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The University of Texas at Austin
Lecture 16
Department of Computer Sciences
Professor Vijaya Ramachandran
Amortized Analysis
CS357: ALGORITHMS, Spring 2006
1
Amortized Analysis
Given a data structure that supports certain operations,
amortized analysis
provides an
upper bound on the average cost of each operation for any sequence of a given length
n
(i.e.,
an upper bound for a worstcase sequence).
By convention, amortized cost is speciFed as cost per operation, and not as cost of a sequence
of operations of a given length.
Although we talk about the ‘average cost per operation’,
no probability is involved.
While performing amortized analysis, we usually assume that the data structure starts from
a canonical start conFguration, which is typically that corresponding to the empty set.
Sometimes we analyze amortized cost per operation by type of operation.
There are three methods commonly used for amortized analysis:
•
Aggregate method
•
Accounting method
•
Potential method
All three methods give the same solution (except that the solution returned by the aggregate
method does not allow for di±erent amortized costs for di±erent types of operations), but
they approach it in di±erent ways.
Example.
Consider a data structure
MStack
that supports the following 3 operations:
•
Push
(
S, x
): add element
x
to set
S
•
Pop
(
S
): remove the most recently added element from
S
•
Multipop
(
S, k
): let
l
= min(
k,

S

); remove the
l
most recently added elements from
S
We implement
Mstack
as a standard stack. Then,
Push
and
Pop
take constant time. But
the third operation takes Θ(
l
) time, since the
Multipop
operation takes
l
units of time to
remove the
l
elements plus a constant time to access the stack and determine the value of
l
.
Question.
What is the amortized cost per operation of an MStack operation, assuming we
start with the empty set
S
?
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Aggregate Analysis
In this type of analysis, we obtain an upper bound
T
(
n
) on the time needed to execute any
sequence of
n
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 Spring '06
 Ramachandran
 Algorithms

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