This document discusses amortized computational complexity and describes three methods for evaluating amortized complexity: the aggregate method, accounting method, and potential method. As an example, it analyzes the amortized complexity of an algorithm that displays the number of toys available in binary format, finding an amortized cost of 2d using each of the three methods. Amortized analysis averages running time over operations to provide a realistic yet robust measure of an algorithm's complexity.
Worst-case analysis is sometimes overly pessimistic.
Amortized analysis of an algorithm involves computing the maximum total number of all operations on the various data structures.
Amortized cost applies to each operation, even when there are several types of operations in the sequence.
In amortized analysis, time required to perform a sequence of data structure operations is averaged over all the successive operations performed. That is, a large cost of one operation is spread out over many operations (amortized), where the others are less expensive.
Therefore, amortized anaysis can be used to show that the average cost of an operation is small, if one averages over a sequence of operations, even though one of the single operations might be very expensive.
Worst-case analysis is sometimes overly pessimistic.
Amortized analysis of an algorithm involves computing the maximum total number of all operations on the various data structures.
Amortized cost applies to each operation, even when there are several types of operations in the sequence.
In amortized analysis, time required to perform a sequence of data structure operations is averaged over all the successive operations performed. That is, a large cost of one operation is spread out over many operations (amortized), where the others are less expensive.
Therefore, amortized anaysis can be used to show that the average cost of an operation is small, if one averages over a sequence of operations, even though one of the single operations might be very expensive.
The amortized analysis method originally emerged as aggregated analysis (Ullmanm, Aho and Hopcroft used a version of it to analyze set representations) to analyze basic set operation in binary trees and union operations. The technique was first formally introduced by Robert Tarjan in his paper Amortized Computational Complexity. It was used to study balanced binary trees and the operation for set representations.
Not just consider one operation, but a sequence of operations on a given data structure.
Average cost over a sequence of operations.
Probabilistic analysis:
Average case running time: average over all possible inputs for one algorithm (operation).
If using probability, called expected running time.
Amortized analysis:
No involvement of probability
Average performance on a sequence of operations, even some operation is expensive.
Guarantee average performance of each operation among the sequence in worst case.
Expense constrained bidder optimization in repeated auctions.
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Optimal bid can be mapped to static auction with a shaded virtual valuation.
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The amortized analysis method originally emerged as aggregated analysis (Ullmanm, Aho and Hopcroft used a version of it to analyze set representations) to analyze basic set operation in binary trees and union operations. The technique was first formally introduced by Robert Tarjan in his paper Amortized Computational Complexity. It was used to study balanced binary trees and the operation for set representations.
Not just consider one operation, but a sequence of operations on a given data structure.
Average cost over a sequence of operations.
Probabilistic analysis:
Average case running time: average over all possible inputs for one algorithm (operation).
If using probability, called expected running time.
Amortized analysis:
No involvement of probability
Average performance on a sequence of operations, even some operation is expensive.
Guarantee average performance of each operation among the sequence in worst case.
Expense constrained bidder optimization in repeated auctions.
the nature of what this budget limit means for the bidders themselves is somewhat of a mystery. There seems to be some risk control element to it, some purely administrative element to it, some bounded-rationality element to it, and more...
A two parameter model for expense constraints in online budgeting problems.
Optimal bid can be mapped to static auction with a shaded virtual valuation.
Paper has more contents: MFE analysis and a finite horizon model.
Solving a “Transportation Planning” Problem through the Programming Language “C”Shahadat Hossain Shakil
Solving a “Transportation Planning” problem through the programming language “C”
Presented by
Yousuf Mahid (0615012)
Shahadat Hossain Shakil (0615020)
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Synchronous loadable up and down counter is a very important block in any complex digital system design. It is not just used for counting, it is also used for phase signal generation, clock division and for initiation of a process.
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What is Complexity Analysis?
What is the need for Complexity Analysis?
Asymptotic Notations
How to measure complexity?
1. Time Complexity
2. Space Complexity
3. Auxiliary Space
How does Complexity affect any algorithm?
How to optimize the time and space complexity of an Algorithm?
Different types of Complexity exist in the program:
1. Constant Complexity
2. Logarithmic Complexity
3. Linear Complexity
4. Quadratic Complexity
5. Factorial Complexity
6. Exponential Complexity
Worst Case time complexity of different data structures for different operations
Complexity Analysis Of Popular Algorithms
Practice some questions on Complexity Analysis
practice with giving Quiz
Conclusion
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Time Space trade-off
Asymptotic time complexity of algorithm and its notation
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Big O-h notation and example
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Big theta notation and its example
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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;
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 .