Amortized Analysis

In general, the running time of an algorithm depends on the efficiency of the data structure that is used to implement it. The algorithm performs, among other things, a sequence of operations on the data structure. In such cases, we are interested in the total time for a sequence of operations. In general, the time to perform a single operation may fluctuate—some operations may be “cheap”, whereas others may be “expensive”. What we want is that the total amount of time for all operations is not too large. Of course, this means that there should not be too many expensive operations. If, for example, each expensive operation is followed by a large number of cheap ones, then on the average, one operation is still cheap. We want to be able to analyze the time for a single operation, averaged over a sequence of operations. In these notes, we introduce the technique of amortized analysis as a tool for achieving this. We illustrate the technique for the implementation of a binary counter and for a special class of balanced binary search trees, the so-called BB[α]-trees.