Bandwidth and pebbling

Abstract

The main results of this paper establish relationships between the bandwidth of a graphG — which is the minimum over all layouts ofG in a line of the maximum distance between images of adjacent vertices ofG — and the ease of playing various pebble games onG. Three pebble games on graphs are considered: the well-known computational pebble game, the “progressive” (i.e., no recomputation allowed) version of the computational pebble game, both of which are played on directed acyclic graphs, and the quite different “breadth-first” pebble game, that is played on undirected graphs. We consider two costs of a play of a pebble game: the minimum number of pebbles needed to play the game on the graphG, and the maximumlifetime of any pebble in the game, i.e., the maximum number of moves that any pebble spends on the graph. The first set of results of the paper prove that the minimum lifetime cost of a play of either of the second two pebble games on a graphG is precisely the bandwidth ofG. The second set of results establish bounds on the pebble demand of all three pebble games in terms of the bandwidth of the graph being pebbled; for instance, the number of pebbles needed to pebble a graphG of bandwidthk is at most min (2k2+k+1, 2k log2|G|); and, in addition, there are bandwidth-k graphs that require 3k−1 pebbles. The third set of results relate the difficulty of deciding the cost of playing a pebble game on a given input graphG to the bandwidth ofG; for instance, the Pebble Demand problem forn-vertex graphs of bandwidthf(n) is in the class NSPACE (f(n) log2n); and the Optimal Lifetime Problem for either of the second two pebble games is NP-complete.