UnitI (1).ppt

1-0
Design and Analysis of Algorithms – Unit I
Chapter 1
Introduction

1-1
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.

1-2
Notion of algorithm
“computer”
Algorithmic solution
problem
algorithm
input output

1-3
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.

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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.

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

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

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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

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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, a mixture of a natural language and
programming language-like constructs.
 flowchart, a method of expressing an algorithm by a
collection of connected geometric shapes containing
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
 Peril of incorrect & inefficient implementation
 Require testing & debugging
 Require code optimizing

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Important Problem Types

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

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

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

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

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

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

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Geometric Problems
 Geometric algorithms deal with geometric objects such as
points, lines, and polygons.
2 class problems:
 The closest pair problem: given n points in the plane, find
the closest pair among them.
 The convex hull problem asks to find the smallest convex
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, another large special area of
applications, are problems that involve mathematical
objects of continuous nature: solving equations and systems
of equations, computing definite integrals, evaluating
functions, and so on.

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Fundamentals of Analysis of
algorithm efficiency

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

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Theoretical analysis of time efficiency
Time efficiency is analyzed by determining the number of
repetitions of the basic operation as a function of input size
 Basic operation: the operation that contributes the most
towards the running time of the algorithm
T(n) ≈ copC(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
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
 Select a specific (typical) sample of inputs
 Use physical unit of time (e.g., milliseconds)
or
Count actual number of basic operation’s executions
 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
• Cworst(n)
 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
• Cbest(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

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

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

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

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

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Values of some important functions as n  

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

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

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

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Big-oh

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

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Big-omega

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

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Big-theta

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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 -notation
and -notation.
Proof. There exist constants c1, c2, n1, n2 such that
t1(n)  c1*g1(n), for all n  n1
t2(n)  c2*g2(n), for all n  n2
Define c3 = c1 + c2 and n3 = max{n1,n2}. Then
t1(n) + t2(n)  c3*max{g1(n), g2(n)}, for all n  n3

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

1-60
Establishing order of growth using limits
lim T(n)/g(n) =
0 order of growth of T(n) < order of growth of g(n)
c > 0 order of growth of T(n) = order of growth of g(n)
∞ order of growth of T(n) > order of growth of g(n)
n→∞

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

1-62
Orders of growth of some important functions
 All logarithmic functions loga n belong to the same class
(log n) no matter what the logarithm’s base a > 1 is
because
 All polynomials of the same degree k belong to the same class:
aknk + ak-1nk-1 + … + a0  (nk)
 Exponential functions an have different orders of growth for
different a’s
 order log n < order n (>0) < order an < order n! < order nn
a
n
n b
b
a log
/
log
log 

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

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

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

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

1-67
Example 2: Element uniqueness problem
T(n) = 0in-2 (i+1jn-1 1)
= 0in-2 n-i-1 = (n-1+1)(n-1)/2
= ( ) comparisons
2
n

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

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

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Example 5: Counting binary digits

1-71
Plan for Analysis of Recursive Algorithms
 Decide on a parameter indicating an input’s size.
 Identify the algorithm’s basic operation.
 Check whether the number of times the basic op. is executed
may vary on different inputs of the same size. (If it may, the
worst, average, and best cases must be investigated
separately.)
 Set up a recurrence relation with an appropriate initial
condition expressing the number of times the basic op. is
executed.
 Solve the recurrence (or, at the very least, establish its
solution’s order of growth) by backward substitutions or
another method.

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

1-73
Solving the recurrence for M(n)
M(n) = M(n-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.

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

1-75
Solving recurrence for number of moves
M(n) = 2M(n-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

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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

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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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