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5. Introduction to Brute Force
Algorithm
“Data is the new oil” this is the new mantra that is ruling the global economy. We live in the digital world,
and every business revolves around data, which translates into profits and helps the industries stay ahead
of their competition. With the rapid digitization, an exponential increase in the app-based business
model, cyber-crimes is a constant threat. One such common activity that hackers perform is Brute force.
Brute Force is a trial and error approach where attackers use programs to try out various combinations to
break into any websites or systems. They use automated software to repetitively generate the User id
and passwords combinations until it eventually
generates the right combination.

6. Brute-Force Search
Brute force search is the most common search algorithm as it does not require any domain knowledge; all
that is required is a state description, legal operators, the initial state and the description of a goal state.
It does not improve the performance and completely relies on the computing power to try out possible
combinations.

The brute force algorithm searches all the positions in the text between 0 and n-m, whether the
occurrence of the pattern starts there or not. After each attempt, it shifts the pattern to the right by
exactly 1 position. The time complexity of this algorithm is O(m*n). If we are searching for n characters in
a string of m characters, then it will take n*m tries.

Let’s see a classic example of a travelling salesman to understand the algorithm in an easy manner.
Suppose a salesman needs to travel 10 different cities in a country, and he wants to determine the
shortest possible routes out of all the possible combinations. Here brute force algorithm simply calculates
the distance between all the cities and selects the shortest one.
Another example is to make an attempt to break the 5 digit password; then brute force may take up to
105
attempts to crack the code.

9. Closest Pair
Problem statement: To find out the two closest points
in a set of n points in the two-dimensional cartesian
plane. There is n number of scenarios where this
problem arises. A real-life example might be in an air
traffic control system where you have to monitor the
planes flying near to each other, and you have to find
out the safest minimum distance these planes should
maintain.

10. Convex Hull
Problem Statement: A convex hull
is the smallest polygon that
contains all the points. The
convex hull of a set s of the point
is the smallest convex polygon
containing s.

11. Convex Hull
The convex hull for this set of points is the convex polygon with vertices at P1, P5, P6, P7, P3.
A line segment P1 and Pn of a set of n points is a part of the convex hull if and only if all the other points
of the set lies inside the polygon boundary formed by the line segment.
Let’s relate it with the rubber band,
Point (x1, y1), (x2,y2) make the line ax+by = c
When a = y2-y1, b = x2-x1 and c = x1*y2 – x2*y1 and divides the plane by ax+by-c < 0 and ax+by-c > 0
So we need to check ax+by-c for the other points.
Brute force solve this problem with the time complexity of O(n 3
)

12. Exhaustive Search
or discrete problems in which there is no known efficient solution, it becomes necessary to test each and
every possible solution sequentially.
Exhaustive search is an activity to find out all the possible solutions to a problem in a systematic manner.
Let’s try to solve the Travelling salesman problem (TSP) using a Brute exhaustive search algorithm.

13. Exhaustive Search
Problem Statement: There are n cities that salesmen need to travel, he wants to find out the shortest
route covering all the cities.
We are considering Hamilton Circuit to solve this problem. If a circuit exists, then any point can start
vertices and end vertices. Once the start vertices are selected, then we only need the order for the
remaining vertices, i.e. n-1
Then there might be (n-1)! Possible combinations and the total cost for calculating the path might be
O(n). thus the total time complexity might be O(n!).

14. Knapsack problem
The knapsack problem is a problem in combinatorial optimization: Given a set of items, each with a weight
and a value, determine the number of each item to include in a collection so that the total weight is less
than or equal to a given limit and the total value is as large as possible.

Recursion by Brute-Force algorithm OR Exhaustive Search.
Approach: A simple solution is to consider all subsets of items and calculate the total weight and value of all
subsets. Consider the only subsets whose total weight is smaller than W. From all such subsets, pick the
maximum value subset.
Optimal Sub-structure : To consider all subsets of items, there can be two cases for every item.
Case 1: The item is included in the optimal subset.
Case 2: The item is not included in the optimal set.

Therefore, the maximum value that can be obtained from ‘n’ items is the max of the following two values.
Maximum value obtained by n-1 items and W weight (excluding nth item).
Value of nth item plus maximum value obtained by n-1 items and W minus the weight of the nth item
(including nth item).
If the weight of ‘nth’ item is greater than ‘W’, then the nth item cannot be included and Case 1 is the only
possibility.