of available pebbles, and T the time necessary for pebbling the graph. A time-space tradeoff of the form ST=Theta(N-squared) is proved for pebbling(using only black pebbles) a special class of permutation graphs that implement the bit reversal permutation. If we are allowed to use black and white pebbles the time-space tradeoff is shown to be of the form T=Theta(N-squared/S-squared)+Theta(N). A time-space tradeoff of the form T=S Theta(N/S)-to the power Theta(N/S) is proved for pebbling a class of graphs constructed by stacking superconcentrators in series. This time-space tradeoff holds whether we use only black or black and white pebbles. A time-space tradeoff of the form T=S 2 to the power 2 to the power Theta(N/s) is proved for pebbling general directed acyclic graphs with only black or black and white pebbles. (Author)