Lecture VI AMORTIZATION

Amortization is the algorithmic idea of distributing computational cost over a period of time. The terminology comes from home mortgages: most Americans pay for a home by taking out a long-term loan called a home mortgage. This loan is repaid monthly, over the period of the loan. In an amortized analysis of an algorithm, we likewise spread the cost of an operation over the entire run of the algorithm. Suppose each run of the algorithm amounts to a sequence of operations on a data structure. For instance, to sort n items, the well-known heapsort algorithm (Lect.III,¶7) is a sequence of n insert’s into an initially empty priority queue, followed by a sequence of n deleteMin’s from the queue until it is empty. Thus if ci is the cost of the ith operation, the algorithm’s running time is ∑2n i=1 ci, since there are 2n priority queue operations in all. In worst case analysis, we ensure that each operation is efficient, say ci = O(log n), leading to the conclusion that the overall algorithm is O(n log n). But an amortization argument might be able to obtain the same bound ∑2n i=1 ci = O(n log n) without ensuring that each ci is logarithmic. In this case, we will say that the amortized cost of each operation is logarithmic. Thus “amortized complexity” is a kind of average complexity although it has nothing to do with probability. Tarjan [11] gave the first systematic account of this topic.