Discrete Mathematical Algorithm, and Data Structure
Discrete Mathematical Algorithm, and Data Structure
Major Components of Mathematics, and Computer Science Explained with the help of C, C++, PHP, Java, C#, Python, and Dart
About the Book
Readers will learn discrete mathematical abstracts as well as its implementation in algorithm and data structures.
I have shown examples using various programming languages.
I have used as C, C++, PHP, Java, C#, Python and Dart. This book combines two major components of Mathematics and Computer Science under one roof.
Firstly, without the core conceptions and tools derived from discrete mathematics, one cannot understand the abstract or the general idea involving algorithm and data structures in Computer Science.
Secondly, the objects of data structures are basically objects of discrete mathematics.
As a result, this book tries to bridge the gap between two major components of Mathematics and Computer Science.
Finally, in any computer science course, studying discrete mathematics is essential, although they are taught separately.
Certainly, there is exception and that proves the rule. Yet, a comprehensive book, combining these two major components, is hard to find out.
Not only that, it is almost impossible to understand one without the help of other.
Consequently, I hope, this book will fill the gap.
Therefore, readers will learn discrete mathematical abstracts as well as its implementation in algorithm and data structures.
Since readers use different programming languages, I try to show examples in different programming languages.
I have used C, C++, PHP, Java, C#, Python and Dart. However, another very basic concept of programing language has motivated me to use such disparate languages.
After all, basic concepts are same in every language. Especially the algorithm and data structure concepts. Although the syntax is different.
As this book combines two major components of Computer Science under one roof, I hope students of Computer Science may find some food for thought.
The book covers the following chapters, you see below, although it’s quite tentative, as it might change a little bit while writing the book.
Table of Contents
1. Introduction to the Discourse
- Is Discrete Mathematics enough to study Computer Science?
- A short Introduction to Discrete Mathematics
- What is Discrete Mathematics
- What is the relationship between Discrete Mathematics and Computer Science
- Introducing necessary conceptions
- QUIZZ and Challenge on Chapter One
2. Introduction to Programming Language and Boolean Algebra
- Logic, Mathematics, and Programming Language
- Introduction to Boolean Algebra
- QUIZZ and Challenge on on Chapter Two
3. De Morgan’s Laws on Boolean Algebra, Logical Expression, and Algorithm
- Logical Expression
- Short Circuit Evaluation
- Syntax, Semantics and Conditional Execution
- Why we need Control Constructs
- Discrete Mathematical Notations and Algorithm
- QUIZZ and Challenge on Chapter Three
4. Data Structures in different Programming languages
- Mean, Median and Mode
- Array, the First Step to Data Structure
- Let us understand some Array features
- Set Theory, Probability and Array
- Skewed Mean, Maximized Median
- Complex Array Algorithm
- QUIZZ and Challenge on Chapter Four
- printing result
5. Data Structures: Abstractions and Implementation
- How objects work with each other
- More Algorithm and Time Complexity
- Introducing Data Structures
- How Calculus and Linear Algebra are Related to this Discourse
- QUIZZ and Challenge on Chapter Five
6. Data Structures in Detail
- Frequently Asked Questions about Data Structures
- Abstract Data Type (ADT)
- Linear Data Structures
- Modeling of a Structure
- ArrayList to overcome limitations of Array
- ArrayList or LinkedList, which is faster?
- Collection Framework in programming languages
- Stack and Queue in Java
- Deque, a high-performance Abstract Data Type
- QUIZZ and Challenge on Chapter Six
- Challenge 1 : Can we test in a program whether the Stack has been overflowed or not?
- Challenge 2 : Can you convert an array to a queue and use all the queue methods to manipulate that array?
- Challenge 3 : Why LinkedList consumes less memory than ArrayList. Write a program and explain why it happens.
- Challenge 4 : Can you give examples of hybrid linear data structure that provides all the capabilities of stacks and queues under one roof?
7. Algorithm, Data Structure, Collection Framework and Standard Template Library (STL)
- Introducing Algorithm Library
- Different types of Algorithms
- Binary Tree and Data Structure
- Collection Framework in Java
- Discrete Mathematical Abstractions and Implementation through Java Collection
- Comparator, Comparable and Iterator
- Standard Template Library in C++
- QUIZZ on Chapter Seven
- Challenge 1 : Standard Template Library (STL) provides many generic versions of standard algorithms that replace our low-level plumbing. Can you show the difference by writing code.
- Challenge 2 : Can you prove with an example how the STL makes a difference when we want to sort a list of data.
- Challenge 3 : Using recursion in programing is closer to our discrete mathematical definitions. Can you prove it?
- Challenge 4 : Recursive algorithm should have a base case. Write a program where we can proceed towards the base case and a condition to stop the recursion.
- Challenge 5 : Can you prove that Discrete Mathematics, data structure and algorithm are inter-connected?
- Challenge 6 : Can you create a general template class and method that will allow to pass any data of your choice.
8. Time Complexity
- Order of n, or O(n)
- Big O Notation
- QUIZZ on Chapter Eight
- Challenge 1 : Data structures are discrete structures and hence, the algorithms are all about discrete structures. Write a program to establish a relationship between algorithms and discrete mathematics
- Challenge 2 : How we can get the total of a series of positive integers that starts from 0 and ends at 5. Can it be done recursive way?
- Challenge 3 : Detect the main problem in the code below and rewrite it in proper way.
9. Set, Symmetric Difference and Propositional Logic
- Why Set is important in Data Structures
- How Symmetric Difference and Propositional Logic combine
- QUIZZ on Chapter Nine
- Challenge 1 : Why the Set implementation ‘HashSet’ is better than the general-purpose Set implementation ‘TreeSet’? Can you compare and prove that?
- Challenge 2 : Can you combine Symmetric Difference and Propositional Logic in one program? You can write it in any programming language.
- Challenge 3 : In discrete mathematical paradigms, Set abstraction is basically chaotic, and unordered. However, no duplicate element is allowed in the world of Set. Can you prove it?
10. Combinatorics and Counting, Permutation and Combinations
- Permutation and Combination
- QUIZZ on Chapter ten
- Challenge 1 : How many ways we can rearrange 5 balls.
- Challenge 2 : Suppose you have a string “abcd”. You are aked to rearrange the String. How many ways could you rearrange the String when repetition is allowed? Here repetition means, you can rearrange this way: bacd, cdba, dbac, etc.
- Challenge 3 : Can you show the difference between two types of permutation,where repetition is not allowed and allowed.
- What Next
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