This document discusses amortized analysis, which averages the time required to perform a sequence of data structure operations over all operations. It presents four methods for amortized analysis: aggregate analysis, the accounting method, the potential method, and dynamic tables. It provides examples of using these methods to analyze incrementing a binary counter. The potential method represents prepaid work as "potential energy" associated with the data structure. It defines a potential function and shows that the amortized cost of each counter incrementation operation is its actual cost plus the change in potential, guaranteeing an amortized cost of O(1) per operation.

1. Design & Analysis of Algorithm
Amortized Analysis
Kamalesh Karmakar
Assistant Professor
Department of C.S.E.
MeghnadSahaInstitute of Technology
Kolkata.

2. Design & Analysis of Algorithm
Inanamortizedanalysis,thetimerequiredtoperformasequenceofdata- structureoperationsisaveragedoveralltheoperationsperformed.Amortizedanalysiscanbeusedtoshowthattheaveragecostofanoperationissmall,ifoneaveragesoverasequenceofoperations,eventhoughasingleoperationwithinthesequencemightbeexpensive.
Amortizedanalysisdiffersfromaverage-caseanalysisinthatprobabilityisnotinvolved;anamortizedanalysisguaranteestheaverageperformanceofeachoperationintheworstcase.
A.Aggregateanalysis
B.Theaccountingmethod
C.Thepotentialmethod
D.Dynamictables
Amortized Analysis

3. Design & Analysis of Algorithm
InAggregateAnalysis,weshowthatforalln,asequenceofnoperationstakesworst-casetimeT(n)intotal.
Intheworstcase,theaveragecost,oramortizedcost,peroperationisthereforeT(n)/n.
Notethatthisamortizedcostappliestoeachoperation,evenwhenthereareseveraltypesofoperationsinthesequence.
Amortized Analysis

4. Design & Analysis of Algorithm
Example:Incrementingabinarycounter
As another example of aggregate analysis, consider the problem of implementing a k-bit binary counter that counts upward from 0. We use an array A[0 . . k −1] of bits, where length[A] = k, as the counter. A binary number x that is stored in the counter has its lowest-order bit in A[0] and its highest-order bit in A[k −1], so that
푥= 푖=0 푘−1퐴푖.2푖
Initially, x = 0, and thus A[i] = 0 for i= 0, 1, . . . , k −1. To add 1(modulo 2k) to the value in the counter, we use the following procedure.
Amortized Analysis

5. Design & Analysis of Algorithm
Amortized Analysis
 The cost of each INCREMENT
operation is linear in the number of
bits flipped.
 A single execution of INCREMENT
takes time (k) in the worst case, in
which array A contains all 1’s.
 Thus, a sequence of n INCREMENT
operations on an initially zero
counter takes time O(nk) in the worst
case.

6. Design & Analysis of Algorithm
Amortized Analysis
Wecantightenouranalysistoyieldaworst-casecostofO(n)forasequenceofnINCREMENT’sbyobservingthatnotallbitsflipeachtimeINCREMENTiscalled. AsFigureshows,A[0]doesflipeachtimeINCREMENTiscalled.
Thenext-highest-orderbit,A[1],flipsonlyeveryothertime:asequenceofnINCREMENToperationsonaninitiallyzerocountercausesA[1]toflip푛/2times.
Similarly,bitA[2]flipsonlyeveryfourthtime,or푛/4timesinasequenceofnINCREMENT’s.Ingeneral,fori=0,1,...,lg푛/2,bitA[i]flips푛/2푖timesinasequenceofnINCREMENToperationsonaninitiallyzerocounter.
Fori>푙푔푛,bitA[i]neverflipsatall.Thetotalnumberofflipsinthesequenceisthus
Theworst-casetimeforasequenceofnINCREMENToperationsonaninitiallyzerocounteristhereforeO(n).Theaveragecostofeachoperation,andthereforetheamortizedcostperoperation,isO(n)/n=O(1).

7. Design & Analysis of Algorithm
Amortized Analysis
Intheaccountingmethodofamortizedanalysis,weassigndifferingchargestodifferentoperations,withsomeoperationschargedmoreorlessthantheyactuallycost.
Theamountwechargeanoperationiscalleditsamortizedcost.
Whenanoperation’samortizedcostexceedsitsactualcost,thedifferenceisassignedtospecificobjectsinthedatastructureascredit.
Creditcanbeusedlaterontohelppayforoperationswhoseamortizedcostislessthantheiractualcost.Thus,onecanviewtheamortizedcostofanoperationasbeingsplitbetweenitsactualcostandcreditthatiseitherdepositedorusedup.
Thismethodisverydifferentfromaggregateanalysis,inwhichalloperationshavethesameamortizedcost.

8. Design & Analysis of Algorithm
Amortized Analysis
 One must choose the amortized costs of operations carefully. If we want analysis
with amortized costs to show that in the worst case the average cost per
operation is small, the total amortized cost of a sequence of operations must be
an upper bound on the total actual cost of the sequence. Moreover, as in
aggregate analysis, this relationship must hold for all sequences of operations. If
we denote the actual cost of the ith operation by ci and the amortized cost of the
ith operation by , we require
for all sequences of n operations. The total credit stored in the data structure is the
difference between the total amortized cost and the total actual cost, or
 By inequality the total credit associated with the data structure must be
nonnegative at all times. If the total credit were ever allowed to become negative,
then the total amortized costs incurred at that time would be below the total
actual costs incurred; for the sequence of operations up to that time, the total
amortized cost would not be an upper bound on the total actual cost. Thus, we
must take care that the total credit in the data structure never becomes negative.

9. Design & Analysis of Algorithm
Amortized Analysis
Example:Incrementingabinarycounter
Asanotherillustrationoftheaccountingmethod,weanalyzetheINCREMENToperationonabinarycounterthatstartsatzero.Asweobservedearlier,therunningtimeofthisoperationisproportionaltothenumberofbitsflipped,whichweshalluseasourcostforthisexample.Letusonceagainuseadollarbilltorepresenteachunitofcost.
Fortheamortizedanalysis,letuschargeanamortizedcostof2dollarstosetabitto1.
Whenabitisset,weuse1dollartopaytoflipthebitbackto0.
TheamortizedcostofINCREMENTcannowbedetermined.Thecostofresettingthebitswithinthewhileloopispaidforbythedollarsonthebitsthatarereset.
Thenumberof1’sinthecounterisnevernegative,andthustheamountofcreditisalwaysnonnegative.Thus,fornINCREMENToperations,thetotalamortizedcostisO(n),whichboundsthetotalactualcost.

10. Design & Analysis of Algorithm
Amortized Analysis
 Instead of representing prepaid work as credit stored with specific objects in the
data structure, the potential method of amortized analysis represents the
prepaid work as “potential energy,” or just “potential,” that can be released to pay
for future operations. The potential is associated with the data structure as a
whole rather than with specific objects within the data structure.
 The potential method works as follows. We start with an initial data structure D0
on which n operations are performed. For each i = 1, 2, . . . , n, we let ci be the
actual cost of the ith operation and Di be the data structure that results after
applying the ith operation to data structure Di−1.
 A potential function ф maps each data structure Di to a real number ф(Di ),
which is the potential associated with data structure Di. The amortized cost of
the ith operation with respect to potential function ф is defined by

11. Design & Analysis of Algorithm
Amortized Analysis
 The amortized cost of each operation is therefore its actual cost plus the increase
in potential due to the operation.The total amortized cost of the n operations is
 If we can define a potential function ф so that (Dn) ≥ (D0), then the total
amortized cost is an upper bound on the total actual cost
 In practice, we do not always know how many operations might be performed.
Therefore, if we require that ф(Di) ≥ ф(D0) for all i , then we guarantee, as in the
accounting method, that we pay in advance. It is often convenient to define ф(D0)
to be 0 and then to show that ф(Di ) ≥ 0 for all i .
 Intuitively, if the potential difference ф(Di ) − ф(Di−1) of the ith operation is
positive, then the amortized cost ci represents an overcharge to the ith operation,
and the potential of the data structure increases. If the potential difference is
negative, then the amortized cost represents an undercharge to the ith operation,
and the actual cost of the operation is paid by the decrease in the potential.

12. Design & Analysis of Algorithm
Amortized Analysis
Example: Incrementing a binary counter
 As another example of the potential method, we again look at incrementing a
binary counter. This time, we define the potential of the counter after the ith
INCREMENT operation to be bi , the number of 1’s in the counter after the ith
operation. Let us compute the amortized cost of an INCREMENT operation.
Suppose that the ith INCREMENT operation resets ti bits. The actual cost of the
operation is therefore at most ti + 1, since in addition to resetting ti bits, it sets at
most one bit to 1.
 If bi = 0, then the ith operation resets all k bits, and so bi−1 = ti = k.
 If bi > 0, then bi = bi−1 −ti +1.
 In either case, bi ≤ bi−1 −ti +1, and the potential difference is
 The amortized cost is therefore

13. Design & Analysis of Algorithm
Amortized Analysis
 If the counter starts at zero, then ф(D0) = 0. Since ф(Di ) ≥ 0 for all i , the total
amortized cost of a sequence of n INCREMENT operations is an upper bound on
the total actual cost, and so the worst-case cost of n INCREMENT operations is
O(n).
 The potential method gives us an easy way to analyze the counter even when it
does not start at zero. There are initially b0 1’s, and after n INCREMENT
operations there are bn 1’s, where 0 ≤ b0, bn ≤ k. (Recall that k is the number of
bits in the counter.)We can rewrite equation as
 We have for all 1 ≤ i ≤ n. Since ф(D0) = b0 and ф(Dn) = bn, the total actual
cost of n INCREMENT operations is
 Note in particular that since b0 ≤ k, as long as k = O(n), the total actual cost is
O(n). In other words, if we execute at least n = (k) INCREMENT operations, the
total actual cost is O(n), no matter what initial value the counter contains.