This document discusses amortized computational complexity and describes several methods for evaluating amortized complexity of algorithms. Specifically, it discusses aggregate, accounting, and potential methods. As an example, it analyzes the amortized complexity of an algorithm that displays the number of toys available in binary format, finding that the amortized cost is 2d using each of the three methods.

1. Amortized Computational Complexity
Abstract : A powerful technique in the complexity analysis of various algorithms is amortization
or averaging over time. Amortized running time is a realistic measure of complexity of a
sequence of operations . It will not provide you with the exact computational time . Instead , it
lets you calculate a tight upper bound and lower bound of algorithms . Based on the result
obtained, varieties of algorithms can be compared for performance analysis . This white-paper
describes several methods of evaluating amortized complexity .
Introduction : In many uses of data structures , a sequence of operations , rather than a single
operation, is performed . So, the final goal is to evaluate the amortized time complexity of that
sequence . The only requirement is that the sum of the amortized complexity of all elements in
a sequence is greater than the sum of the actual complexities of the elements . So, we can
conclude :
Sum(amortized(i)) >= Sum(actual(i)) ........... (1.1)
Relative to actual cost and amortized cost of an operation in a sequence of n operations , we
define a potential function as below :
P(i) = amortized(i) - actual (i) - P(i-1) ...........(1.2)
Taking sum of the equation , we obtain :
P(n) = Sum(amortized(i)) - Sum(actual(i)) ..........(1.3)
From (1.1) & (1.3) , we conclude :
P(n) >= 0 ........(1.4)
To make the idea of amortization and motivation behind it more concrete , let us consider the
example of evaluating the amortized complexity of an algorithm , which displays number of toys
available in binary format . To update every digit of the display the algorithm takes d unit of
time . Assuming that the number of current toys is N and initial no. of toys is 0, we need to
evaluate the amortized complexity of the algorithm used to display the number of toys . There
are three popular methods to arrive at the amortized complexity of the operations, namely , (1)
aggregate method , (2) accounting method and (3) potential method .

2. Aggregate Method : In the aggregate method ,we determine the
UpperBoundOnSumOfActualCosts(N) for the sum of the actual costs of the n operations.The
amortized cost is amounted to UpperBoundOnSumOfActualCosts(N) /N .
Let N be the number of toys manufactured ; the number of digits used to display the number of
toys is n or 'Base-2 logN' . The least significant bit of the display has been changed N times ; the
second digit from the right changes once for every 2 toys ; the third digit from the lright changes
once for every 4 times and so on...
So, the aggregate number of digits that have been changed is bounded by n (1+1/2+1/4+...)d or
d2n. So, the amortized cost is amounted to 2n/n = 2d.
Accounting Method : To proceed using accounting method , we must first assign an amortized
cost and then prove that this assignment satisfies eq - (1.1). Generally we start by assigning an
amortized cost obtained by a good guess . Once we have shown this , we obtain an upper bound
on the cost of any operation sequence by computing Sum(f(i)*amortized(i)) , where f(i) be the
frequency of the operation and amortized(i) the amortized cost of the operation .
Suppose , we assign a guessed amortized cost of 2d for each display change . Then the potential
function
P(n) = Sum(amortized(i)) - Sum(actual(i))
= 2dn - d(1+1/2+1/4+.......)
> = 0;
So, the accounting method proved that the amortized cost of the display can be reduced to 2d.

3. Potential Method : In this approach , we postulate a function for the analysis and use the
function to satisfy eq - (1) .
Consider the example given in the introduction section . When the first toy is available , we can
use the amortized cost 2d to pay for the update , in which d amount is spent and remaining d
amount is retained as a credit to the first least significant bit. Following the update of the second
digit , the amount spent is 2d*2 in which 3d is spent to update two least significant digits and
the second least significant bit now has credit d . Proceeding in this way , we can sum up credit
on each digit of the display always equal (2d-d)v , where 2d be the amortized cost , d the actual
cost and v the value of the digit. So, we should use the potential function
P(n) = (2d-d)*Sum(v(i)) , where v(i) = digit value of the i-th digit
Let q be the number of 1’s at the rightmost places and the j th display and it has been changed
from jth to (j+1) th no.So, when the display changes from j th to j+1 th value, where j = 1111011
the potential change incurred
P(n) = d(1-q*1) ;
So the amortized cost for the display change = actual cost + p(n) = (q+1)d +P(n) = (q+1)d + d(1-
2*1) = 2d .

4. Conclusion : A worst case analysis ,in which we sum the worst-case time of individual
operations, may be unduly pessimistic ; it ignores correlated effect on data structures. On the
other hand, average case analysis can be inaccurate , as the probabilistic assumptions needed to
carry out the analysis may be false . In such a scenario, amortized analysis , in which we average
the running time over per operation , yield an answer that is both realistic and robust .