This chapter discusses the fundamentals of analyzing algorithm efficiency. It covers measuring an algorithm's input size, units for measuring running time such as counting basic operations, orders of growth for analyzing efficiency as input size increases, and analyzing worst-case, best-case, and average-case running times. The goal is to determine an algorithm's time and space complexity based on how its running time and memory usage grow relative to the input size.

1. The Design and
Analysis of
Algorithms
by Anany Levitin

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CHAPTER 2:CHAPTER 2: Fundamentals of the
Analysis of Algorithm Efficiency
 The Analysis Framework
 Asymptotic Notations and Basic Efficiency Classes
 Mathematical Analysis of Nonrecursive Algorithms
 Mathematical Analysis of Recursive Algorithms
 Empirical Analysis of Algorithms
 Algorithm Visualization

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4. Analysis of Algorithm Efficiency
 Two kinds of efficiency:
Time efficiency
Space efficiency
 Time efficiency - Time complexity
How fast an algorithm in question runs.
 Space efficiency - Space complexity
The amount of memory units required by the
algorithm in addition to the space needed for its
input and output.
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5. The Analysis Framework
 Measuring an Input’s Size
 Units for Measuring Running Time
 Orders of Growth
 Worst-Case, Best-Case, and Average-
Case Efficiencies
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6. Measuring an Input’s Size
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7. Measuring an Input’s Size
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8. Measuring an Input’s Size
 Almost all algorithms run longer on larger
inputs
 It is logical to investigate an algorithm’s
efficiency as a function of some parameter n
indicating the algorithm’s input size
 Examples:
 Searching/ Finding Smallest or largest
 Evaluating Polynomial Equation
 Matrix Manipulation
 Spell Checking
 Primality Testing 8

9. Units for Measuring Running Time
 Concerns with units for measurement of an
algorithm’s running time
 Standard unit of time measurement—a second, or
millisecond cannot be used
 Reasons:
 Dependence on the speed of a particular computer
 Dependence on the quality of a program implementing
the algorithm
 Depends on the compiler used in generating the
machine code
 Difficulty of clocking the actual running time of the
program
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10.  Solution:
Identify the most important operation of
the algorithm, called the basic operation
in an algorithm
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Units for Measuring Running Time

11.  Basic Operation
 It is the operation contributing the most to the total
running time of an algorithm
 It is the most time consuming operation in the
algorithm’s innermost loop
 Example:
 Sorting - comparing elements (keys)
Mathematical problems – all four arithmetical
operations
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Units for Measuring Running Time

12.  Analysis of an algorithm’s time efficiency
suggests measuring it by counting the number
of times the algorithm’s basic operation is
executed on inputs of size n.
 Answer Questions Like:
 How much faster would this algorithm run on a
machine that is 10 times faster than the one we have?
 How much longer the algorithm will take if the size of
the input is doubled?
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Units for Measuring Running Time

13. Orders of Growth
 Smaller inputs are not considered in calculating the
efficiency of an algorithm
 A difference in running time of very large inputs really
distinguish an efficient algorithm from an inefficient
one
 Magnitude of the input has a profound significance for
the analysis
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Orders of Growth with input size ‘n’

15. Worst-Case, Best-Case & Average-Case
Efficiencies
 There are many algorithms for which running
time depends not only on an input size but
also on the specifics of a particular input.
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16. Best-case efficiency
 The best-case efficiency of an
algorithm is its efficiency for the best-
case input of size n, which is an input (or
inputs) of size n for which the algorithm
runs the fastest among all possible inputs
of that size.
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17. Worst-case efficiency
 The worst-case efficiency of an
algorithm is its efficiency for the worst-
case input of size n, which is an input (or
inputs) of size n for which the algorithm
runs the longest among all possible inputs
of that size.
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18. Average-case efficiency
 The average-case efficiency cannot be
obtained by taking the average of the
worst-case and the best-case efficiencies.
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19. Amortized Efficiency
 It applies not to a single run of an
algorithm but rather to a sequence of
operations performed on the same data
structure.
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20. Recapitulation of the Analysis
Framework
 Input size
 Time efficiency is measured by counting
the number of times the algorithm’s
 basic operation is executed. Space
efficiency is measured by counting the
 number of extra memory units consumed
by the algorithm
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21.  The efficiencies of some algorithms may
differ significantly for inputs of the
 same size. For such algorithms, we need
to distinguish between the worst-case,
 average-case, and best-case efficiencies.
 The framework’s primary interest lies in
the order of growth of the algorithm’s
 running time (extra memory units
consumed) as its input size goes to
infinity.
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22. Have a SPLENDID Day..!!!
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