Amortized complexity
- 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; (P(n) converges to ‘0’ ,. Hence , it fulfills the requirement of amortized complexity ) . 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 .