DAA_UnitIDAA_UnitIDAA_UnitIDAA_UnitIDAA_UnitIDAA_UnitI.ppt

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Design and Analysis of Algorithms – Unit I
Chapter 1
Chapter 1
Introduction
Introduction

1-2
Algorithm
 An Algorithm is a sequence of unambiguous
instructions for solving a problem,
 i.e., for obtaining a required output for any
legitimate input in a finite amount of time.

1-3
Notion of algorithm
“computer”
Algorithmic solution
problem
algorithm
input output

1-4
PSEUDOCODE
 Pseudocode (pronounced SOO-doh-kohd) is a detailed yet readable
description of what a computer program or algorithm must do, expressed in
a formally-styled natural language rather than in a programming language.
 It is sometimes used as a detailed step in the process of developing a
program.
 It allows programmers to express the design in great detail and provides
programmers a detailed template for the next step of writing code in a
specific programming language.

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Formatting and Conventions in Pseudocoding
 INDENTATION in pseudocode should be identical to its
implementation in a programming language. Try to indent
at least four spaces.
 The pseudocode entries are to be cryptic, AND SHOULD
NOT BE PROSE. NO SENTENCES.
 No flower boxes in pseudocode.
 Do not include data declarations in pseudocode.

1-6
Some Keywords That Should be Used
 For looping and selection,
• Do While...EndDo;
• Do Until...Enddo;
• Case...EndCase;
• If...Endif;
• Call ... with (parameters); Call; Return ....; Return;
When; Always use scope terminators for loops and
iteration.

1-7
Some Keywords …
 As verbs, use the words
• generate, Compute, Process,
• Set, reset,
• increment,
• calculate,
• add, sum, multiply, ...
• print, display,
• input, output, edit, test , etc.

1-8
Methods of finding GCD
M - 1
M - 2
M - 3

1-9
Euclid’s Algorithm
Problem: Find gcd(m,n), the greatest common divisor of two
nonnegative, not both zero integers m and n
Examples: gcd(60,24) = 12, gcd(60,0) = 60, gcd(0,0) = ?
Euclid’s algorithm is based on repeated application of equality
gcd(m,n) = gcd(n, m mod n)
until the second number becomes 0, which makes the problem
trivial.
Example: gcd(60,24) = gcd(24,12) = gcd(12,0) = 12

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Two descriptions of Euclid’s algorithm
Step 1 If n = 0, return m and stop; otherwise go to Step 2
Step 2 Divide m by n and assign the value fo the remainder to r
Step 3 Assign the value of n to m and the value of r to n. Go to
Step 1.
while n ≠ 0 do
r ← m mod n
m← n
n ← r
return m

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Other methods for computing gcd(m,n)
Consecutive integer checking algorithm
Step 1 Assign the value of min{m,n} to t
Step 2 Divide m by t. If the remainder is 0, go to Step 3;
otherwise, go to Step 4
Step 3 Divide n by t. If the remainder is 0, return t and stop;
otherwise, go to Step 4
Step 4 Decrease t by 1 and go to Step 2

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Other methods for gcd(m,n) [cont.]
Middle-school procedure
Step 1 Find the prime factorization of m
Step 2 Find the prime factorization of n
Step 3 Find all the common prime factors
Step 4 Compute the product of all the common prime factors
and return it as gcd(m,n)
Is this an algorithm?

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Sieve of Eratosthenes
Input: Integer n ≥ 2
Output: List of primes less than or equal to n
for p ← 2 to n do A[p] ← p
for p ← 2 to n do
if A[p]  0 //p hasn’t been previously eliminated from the list
j ← p* p
while j ≤ n do
A[j] ← 0 //mark element as eliminated
j ← j + p
Example: 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

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Termination of Euclid’s Algorithm
 The second number of the pair gets smaller with each
iteration and cannot become negative:
 Indeed, the new value of n is r = m mod n, which is always
smaller than n.
 Eventually, r becomes zero, and the algorithms stops.

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Compute the GCD of 120 and 23.

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Fundamentals of Algorithmic
Problem Solving

1-17
Fundamentals of Algorithmic Problem
Solving
 Algorithm = Procedural Solutions to Problem
 NOT an answer, BUT rather specific instructions of
getting answers.
 Therefore, requires steps in designing and analyzing an
algorithm

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Algorithm Design & Analysis Process

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Step 1: Understand the Problem
 Before designing an algorithm - understand completely the
problem given.
 Read the problem’s description carefully and ask
questions if you have any doubts about the problem,
 Do a few small examples by hand, think about special
cases, and ask questions again if needed.

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 An input to an algorithm specifies an instance of the
problem the algorithm solves.
 It is very important to specify exactly the range of instances
the algorithm needs to handle.
 Failing which – the algorithm works correctly for some
inputs , but crashes on some boundary values.
 Remember that a correct algorithm is not one that works
most of the time but one that works correctly for all
legitimate inputs.
Step 1: Understand the Problem

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Step 2: Ascertaining the capabilities
of a computational device
 Algorithms designed to be executed on machines that
executes intstructions one after another are called
sequential algorithms.
 Algorithms that take advantage of computers that can
execute operations concurrently are called parallel
algorithms.

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Step 3: Choosing between Exact &
Approximate Problem Solving
 Solving the problem exactly - Exact algorithms
 Solving the problem approximately - Approximation algorithms
 Why approximation algorithms?
1. Problems cannot be solved exactly.
Eg. Extracting square roots, solving non-linear equations
2. Available exact algorithms are unacceptably slow because of
problem’s complexity
Eg. Traveling Salesman Problem
3. Approx. Algs can be a part of algorithms that solve the
problem exactly.

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Step 4: Deciding on Appropriate Data
Structures
 In the new world of object-oriented programming, data
structures remain important for both design and analysis of
algorithms.
 However, we will assume a very basic data structure for
now and concentrate on the algorithm side.

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Step 5: Algorithm Design Techniques
 An algorithm design technique (or “strategy” or
“paradigm”) is a general approach to solving problems
algorithmically that is applicable to a variety of problems
from different areas of computing.
 Eg. Brute force, Divide-and-Conquer, Transform-and-
Conquer
 Importance:
1. Provide guidance for designing algorithms for new
problems.
2. To classify algorithms according to an underlying
design idea.

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Step 6: Methods of Specifying an
Algorithm
 Pseudocode,
Pseudocode, a mixture of a natural language and
a mixture of a natural language and
programming language-like constructs.
programming language-like constructs.
 flowchart
flowchart, a method of expressing an algorithm by a
, a method of expressing an algorithm by a
collection of connected geometric shapes containing
collection of connected geometric shapes containing
descriptions of the algorithm’s steps.
descriptions of the algorithm’s steps.

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Step 7: Proving an Algorithm’s
Correctness
 Prove algorithm’s correctness = prove that the algorithm
yields a required result for every legitimate input in a finite
amount of time.
 For an approximation algorithm, correctness means to be
able to show that the error produced by the algorithm does
not exceed a predefined limit.

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Step 8: Analyzing an Algorithm
1. Efficiency
Time efficiency indicates how fast the algorithm runs.
space efficiency indicates how much extra memory the
algorithm needs.
2. Simplicity
3. Generality
Design an algorithm for a problem posed in more
general terms.
Design an algorithm that can handle a range of inputs
that is natural for the problem at hand.

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Step 9: Coding the algorithm
 More than implementation
More than implementation
 Peril of incorrect & inefficient implementation
Peril of incorrect & inefficient implementation
 Require testing & debugging
Require testing & debugging
 Require code optimizing
Require code optimizing

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Important Problem Types

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Important Problem Types
 Sorting
Sorting
 Searching
Searching
 String processing
String processing
 Graph problems
Graph problems
 Combinatorial problems
Combinatorial problems
 Geometric problems
Geometric problems
 Numerical problems
Numerical problems

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Sorting
 The
The sorting problem
sorting problem asks us to rearrange the items of a
asks us to rearrange the items of a
given list in ascending order.
given list in ascending order.
 we usually need to
we usually need to
sort lists of numbers,
sort lists of numbers,
characters from an alphabet,
characters from an alphabet,
character strings,
character strings,
 records similar to those maintained by schools about
records similar to those maintained by schools about
their students,
their students,
libraries about their holdings,
libraries about their holdings,
companies about their employees.
companies about their employees.

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Searching
 The
The searching problem
searching problem deals with finding a given value,
deals with finding a given value,
called a
called a search key
search key, in a given set (or a multiset, which
, in a given set (or a multiset, which
permits several elements to have the same value).
permits several elements to have the same value).

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String Processing
 A
A string
string is a sequence of characters from an alphabet.
is a sequence of characters from an alphabet.
 String of particular interest:
String of particular interest:
1. Text string – comprises letters, numbers, and
1. Text string – comprises letters, numbers, and
special characters
special characters
2. Bit string – comprises zeros and ones
2. Bit string – comprises zeros and ones
3. Gene sequence
3. Gene sequence
 Mainly
Mainly string matching problem
string matching problem: searching for a given
: searching for a given
word in a text
word in a text

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Graph Problems
 A
A graph
graph can be thought of as a collection of points called
can be thought of as a collection of points called
vertices, some of which are connected by line segments
vertices, some of which are connected by line segments
called edges.
called edges.
 Used for modeling a wide variety of real-life applications.
Used for modeling a wide variety of real-life applications.
 Basic graph algorithms include:
Basic graph algorithms include:
1.
1. Graph traversal algorithms
Graph traversal algorithms - How can one visit all
- How can one visit all the
the
points in a network?
points in a network?
2.
2. Shortest-path algorithms
Shortest-path algorithms - What is the best
- What is the best
Introduction route between two cities?
Introduction route between two cities?
3.
3. Topological sorting
Topological sorting for graphs with directed edges
for graphs with directed edges

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Combinatorial Problems
 combinatorial problems:
combinatorial problems: problems that ask (explicitly or
problems that ask (explicitly or
implicitly) to find a combinatorial object—such as a
implicitly) to find a combinatorial object—such as a
permutation, a combination, or a subset—that satisfies
permutation, a combination, or a subset—that satisfies
certain constraints and has some desired property (e.g.,
certain constraints and has some desired property (e.g.,
maximizes a value or minimizes a cost).
maximizes a value or minimizes a cost).
1. Combinatorial grows extremely fast with problem size
1. Combinatorial grows extremely fast with problem size
2. No known algorithm solving most such problems
2. No known algorithm solving most such problems
exactly in an acceptable amount of time.
exactly in an acceptable amount of time.

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Geometric Problems
 Geometric algorithms
Geometric algorithms deal with geometric objects such as
deal with geometric objects such as
points, lines, and polygons.
points, lines, and polygons.
2 class problems:
2 class problems:
 The
The closest pair problem
closest pair problem: given
: given n
n points in the plane, find
points in the plane, find
the closest pair among them.
the closest pair among them.
 The
The convex hull problem
convex hull problem asks to find the smallest convex
asks to find the smallest convex
polygon that would include all the points of a given set. If
polygon that would include all the points of a given set. If

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Convex hull problem

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Numerical Problems
 Numerical problems
Numerical problems, another large special area of
, another large special area of
applications, are problems that involve mathematical
applications, are problems that involve mathematical
objects of continuous nature: solving equations and systems
objects of continuous nature: solving equations and systems
of equations, computing definite integrals, evaluating
of equations, computing definite integrals, evaluating
functions, and so on.
functions, and so on.

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Fundamentals of Analysis of algorithm efficiency

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Analysis of algorithms
Analysis of algorithms
 Issues:
Issues:
• correctness
correctness
• time efficiency
time efficiency
• space efficiency
space efficiency
• optimality
optimality
 Approaches:
Approaches:
• theoretical analysis
theoretical analysis
• empirical analysis
empirical analysis

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Theoretical analysis of time efficiency
Theoretical analysis of time efficiency
Time efficiency is analyzed by determining the number of
Time efficiency is analyzed by determining the number of
repetitions of the
repetitions of the basic operation
basic operation as a function of
as a function of input size
input size
 Basic operation
Basic operation: the operation that contributes the most
: the operation that contributes the most
towards the running time of the algorithm
towards the running time of the algorithm
T
T(
(n
n)
) ≈
≈ c
cop
opC
C(
(n
n)
)
running time execution time
for basic operation
or cost
Number of times
basic operation is
executed
input size
Note: Different basic operations may cost differently!

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Input size and basic operation examples
Input size and basic operation examples
Problem Input size measure Basic operation
Searching for key
in a list of n items
Number of list’s
items, i.e. n
Key comparison
Multiplication of
two matrices
Matrix dimensions or
total number of
elements
Multiplication of
two numbers
Checking primality
of a given integer n
n’size = number of
digits (in binary
representation)
Division
Typical graph
problem
#vertices and/or
edges
Visiting a vertex
or traversing an
edge

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Empirical analysis of time efficiency
Empirical analysis of time efficiency
 Select a specific (typical) sample of inputs
Select a specific (typical) sample of inputs
 Use physical unit of time (e.g., milliseconds)
Use physical unit of time (e.g., milliseconds)
or
or
Count actual number of basic operation’s executions
Count actual number of basic operation’s executions
 Analyze the empirical data
Analyze the empirical data

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Efficiencies
 Worst Case Efficiency:
• Is its efficiency for the worst case input of size n, which
is an input of size n for which the algorithm runs the
longest among all possible inputs of that size
• C
Cworst
worst(
(n
n)
)
 Best-case efficiency:
Best-case efficiency:
• Is its efficiency for the worst case input of size n, which
is an input of size n for which the algorithm runs the
fastest among all possible inputs of that size
• C
Cbest
best(
(n
n)
)

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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

1-46
Best-case, average-case, worst-case
Best-case, average-case, worst-case
For some algorithms, efficiency depends on form of input:
For some algorithms, efficiency depends on form of input:
 Worst case: C
Worst case: Cworst
worst(
(n
n) – maximum over inputs of size
) – maximum over inputs of size n
n
 Best case: C
Best case: Cbest
best(
(n
n) – minimum over inputs of size
) – minimum over inputs of size n
n
 Average case: C
Average case: Cavg
avg(
(n
n) – “average” over inputs of size
) – “average” over inputs of size n
n
• Number of times the basic operation will be executed on typical
Number of times the basic operation will be executed on typical
input
input
• NOT the average of worst and best case
NOT the average of worst and best case
• Expected number of basic operations considered as a random
Expected number of basic operations considered as a random
variable under some assumption about the probability
variable under some assumption about the probability
distribution of all possible inputs. So, avg = expected under
distribution of all possible inputs. So, avg = expected under
uniform distribution.
uniform distribution.

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Example: Sequential search
Example: Sequential search
 Worst case
Worst case
 Best case
Best case
 Average case
Average case
n key comparisons
1 comparisons
(n+1)/2, assuming K is in A

1-48
Types of formulas for basic operation’s count
Types of formulas for basic operation’s count
 Exact formula
Exact formula
e.g., C(
e.g., C(n
n) =
) = n
n(
(n
n-1)/2
-1)/2
 Formula indicating order of growth with specific
Formula indicating order of growth with specific
multiplicative constant
e.g., C(
e.g., C(n
n)
) ≈
≈ 0.5
0.5 n
n2
2
 Formula indicating order of growth with unknown
Formula indicating order of growth with unknown
e.g., C(
e.g., C(n
n)
) ≈
≈ cn
cn2
2

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Order of growth
Order of growth
 Most important: Order of growth within a constant multiple
Most important: Order of growth within a constant multiple
as
as n
n→∞
→∞
 Example:
Example:
• How much faster will algorithm run on computer that is
How much faster will algorithm run on computer that is
twice as fast?
twice as fast?
• How much longer does it take to solve problem of double
How much longer does it take to solve problem of double
input size?
input size?

1-50
Values of some important functions as
Values of some important functions as n
n 
 


1-51
Asymptotic Notations
 O (Big-Oh)-notation
 Ω (Big-Omega) -notation
 Θ (Big-Theta) -notation

1-52
Asymptotic order of growth
Asymptotic order of growth
A way of comparing functions that ignores constant factors and
A way of comparing functions that ignores constant factors and
small input sizes
small input sizes (because?)
(because?)
 O(
O(g
g(
(n
n)): class of functions
)): class of functions f
f(
(n
n) that grow
) that grow no faster
no faster than
than g
g(
(n
n)
)
 Θ
Θ(
(g
g(
(n
f(
(n
n) that grow
) that grow at same rate
at same rate as
as g
g(
(n
n)
)
 Ω
Ω(
(g
g(
(n
f(
(n
n) that grow
) that grow at least as fast
at least as fast as
as g
g(
(n
n)
)

1-53
O-notation
O-notation
Definition:
Definition: A function t
A function t(
(n
n) is said to be in O(
) is said to be in O(g
g(
(n
n)), denoted t
)), denoted t(n)
(n)

 O(g(n)) is bounded above by some constant multiple of g(n)
O(g(n)) is bounded above by some constant multiple of g(n)
for all large n,
for all large n, i.e., there exist positive constant
i.e., there exist positive constant c
c and non-
and non-
negative integer
negative integer n
n0
0 such that
such that
f
f(
(n
n)
) ≤
≤ c g
c g(
(n
n) for every
) for every n
n ≥
≥ n
n0
0

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Big-oh
Big-oh

1-55

-notation
-notation
 Formal definition
Formal definition
• A function
A function t(n)
t(n) is said to be in
is said to be in 
(g(n)),
(g(n)), denoted
denoted t(n)
t(n) 


(g(n)),
(g(n)), if
if t(n)
t(n) is bounded below by some constant
is bounded below by some constant
multiple of
multiple of g(n)
g(n) for all large
for all large n
n, i.e.,
, i.e., if there exist some
if there exist some
positive constant c and some nonnegative integer
positive constant c and some nonnegative integer n
n0
0
such that
such that
t(n)
t(n) 
 cg(n) for all n
cg(n) for all n 
 n
n0
0

1-56
Big-omega
Big-omega

1-57

-notation
-notation
 Formal definition
Formal definition
• A function
A function t(n)
t(n) is said to be in
is said to be in 
(g(n)),
(g(n)), denoted
denoted t(n)
t(n) 


(g(n)),
(g(n)), if
if t(n)
t(n) is bounded both above and below by
is bounded both above and below by
some positive constant multiples of
some positive constant multiples of g(n)
g(n) for all large
for all large
n
n, i.e.,
, i.e., if there exist some positive constant c
if there exist some positive constant c1
1 and c
and c2
2
and some nonnegative integer
and some nonnegative integer n
n0
0 such that
such that
c
c2
2 g(n)
g(n) 
 t(n)
t(n) 
 c
c1
1 g(n) for all n
g(n) for all n 
 n
n0
0

1-58
Big-theta
Big-theta

1-59
Theorem
Theorem
 If t1(n)  O(g1(n)) and t2(n)  O(g2(n)), then
t1(n) + t2(n)  O(max{g1(n), g2(n)}).
• The analogous assertions are true for the
The analogous assertions are true for the 
-notation
-notation
and
and 
-notation.
-notation.
Proof. There exist constants c1, c2, n1, n2 such that
t
t1
1(n)
(n) 
 c
c1
1*
*g
g1
1(n),
(n), for all
for all n
n 
 n
n1
1
t
t2
2(n)
(n) 
 c
c2
2*
*g
g2
2(n),
(n), for all
for all n
n 
 n
n2
2
Define
Define c
c3
3 = c
= c1
1 + c
+ c2
2 and
and n
n3
3 = max{n
= max{n1
1,n
,n2
2}. Then
}. Then
t
t1
1(n) + t
(n) + t2
2(n)
(n) 
 c
c3
3*
*max{g
max{g1
1(n), g
(n), g2
2(n)},
(n)}, for all
for all n
n 
 n
n3
3

1-60
Some properties of asymptotic order of growth
Some properties of asymptotic order of growth
 f
f(
(n
n)
) 
 O(
O(f
f(
(n
n))
))
 f
f(
(n
n)
) 
 O(
O(g
g(
(n
n)) iff
)) iff g
g(
(n
n)
) 
(
(f
f(n))
(n))
 If
If f
f (
(n
n)
) 
 O(
O(g
g (
(n
n)) and
)) and g
g(
(n
n)
) 
 O(
O(h
h(
(n
n)) , then
)) , then f
f(
(n
n)
) 
 O(
O(h
h(
(n
n))
))
Note similarity with
Note similarity with a
a ≤
≤ b
b
 If
If f
f1
1(
(n
n)
) 
 O(
O(g
g1
1(
(n
n)) and
)) and f
f2
2(
(n
n)
) 
 O(
O(g
g2
2(
(n
n)) , then
)) , then
f
f1
1(
(n
n) +
) + f
f2
2(
(n
n)
) 
 O(max{
O(max{g
g1
1(
(n
n),
), g
g2
2(
(n
n)})
)})
Also,
Also, 
1
1
i
i
n
n 
(
(f
f(
(i
i)) =
)) = 
 (
(
1
1
i
i
n
n f
f(
(i
i))
))

1-61
Establishing order of growth using limits
Establishing order of growth using limits
lim
lim T
T(
(n
n)/
)/g
g(
(n
n) =
) =
0
0 order of growth of T
T(
(n)
n) < order of growth of g
g(
(n
n)
)
c
c > 0
> 0 order of growth of T
T(
(n)
n) = order of growth of g
g(
(n
n)
)
∞
∞ order of growth of T
T(
(n)
n) > order of growth of g
g(
(n
n)
)
n
n→∞
→∞

1-62
L’Hôpital’s rule and Stirling’s formula
L’Hôpital’s rule and Stirling’s formula
L’Hôpital’s rule: If
L’Hôpital’s rule: If lim
limn
n
 f
f(
(n
n) =
) = lim
limn
n
 g(n
g(n) =
) = 
 and
and
the derivatives
the derivatives f
f´,
´, g
g´ exist, then
´ exist, then
Stirling’s formula:
Stirling’s formula: n
n!
! 
 (2
(2
n
n)
)1/2
1/2
(
(n
n/e)
/e)n
n
f
f(
(n
n)
)
g
g(
(n
n)
)
lim
lim
n
n

=
f
f ´(
´(n
n)
)
g
g ´(
´(n
n)
)
lim
lim
n
n

Example: log
Example: log n
n vs.
vs. n
n
Example: 2
Example: 2n
n
vs.
vs. n
n!
!

1-63
Orders of growth of some important functions
Orders of growth of some important functions
 All logarithmic functions log
All logarithmic functions loga
a n
n belong to the same class
belong to the same class

(log
(log n
n)
) no matter what the logarithm’s base
no matter what the logarithm’s base a
a > 1 is
> 1 is
because
because
 All polynomials of the same degree
All polynomials of the same degree k
k belong to the same class:
belong to the same class:
a
ak
kn
nk
k
+
+ a
ak
k-1
-1n
nk
k-1
-1
+ … +
+ … + a
a0
0 
 
(
(n
nk
k
)
)
 Exponential functions
Exponential functions a
an
n
have different orders of growth for different
have different orders of growth for different
a
a’s
’s
 order
order log
log n <
n < order
order n
n

(
(
>0) < order
>0) < order a
an
n
< order
< order n
n! < order
! < order n
nn
n
a
n
n b
b
a log
/
log
log 

1-64
Basic asymptotic efficiency classes
Basic asymptotic efficiency classes
1
1 constant
constant
log
log n
n logarithmic
logarithmic
n
n linear
linear
n
n log
log n
n n-
n-log
log-n
-n
n
n2
2
quadratic
quadratic
n
n3
3
cubic
cubic
2
2n
n
exponential
exponential
n
n!
! factorial
factorial

1-65
Plan for analyzing nonrecursive algorithms
Plan for analyzing nonrecursive algorithms
General Plan for Analysis
General Plan for Analysis
 Decide on parameter
Decide on parameter n
n indicating
indicating input size
input size
 Identify algorithm’s
Identify algorithm’s basiyc operation
basiyc operation
 Determine
Determine worst
worst,
, average
average, and
, and best
best cases for input of size
cases for input of size n
n
 Set up a sum for the number of times the basic operation is
Set up a sum for the number of times the basic operation is
executed
executed
 Simplify the sum using standard formulas and rules
Simplify the sum using standard formulas and rules

1-66
Useful summation formulas and rules
Useful summation formulas and rules

l
l
i
i
n
n1 = 1+1+…+1 =
1 = 1+1+…+1 = n
n -
- l
l + 1
+ 1
In particular,
In particular, 
l
l
i
i
n
n1 =
1 = n
n - 1 + 1 =
- 1 + 1 = n
n 
 
(
(n
n)
)

1
1
i
i
n
n i
i = 1+2+…+
= 1+2+…+n
n =
= n
n(
(n
n+1)/2
+1)/2 
 n
n2
2
/2
/2 
 
(
(n
n2
2
)
)

1
1
i
i
n
n i
i2
2
= 1
= 12
2
+2
+22
2
+…+
+…+n
n2
2
=
= n
n(
(n
n+1)(2
+1)(2n
n+1)/6
+1)/6 
 n
n3
3
/3
/3 
 
(
(n
n3
3
)
)

0
0
i
i
n
n a
ai
i
= 1
= 1 +
+ a
a +…+
+…+ a
an
n
= (
= (a
an
n+1
+1
- 1)/(
- 1)/(a
a - 1) for any
- 1) for any a
a 
 1
1
In particular,
In particular, 
0
0
i
i
n
n 2
2i
i
= 2
= 20
0
+ 2
+ 21
1
+…+ 2
+…+ 2n
n
= 2
= 2n
n+1
+1
- 1
- 1 
 
(2
(2n
n
)
)

(
(a
ai
i ±
± b
bi
i ) =
) = 
a
ai
i ±
± 
b
bi
i 
ca
cai
i =
= c
c
a
ai
i 
l
l
i
i
u
ua
ai
i =
= 
l
l
i
i
m
ma
ai
i +
+ 
m
m+1
+1
i
i
u
ua
ai
i

1-67
Example 1: Maximum element
Example 1: Maximum element
T(n) = 
1
1
i
i
n-1
n-1 1 = n-1 =
1 = n-1 = 
(
(n
n)
) comparisons

1-68
Example 2: Element uniqueness problem
Example 2: Element uniqueness problem
T(n) = 
0
0
i
i
n-2
n-2 (
(
i+1
i+1
j
j
n-1
n-1 1)
=
= 
0
0
i
i
n-2
n-2 n-i-1 = (n-1+1)(n-1)/2
= 
( )
( ) comparisons
2
n

1-69
Example 3: Matrix multiplication
Example 3: Matrix multiplication
T(n) = 
0
0
i
i
n-1
n-1 
0
0
i
i
n-1
n-1 n
= 
0
0
i
i
n-1
n-1 
(
( )
= 
(
( ) multiplications
2
n
3
n

1-70
Example 4: Gaussian elimination
Example 4: Gaussian elimination
Algorithm
Algorithm GaussianElimination
GaussianElimination(
(A
A[0..
[0..n
n-
-1,0..
1,0..n
n])
])
//Implements Gaussian elimination on an
//Implements Gaussian elimination on an n-
n-by
by-
-(
(n
n+1) matrix
+1) matrix A
A
for
for i
i 
 0
0 to
to n
n -
- 2
2 do
do
for
for j
j 
 i
i + 1
+ 1 to
to n
n -
- 1
1 do
do
for
for k
k 
 i
i to
to n
n do
do
A
A[
[j
j,
,k
k]
] 
 A
A[
[j
j,
,k
k]
] -
- A
A[
[i
i,
,k
k]
] 
 A
A[
[j
j,
,i
i] /
] / A
A[
[i
i,
,i
i]
]
Find the efficiency class and a constant factor improvement.
Find the efficiency class and a constant factor improvement.
for
for i
i 
 0
0 to
to n
n -
- 2
2 do
do
for
for j
j 
 i
i + 1
+ 1 to
to n
n -
- 1
1 do
do
B
B 
 A
A[
[j,i
j,i] /
] / A
A[
[i
i,
,i
i]
]
for
for k
k 
 i
i to
to n
n do
do
A
A[
[j
j,
,k
k]
] 
 A
A[
[j
j,
,k
k]
] – A
– A[
[i,k
i,k]
] * B
* B

1-71
Example 5: Counting binary digits
Example 5: Counting binary digits

1-72
Plan for Analysis of Recursive Algorithms
Plan for Analysis of Recursive Algorithms
 Decide on a parameter indicating an input’s size.
Decide on a parameter indicating an input’s size.
 Identify the algorithm’s basic operation.
Identify the algorithm’s basic operation.
 Check whether the number of times the basic op. is executed
Check whether the number of times the basic op. is executed
may vary on different inputs of the same size. (If it may, the
may vary on different inputs of the same size. (If it may, the
worst, average, and best cases must be investigated separately.)
worst, average, and best cases must be investigated separately.)
 Set up a recurrence relation with an appropriate initial
Set up a recurrence relation with an appropriate initial
condition expressing the number of times the basic op. is
condition expressing the number of times the basic op. is
executed.
executed.
 Solve the recurrence (or, at the very least, establish its
Solve the recurrence (or, at the very least, establish its
solution’s order of growth) by backward substitutions or
solution’s order of growth) by backward substitutions or
another method.
another method.

1-73
Example 1: Recursive evaluation of
Example 1: Recursive evaluation of n
n!
!
Definition:
Definition: n
n ! = 1
! = 1 
 2
2 
 …
… 
(
(n-
n-1)
1) 
 n
n for
for n
n ≥
≥ 1 and 0! = 1
1 and 0! = 1
Recursive definition of
Recursive definition of n
n!:
!: F
F(
(n
n) =
) = F
F(
(n-
n-1)
1) 
 n
n for
for n
n ≥
≥ 1 and
1 and
F
F(
(0) = 1
0) = 1
Size:
Size:
Basic operation:
Basic operation:
Recurrence relation:
Recurrence relation:
n
multiplication
M(n) = M(n-1) + 1
M(0) = 0

1-74
Solving the recurrence for M(
Solving the recurrence for M(n
n)
)
M(
M(n
n) = M(
) = M(n
n-1) + 1, M(0) = 0
-1) + 1, M(0) = 0
M(n) = M(n-1) + 1
= (M(n-2) + 1) + 1 = M(n-2) + 2
= (M(n-3) + 1) + 2 = M(n-3) + 3
…
= M(n-i) + i
= M(0) + n
= n
The method is called backward substitution.

1-75
Example 2: The Tower of Hanoi Puzzle
Example 2: The Tower of Hanoi Puzzle
1
2
3
Recurrence for number of moves:
Recurrence for number of moves: M(n) = 2M(n-1) + 1

1-76
Solving recurrence for number of moves
Solving recurrence for number of moves
M(
M(n
n) = 2M(
) = 2M(n
n-1) + 1, M(1) = 1
-1) + 1, M(1) = 1
M(n) = 2M(n-1) + 1
= 2(2M(n-2) + 1) + 1 = 2^2*M(n-2) + 2^1 + 2^0
= 2^2*(2M(n-3) + 1) + 2^1 + 2^0
= 2^3*M(n-3) + 2^2 + 2^1 + 2^0
= …
= 2^(n-1)*M(1) + 2^(n-2) + … + 2^1 + 2^0
= 2^(n-1) + 2^(n-2) + … + 2^1 + 2^0
= 2^n - 1

1-77
Tree of calls for the Tower of Hanoi Puzzle
Tree of calls for the Tower of Hanoi Puzzle
n
n-1 n-1
n-2 n-2 n-2 n-2
1 1
... ... ...
2
1 1
2
1 1
2
1 1
2

1-78
Example 3: Counting #bits
Example 3: Counting #bits
A( ) = A( ) + 1, A( ) = 1 (using the Smoothness Rule)
= (A( ) + 1) + 1 = A( ) + 2
= A( ) + i
= A( ) + k = k + 0
=
k
2 1
2 
k 0
2
2
2 
k
n
2
log
2
2 
k
i
k
2
k
k
2
A(n) = A( ) + 1, A(1) = 0
 
2
/
n

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