在摊销分析中使用潜力和会计方法

Question

I have read about the accounting and the potential methods but I still having trouble solving such questions:

Answer 1

Intuition

By analyzing for each operation its expensive case and cheap case you can build an idea of how to charge each operation (then you may need some small fixes to 'fix' the numbers)

By looking at the important parameter of the data structure you can build a potential function (here for example it is the number of the elements in s2)

• Also, seeing many examples really helps here

The accounting method:

Lets charge each insertion two coins: one will be used to pop the element from s2, the second will be used to insert the element to s1 and Lets charge each deletion one coin that will be used to delete the element from s1.

Then note that the credit in the bank cannot be negative because if s1 is not empty then we are paying only for the actual cost of the operation (which is the only deletion from s1) and if s1 is empty then the operations are popping from s2 and inserting to s1 which we already charged for. ( it is important to note that we assume we start from empty data structure, OR, that we got the proper credit for every initial state of the data structure )

Now you can see that for each set of operations our total credit of 3n covers the actual cost of the operations (and we got a margin of the credit of the remaining elements in s2) and hence these are valid amortized bounds.

To solve with potential function lets consider the function:

Potential(DataStructure) = 2 * Number of elements in s2

Recall that we compute like this:

amortized cost of OP.A = actual cost of OP.A + Potential(DS after)-Potential(DS before)

So we got (denote number of elements in s2 in k):

For insertion: 1 + 2(k+1)-2k = 3

For cheap deletion: 1 + 2k - 2k = 1

For Expensive deletion: 1+2k + 0 - 2k = 0 (the actual cost is 1+2k because we pop from s2 and insert to s1 each element and then pop one element from s1)