Potential Method in data structure
Potential (Physicist’s) Method
Suppose we can define a potential function Φ (read “Phi”) on states of a data structure with the following properties:
- Φ(h0) = 0, where h0 is the initial state of the data structure.
- Φ(ht) ≥ 0 for all states ht of the data structure occurring during the course of the computation.
Intuitively, the potential function will keep track of the precharged time at any point in the computation. It measures how much saved-up time is available to pay for expensive operations. It is analogous to the bank balance in the banker’s method. But interestingly, it depends only on the current state of the data structure, irrespective of the history of the computation that got it into that state.
We then define the amortized time of an operation as
c + Φ(h‘) − Φ(h),
where c is the actual cost of the operation and h and h‘ are the states of the data structure before and after the operation, respectively. Thus the amortized time is the actual time plus the change in potential. Ideally, Φ should be defined so that the amortized time of each operation is small. Thus the change in potential should be positive for low-cost operations and negative for high-cost operations.
Now consider a sequence of n operations taking actual times c0, c1, c2, …, cn−1 and producing data structures h1, h2, …, hn starting from h0. The total amortized time is the sum of the individual amortized times:
(c0 + Φ(h1) − Φ(h0)) + (c1 + Φ(h2) − Φ(h1)) + … + (cn−1 + Φ(hn) − Φ(hn−1))
= c0 + c1 + … + cn−1 + Φ(hn) − Φ(h0)
= c0 + c1 + … + cn−1 + Φ(hn).
Therefore the amortized time for a sequence of operations overestimates of the actual time by Φ(hn), which by assumption is always positive. Thus the total amortized time is always an upper bound on the actual time.
For dynamically resizable arrays with resizing by doubling, we can use the potential function
Φ(h) = 2n − m,
where n is the current number of elements and m is the current length of the array. If we start with an array of length 0 and allocate an array of length 1 when the first element is added, and thereafter double the array size whenever we need more space, we have Φ(h0) = 0 and Φ(ht) ≥ 0 for all t. The latter inequality holds because the number of elements is always at least half the size of the array.
Now we would like to show that adding an element takes amortized constant time. There are two cases.
- If n < m, then the actual cost is 1, n increases by 1, and m does not change. Then the potential increases by 2, so the amortized time is 1 + 2 = 3.
- If n = m, then the array is doubled, so the actual time is n + 1. But the potential drops from n to 2, so amortized time is n + 1 + (2 − n) = 3.
In both cases, the amortized time is O(1).
The key to amortized analysis with the physicist’s method is to define the right potential function. The potential function needs to save up enough time to be used later when it is needed. But it cannot save so much time that it causes the amortized time of the current operation to be too high.