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# Amortized Cost of Meld Operation

Calculating the amortized cost of meld operation is a difficult task. The major difficulty is in accumulating for the wide variations in the costs of an operation performed at diﬀerent points in a random sequence of operations. Although our design goal is affected by the costs of sequence of operations, deﬁning the notion of amortized cost of an operation in terms of the costs of sequences of operations leads nothing. Implementing a potential function to oﬀ set the variations in the actual costs is a perfect way of handling the situation.In the next topic we discuss the notion of amortized cost.

Let B be an Abstract data type(ADT) with basic operations P = {P_{1}, P_{2},……, P_{k}} and let DS be a data structure implementing B. Let F be a potential function specified on the conﬁgurations of the data structures to non-negative real number. Let further that F(Φ) = 0. Let DS^{j} specify a conﬁguration we obtain if we perform an operation Pk on a conﬁguration DS and let C denote the actual cost of performing Pk on DS.

Then, the amortized cost of Pk operating on DS, denoted as a(Pk, DS), is given by

a(P_{k}, DS) = C + F(DS^{j}) – F (DS)

If a(P_{k}, DS)≤ cjg(m) for all conﬁguration DS of size m, then we conclude that the amortized cost of Pk is O(g(m)).

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