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 least significant bit changes once for every 2 toys ; the third least significant bit 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 2dN/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
toy , 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 and the first least significant bit has credit 0 .
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) = face 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) - P(N-1)= d(1-q*1) ;
So the amortized cost for the display change = actual cost + P(N) - P(N-1)= (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 .
