The Accommodating Function - A Generalization of the Competitive Ratio

A new measure, the accommodating function, for the quality of on-line algorithms is presented. The accommodating function, which is a generalization of both the competitive ratio and the accommodating ratio, measures the quality of an on-line algorithm as a function of the resources that would be suucient for some algorithm to fully grant all requests. More precisely, if we have some amount of resources n, the function value at is the usual ratio (still on some xed amount of resources n), except that input sequences are restricted to those where all requests could have been fully granted by some algorithm if it had had the amount of resources n. The accommodating functions for three speciic on-line problems are investigated: a variant of bin-packing in which the goal is to maximize the number of objects put in n bins, the seat reservation problem, and the problem of optimizing total ow time when preemption is allowed. We also show that when trying to distinguish between two algorithms , the decision as to which one performs better cannot necessarily be made from the competitive ratio or the accommodating ratio alone. 1 For the variant of bin-packing considered, we show that Worst-Fit has a strictly better competitive ratio than First-Fit, while First-Fit has a strictly better accommodating ratio than Worst-Fit.