Design and Analysis of Algorithm Brute Force 1.ppt

Department of Computer Science
COMSATS University, Islamabad
Design and Analysis of Algorithm
Tanveer Ahmed Siddiqui

Algorithm Design and Analysis Process
Understand the problem
Decide on : algorithm
design techniques etc.
Design an algorithm
Prove correctness
Analyze efficiency etc
Code the algorithm
Decide on : algorithm
design techniques etc.
Recap and Today Covered

Reading Material
Read Chapter 3
Brute Force and Exhaustive Search

 How to Design Algorithm using Brute Force(BF) Approach
 Swapping two numbers
 Computing an (a > 0, n a nonnegative integer)
 Computing n!
 Add n numbers
 Decimal to Binary conversion
 Finding Maximum/Minimum Element in a list
 Multiply two n by n Matrices
 Selection Sort
 Bubble Sort
 Sequential Search/Linear Search
 Conversion of a 2D array into 1D array
 Find the value of polynomial
Objectives

 Designing an algorithm is essentially a
creative effort containing all the ingredients
of a thriller:
 Adventure
 Excitement
 Challenge
 Suspense.
Designing an Algorithm

Designing an algorithm is easy if you
know
design
techniques

 An algorithm design techniques (or “strategy”
or “paradigm”) is a general approach to solve
problems algorithmically
 It can be applicable to a variety of problems
from different area of computing.
 The design approach depends mainly on the
model chosen.
What is algorithm designing technique?

 They provide us guidance in designing
algorithms for new problems
 They represent a collection of tools useful for
applications
Why do we need to know such techniques?

Department of Computer Science 9
 Brute force
 Decrease and conquer
 Divide and conquer
 Greedy technique
 Dynamic programming
 Backtracking
What are the most used techniques?

 How you solve this puzzle:
 Given a 3×3 board with 8 tiles (every tile has
one number from 1 to 8) and one empty space.
The objective is to place the numbers on tiles to
match the final configuration using the empty
space. We can slide four adjacent (left, right,
above, and below) tiles into the empty space.
 Initial state : any configuration
 Goal state : tiles in a specific order
10
Motivation Discussion

Motivation Discussion
State Space

What is Brute Force?
 A straightforward approach to solving a problem,
usually directly based on the problem’s statement
and definitions of the concepts involved.
 “Brute Force” means “Just do it!”
 Generally, it involved iterating through all possible
solutions until a valid one is found.
 Other Names
 Generate and Test
 Exhaustive Search
 Can you name some problems that can be solved
with BF
 Examples:
12

 How many examples do you know?
 Swapping two numbers
 Computing an (a > 0, n a nonnegative integer)
 Computing n!
 Add n numbers
 Decimal to Binary conversion
 Finding Maximum/Minimum Element in a list
 Multiply two n by n Matrices
 Selection Sort
 Bubble Sort
 Sequential Search/Linear Search
 Conversion of a 2D array into 1D array
 Find the value of polynomial
13
Known examples of Brute Force

Designing
Algorithms
using Brute
Force

 Problem: Design an algorithm that Exchange the
values of two variables say x and y.
 Solution:
 What is input
 Two numbers say x and y.
 What should be the output
 Interchange the value of x and y
 Which designing technique?
 Brute-Force
 Logic/Idea
Example-1

 Logic/Idea
 Take a temporary variable temp to swap the
numbers.
 The contents of the first variable is copied into the
temp variable.
 Then, the contents of second variable is copied to the
first variable.
 Finally, the contents of the temp variable is copied
back to the second variable which completes the
swapping process.
Example-1
Do you have another
idea to solve
exchange problem?

 Do you have another idea to solve exchange
problem?
Example-1

Brute Force Example
 In RSA (Ron Rivest, Adi Shamir, and Leonard
Adleman) encryption algorithm we need to
compute an mod m for a > 1 and large n.
 Design an algorithm that computes an using BF
 Solution:
 What is input
 Two numbers say x and n.
 Interchange the value of xn
 Brute-Force
 Logic/Idea
18
Example-2: Computing an

Brute Force Example
 Logic/Idea: x n =
 First response: Multiply 1 by x n times which is
the “Brute Force” approach.
19
Computing an
x × x × … … × x
n times

Induction Examples (4/4)
3.3 Mathematical Induction
Example 3: Factorial of positive integer
 We want to compute n! = 1 × 2 × … … × n
Is there a
connection?

 Problem: Design an algorithm for finding the
binary representation of a positive decimal
integer.
 Solution:
 What is input
 Positive integer n.
 Binary string bk bk bk-1 bk-2………….. b1 b0
 Brute-Force
 Logic/Idea
Example 4: Decimal to Binary Conversion

 Problem: Design an algorithm for finding the
binary representation of a positive decimal
integer.
Representation
Example 4: Decimal to Binary Conversion

 Problem: Design an algorithm that count
numbers of bits in a binary representation of a
given number n.
Representation
Your Turn

Brute Force Examples
Some Examples

 Problem: Design an algorithm that find the
maximum number in a given array of n elements.
 Solution:
 What is input
 An array A of n numbers e.g.
 The value of largest element in A. e.g., 10
 Brute-Force
 Logic/Idea
4 1 8 9 3 7 10 2 3 1
Example-6: Largest element

 Problem: Design an algorithm that find maximum
number in a given 2D array of nxn elements.
 Solution:
Your Turn

 Problem: Design an algorithm that multiply two matrices
A and B
 Solution:
 What is input
 Two 2D array A and B of compatible size.
 A matrix C such that C = AB.
 Brute-force
 Logic/Idea?
55 99 6 29 1 7 3
Example-7

Brute Force Examples
Example 7: Matrix Multiplication

 Problem: Design an algorithm that find a key in a
given array of n elements.
 Solution:
 What is input
 An array A of n numbers and a key say K
 The location of key if found otherwise return false
 Brute-Force
 Logic/Idea
Example-8: Searching

Brute-Force Polynomial Evaluation
Problem: Design an algorithm that compute the
value of polynomial
p(x) = anxn + an-1xn-1 +… + a1x1 + a0
at a point x = x0
Example-9

Brute-Force Polynomial Evaluation
Problem: Find the value of polynomial
p(x) = anxn + an-1xn-1 +… + a1x1 + a0
at a point x = x0
Brute-force algorithm
p  0.0
for i  n downto 0 do
power  1
for j  1 to i do //compute xi
power  power  x
p  p + a[i]  power
return p
Example-9

Polynomial Evaluation: Improvement
We can do better by evaluating from right to left:
Better brute-force algorithm
p  a[0]
power  1
for i  1 to n do
power  power  x
p  p + a[i]  power
return p
Example-9

 Problem: Design an algorithm convert a 2D array
into 1D array
 Solution:
 What is input
 An 2D-array A[i][j]. e. g. int [5][5]
 You could picture the conversion
 to the corresponding 1-D array
 like this:
Example-10
:

 Brute-Force
 Logic/Idea
 A 1-D array looks like this:int [5] :
 You could picture the conversion to the corresponding 1-D array
like this
Example-10
:

 Logic/Idea
 But an alternative way of thinking about it is to
picture the original array, but re-labelled - like this
Example-10
:
2-D array index [i][j] => 1-D array index [i*5 + j]

Example-10
ALGORITHM Convert2d_into_1D(arr2D[i][j], arr1D [n*m])
Input:
OutPut:
for (i = 0; i < n; ++i)
{
for (j = 0; j < m; ++j)
{ // mapping 2D array to 1D array
arr1D[i * m + j] = arr2D[i][j];
}
}
2-D array index [i][j] => 1-D array index [i*5 + j]

 Numerical problems, searching, sorting, etc.
 Acceptable efficiency
 Can be used for large problem instances
 Combinatorial problems
 Exhaustive search
 Set of candidate solutions grows very fast
 Used only for reduced size instances.
 Let us apply Brute force approach for solving
sorting problem.
38
Where to Apply?

CONCLUSION

 The (most) straightforward approach for solving
a problem.
 “Brute Force” means “Just do it!”
 Directly based on
 The problem statement
 The definitions involved
 Generally, it involved iterating through all possible
solutions until a valid one is found.
 Other Names
 Generate and Test
 Exhaustive Search
40

 Strengths
 Simplicity
 Applicable to different kinds of problems
 Weaknesses
 (Very!) Low efficiency in some cases
 Useful only for instances of (relatively) small size!
41
Strength and Weakness of Brute Force

 Algorithms should be learn at a higher level
with linkages to prior knowledge and with each
other so as to see how one algorithm is different
from the rest and what it does and does not
perform, and why?
 In this integrative approach of learning the
initial investment is large but it is essential for
meaningful learning
How we do it?

 Intuition First & Formalism Later
 Common & Small Building Blocks
 Design of Cruder versions providing a ladder
 Visual patterns or puzzles
 We do not tell the story of an algorithm?
 The interesting story connects all characters and
provides a panoramic picture
 Platforms for Comparisons
Ladder
How we do it?

Design and Analysis of Algorithm Brute Force 1.ppt

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