Sanjib Sinha - Discrete Mathematical Algorithm, and Data Structure

As you know from the title, this book covers a wide range of three topics, in fact, too wide range of three topics, on which you will find books separately. You will find many good books on Discrete Mathematics, you will find many good books on Algorithm and Data Structures, as well.

Keeping that fact in mind, we are writing a book which tries to discover the relations between three well known topics, and, moreover, how they influence each other.

We can compare this relationship, such as, a relation between the writer and readers. Readers always reconstruct a book while reading, and they write it in a new way in their mind. Each time a book is read, it is actually written again.

We can consider this book as a confluence of the three distinct topics, such as Discrete Mathematics, Algorithm and Data Structures.

Let us start with Discrete Mathematics first.

What is Discrete Mathematics? We need to know that first. Second, we need to know what is the relationship of Discrete Mathematics to Computer Science. Finally, we want to know how we can implement the Discrete Mathematical concepts into various fields of Computer Science, such as Algorithm and Data Structures.

Here, we are primarily concerned with Algorithm and Data Structure. These topics sit at the core of any Computer Science curriculum. Without understanding these two components properly, we can not claim that we are students of Computer Science!

Data Structures are most fundamental and building blocks of computer science. Understanding Data Structures is an essential thing to design and build efficient software. At the same way, we can say, that knowing Algorithm is also compulsory to understand these fundamental blocks of Computer Science.

Therefore, I have chosen these two topics and try to implement Discrete Mathematical concepts into them so that our understanding might take a proper shape.

In this chapter, our first and foremost concern is to know what is Discrete Mathematics. However, before starting the book, there is an important question.

Our answer is NO.

There are some other topics that we must know now, simultaneously, or later, to get a proper understanding of Computer Science. If we love our subject, then we will try to understand how Calculus, Linear Algebra and as a whole, Mathematics is related to Computer Science.

We are not pushing ourselves to become a mathematician. That we are not going to suggest.

We want to study computer science.

Nevertheless, we also want to know all the necessary concepts that are related to computer science.

Before moving on to our real discourse, let us know what the word Mathematics means to us. Mathematics includes the topics, such as quantity (number theory), structure (algebra), space (geometry), and change (mathematical analysis).

As we see, as a programmer, we also deal with such topics. We deal with numbers, all types of numbers, so an introduction to number theory should be included in our study. We need to study structures (remember data structures); therefore, a knowledge of algebra is important where calculus and linear algebraic equations might play a vital role.

Therefore we need to know how to find area and volume of two moving objects, how to find the slope of a line, or need to know about linear algebraic concepts like matrices and vectors.

On the other hand, Computer Programming involves tasks like data analysis, generating algorithms, making algorithms more accurate with less resource consumption, and finally we need to implement algorithms in a chosen programming language. These processes, as a whole, are known as coding. Knowledge of specialized algorithms and formal logic may be enough to be a good programmer.

However, if we want to change the word good to another word great then we must equip ourselves with some mathematical knowledge that are not directly related to Discrete Mathematics.

Although they are not directly related, they are indirectly related. The study of calculus starts with the relationship between velocity and distance; and, they are ordered pair of sets, which is very much a discrete mathematical conceptions.

Calculus is a mathematical study of continuous change; however, computer scientific study is severely discrete, it deals with distinct objects.

Besides, linear algebraic conceptions are fundamental in many areas where line, plane and spaces are involved. We need to know them as well. If not now, then we should know them later, of course.

We should always remember that different branches of mathematical conceptions are key components in computation. We should not forget that truth as long as we think ourselves as a student of computer science.

What is Discrete Mathematics? We need to know that first. Second, we need to know what is the relationship of Discrete Mathematics to Computer Science. Finally, we want to know how we can implement the Discrete Mathematical concepts into various fields of Computer Science.

Here, we are primarily concerned with Algorithm and Data Structure. These topics sit at the core of any Computer Science curriculum. Without understanding these two components properly, we can not claim that we are students of Computer Science!

Therefore, I have chosen these two topics and try to implement Discrete Mathematical concepts into them so that our understanding might take a proper shape.In this chapter, our first and foremost concern is to know what is Discrete Mathematics.

Discrete Mathematics is a branch of Mathematics and, to be very particular, it is the study of Mathematical structures, and objects that are discrete.

In that sense, people, animal, chair and everything we see around us, fall under this mathematical study. Why? Because, they are discrete, distinct. They are not continuous. Think about real numbers that consist of rational and irrational numbers, or a number line, which is continuous; so that is not discrete mathematics. Rather, natural numbers (N), like 0, 1, 2, etc, which are discrete fall under this category.

Discrete Mathematical conceptions handle with distinct and separated values. The opposite of discrete mathematics is continuous mathematics, such as calculus or Euclidean Geometry. However, Euclidean Algorithm that helps us find the greatest common divisor of two positive natural numbers (GCD), is very much discrete mathematics. We will see to an example in a minute.

Have you found any contradiction in the above statement?

I have found one; and before proceeding further, I think, we should make it clear first. We have just said that real numbers, or a number line where between 1 and 2, there are infinite numbers, cannot be discrete. However, when we say, positive natural numbers like 1, 2, 3, which are distinct and separated, belong to discrete mathematics. 1, 2 or 51, every positive natural numbers are also real numbers. Right? They belong to the number line. Right?Then where we stand? We are contradicting ourselves.