Potential Method

Credit adds to ``potential'' of whole data structure instead of to individual objects.

Definitions

• D₀ = initial data structure
• c_i = actual cost of ith operation resulting in data structure D_i after
  operating on D_i-1 (i = 1, ..., n).
• = real number potential associated with data structure D_i.
  $\Phi$ represents the potential function.
• = amortized cost of ith operation with respect to $\Phi$
  = actual cost + potential increase
  = c_i + .
• The total amortized cost is

$$\sum_{i=1}^{n} \hat{c_i} \;=\; \sum_{i=1}^{n} (c_i \;+\; \Phi(D_i) \;-\; \Phi(D_{i-1}))$$

This is a telescoping series. $\sum_{i=1}^n (a_i - a_{i-1}) = a_n - a_0$

$$=\; \sum_{i=1}^{n} c_i \;+\; \Phi(D_n) \;-\; \Phi(D_0).$$

If , then the amortized cost is an upper bound on the actual cost.

However, we do not know n.

If for all i, then we always pay in advance.

Let , thus we want .

This is similar to the accounting method since we credit potential when is positive and debit potential when is negative.