UNIT-2

1. UNIT-II
BRUTE FORCE AND DIVIDE-
AND-CONQUER
Brute Force: String Matching-Closest-Pair and
Convex-Hull Problems-Exhaustive Search -
Traveling Salesman Problem - Knapsack
Problem - Assignment problem.

2. BRUTE FORCE:
Brute Force is a straightforward approach of solving the
problem.
Works Directly on the problem statement and definitions of
concepts that are directly involved in the problem.
In short the method has “JUST DO IT” approach.
Examples: Selection sort, Bubble Sort, Sequential Search,
Depth first search, Breadth first search

3. Examples:
For a given number a and a non negative integer n,
a^n=a*a*a*……*a for n times
Computing n! =1*2*3*…….n
Performing multiplication of two matrices : C=AB
Searching a key value from given list of elements.

4. Advantages Of Brute Force:
•Simple and easy to understand
•Easy to implement
•Guarantees correct solution
•Applicable to many problems
•No complex data structures required
•Useful for small input sizes
•Serves as a baseline for optimized algorithms

5. String Matching
String Matching algorithms are used in text processing.
String matching means finding one or more generally all the occurrences
of a string in the text. These occurrences are called as pattern.
Also called as Pattern matching algorithms.

6. String Matching
Naïve algorithm
Rabin Karp algorithm
Knuth Morris pratt
Boyer Moore

7. Naïve Algorithm
Simplest method
This algorithm performs a checking at all positions in the text
between 0 to n-m, whether an occurrence of the pattern starts there
or not.
Then after each attempt, it shifts the pattern by exactly one
position to the right.
If match is found then it returns otherwise the matching process
is continued by shifting one character to the right.
If there is no match at all the text for the given pattern even then
we have to do n comparisons.

15. Homework:
1) Show the comparisons the Naïve-
String matcher makes for the pattern p={1001} in the text
T={0000100010010}
2)Solve the following using Brute-Force algorithm: Find
whether the given string follows specified pattern and return 0
or 1.
a)Pattern:”abba”input:”redblueredblue” should return 1
b) Pattern:”aaaa”input:”asdasdasdasd” should return 1
c) Pattern:”aabb”input:”xyzabcxyzabc” should return 0

16.  How many comparisons (both successful and unsuccessful) will be
made by the Brute force string matching algorithm in searching for each
of the following patterns in the binary text of 1000(Thousand) zeros?
A)00001 b)10000
Total unsuccessful searches=n-m+1=1000-5+1=996
Total number of comparisons(C)= Comparisons made by pattern
* Total unsuccessful searches
A) C= 5(Total character compared) *996(1000 zeros-
4matched zeros)
C=4980

17. B) C= 1(Total character compared) *996
C=996

18. Rabin Karp algorithm
The Rabin–Karp algorithm is an efficient string matching
(pattern searching) algorithm that uses hashing to find
occurrences of a pattern in a given text.

20. HASH VALUE
CCB-86
CBB-32
BBC-46
BCC-100
CCC-87
CCB-86
CBG-37
BGH-18
GHC-12
HCC-22
CCB-86
CBJ-40

21. Knuth Morris Pratt Algorithm
The Knuth–Morris–Pratt (KMP) algorithm is an efficient
string matching algorithm that avoids unnecessary
comparisons by using information from previous matches.
When a mismatch occurs, KMP does not restart matching
from the beginning of the pattern.
Instead, it uses a preprocessed array called LPS (Longest
Prefix Suffix) to decide how far the pattern can be shifted.

26. Boyer Moore
The Boyer–Moore algorithm is a highly efficient string matching (pattern
searching) algorithm that compares the pattern with the text from right to left and
skips sections of the text using smart shift rules.
THIS IS A TEST (TEXT) TEST(PATTERN)
TEST
T=max(1,4-0-1)= 3 = 1
E=max(1,4-1-1)=2
S=max(1,4-2-1)=1
T=max(1,4-3-1)=1 T=1update using recent value


27. Closest-Pair
The Closest pair problem is finding the two closest points
from the set of n points.
For two dimensional case the distance between two pints is
denoted by Euclidean distance.
The basic operation is computing Euclidian distance between
two points.

28. Closest-Pair

29. Convex Hull Problem
Given a set {P1,P2,P3,….Pn} of points in the plane, the convex hull H(s)
is the smallest convex polygon in the plane that contains all of the points
of S.The set S is called as convex set.
A polygon is convex if and only if any two points from the set
forming a line segment with end points entirely within the polygon.

30. The convex hull of a set S points is the smallest convex set
containing S.
If S is a set of two points its convex hull is the line segment
connecting these points. If S is a set of three points then its
convex hull is the triangle
 A polygon is convex if and only if any two points from
the set forming a line segment with end points entirely within
the polygon.
Convex hull problem is the problem of constructing the
convex hull for a given set S of n points.

32. The Points
{1,2,3,4,5,6}
are called
Extreme Points.

33. Definition Of Extreme Points
An extreme point of a convex set is a point which
is not a middle point of any line segment with end
points in the set.

34. Exhaustive Search
Exhaustive Search is a method in which solution is
obtained by searching each element of given
problem
Exhaustive Search makes use of straightforward
Brute Force’s approach.
Travelling Salesman Problem
Knapsack Problem
Assignment Problem

35. Travelling Salesman Problem
Travelling Salesman Problem is a famous problem in the
graph theory.
It can be stated as Consider that there are n cities and
traveling salesman has to visit each city exactly once and
has to return to the city from where he has started.
To model this problem weighted graph can be used.
The vertices of such graph represent cities and the edge
weight specifies the distance between the cities.

36. Travelling Salesman Problem….
Travelling Salesman Problem can also be stated as
finding shortest Hamiltonian circuit of a graph.
Shortest Hamiltonian circuit is a cycle in the given
graph such that all the vertices of the graph can be
visited only once.
The tour is obtained in such a way has shortest
distance.

37. Consider the graph
This is a weighted graph in
which weights along the
edges represent the
distances among the cities.
To find Hamiltonian circuit (i.e) the path in which
each city is visited exactly once and returning to the
city from which it has started initially

39. Find all the solution to the travelling salesman
problem by exhaustive search. Give the optimal.

40. Knapsack Problem
Knapsack Problem which can be solved using exhaustive search.
It can be stated as follows:
Suppose that there are n objects from i=1,2,3,…….n. Each
object i has some weight wi and values associated with each
object is vi.
And capacity of knapsack is W.A thief has to pickup the most
valuable objects to fill the knapsack to its capacity

41. Knapsack Problem

42. ITE
M
WEIGHT (
∑ Wi )
VALUES (
∑ Vi )
1 2 $1
2 3 $2
3 4 $8
4 5 $6
1,2 2+3=5 1+2=$3
1,3 2+4=6 1+8=$9
1,4 2+5=7 1+6=$7
2,3 3+4=7 2+8=$10(Not
Feasible as
exceeding W)
1,2,3 2+3+4=9 Not Feasible as
exceeding W
1,2,4 2+3+5=10 Not Feasible as
exceeding W
2,3,4 3+4+5=12 Not Feasible as
exceeding W
Thus by exhaustive search method ,we get
selection of item 2 and item 3 for
knapsack. Each element of the problem
domain has to be searched for obtaining
solution. These problems are also called as
NP hard problems.

43. Assignment problem
Consider that there are n people who need to be assigned to execute n jobs
i.e only one job at a time.
Then problem is to find such assignment that gives smallest total cost.
The cost can be computed as cost C[i,j] i th person assigned to j th job.

44. The cost can be obtained by assigning the jobs in various
combinations as n!=4!=4*3*2*1=24
Combinations
(Job 1,2,3,4)
Cost
1,2,3,4 10+5+2+5=22
1,2,4,3 10+5+9+10=34
1,3,4,2 10+4+9+7=30
1,3,2,4 10+4+9+5=28
1,4,2,3 10+8+9+10=37
1,4,3,2 10+8+2+7=27
2,1,3,4 3+7+2+5=17
2,1,4,3 3+1+10+9=23
2,3,1,4 3+9+8+5=25
2,3,4,1 3+9+10+9=31
2,4,1,3 3+7+8+9=27
2,4,3,1 3+7+2+9=21

45. Home Work: