Amortized Analysis of Algorithms

Question

I am currently reading amortized analysis. I am not able to fully understand how it is different from normal analysis we perform to calculate average or worst case behaviour

Answer 1

average - a probabilistic analysis, the average is in relation to all of the possible inputs, it is an estimate of the likely run time of the algorithm.

amortized - non probabilistic analysis, calculated in relation to a batch of calls to the algorithm.

example - dynamic sized stack:

say we define a stack of some size, and whenever we use up the space, we allocate twice the old size, and copy the elements into the new location.

overall our costs are:

• O(1) per insertion \ deletion

• O(n) per insertion ( allocation and copying ) when the stack is full

so now we ask, how much time would n insertions take?

one might say O(n^2), however we don't pay O(n) for every insertion. so we are being pessimistic, the correct answer is O(n) time for n insertions, lets see why:

lets say we start with array size = 1.

ignoring copying we would pay O(n) per n insertions.

now, we do a full copy only when the stack has these number of elements:

1,2,4,8,...,n/2,n

for each of these sizes we do a copy and alloc, so to sum the cost we get:

const*(1+2+4+8+...+n/4+n/2+n) = const*(n+n/2+n/4+...+8+4+2+1) <= const*n(1+1/2+1/4+1/8+...)

where (1+1/2+1/4+1/8+...) = 2

so we pay O(n) for all of the copying + O(n) for the actual n insertions

O(n) worst case for n operation -> O(1) amortized per one operation.