## The Mastery of Computer Programming: Primary Algorithms - Sykalo Eugene 2023

# Basic techniques

Arithmetic Algorithms

## Introduction to Algorithms

Algorithms are step-by-step procedures used to solve problems or perform tasks. They are at the heart of computer programming and are used to process and manipulate data. Algorithms can be classified into different types, such as sorting, searching, and encryption.

Sorting algorithms are used to arrange data in a specific order, such as alphabetical or numerical. Searching algorithms are used to find a specific item within a collection of data. Encryption algorithms are used to encode and decode data to keep it secure.

Algorithms are important in computer programming because they allow us to solve complex problems efficiently. By breaking down a problem into smaller, more manageable parts, we can design algorithms that can solve the problem in a reasonable amount of time.

There are different ways to measure the efficiency of an algorithm, such as its time complexity and space complexity. Time complexity refers to the amount of time an algorithm takes to complete as the input size increases. Space complexity refers to the amount of memory an algorithm uses to complete the task.

## Algorithm Design Techniques

Algorithm design is the process of creating an efficient algorithm to solve a given problem. There are several techniques that are commonly used to design algorithms, including:

### Divide and Conquer

This technique involves breaking down a problem into smaller subproblems, solving each subproblem independently, and then combining the solutions to solve the original problem. This approach is often used in sorting and searching algorithms.

### Dynamic Programming

Dynamic programming is a technique used to solve problems that can be broken down into smaller subproblems. It involves solving each subproblem only once and storing the results to avoid redundant computations. This technique is often used in optimization problems.

### Greedy Algorithms

Greedy algorithms make locally optimal choices at each step of the algorithm in the hope of finding a global optimum. This approach is often used in optimization problems and can be very efficient, but it does not always guarantee the optimal solution.

### Backtracking

Backtracking is a technique used to find all possible solutions to a problem by exploring all possible paths. It involves choosing a path, exploring it to see if it leads to a solution, and backtracking if it does not. This approach is often used in combinatorial problems.

### Branch and Bound

Branch and bound is a technique used to solve optimization problems by exploring all possible solutions and keeping track of the best solution found so far. It involves dividing the problem into smaller subproblems, exploring each subproblem, and pruning the search space based on the current best solution.

Each of these techniques has its own strengths and weaknesses, and the choice of technique depends on the problem at hand. By understanding these techniques, you can design efficient algorithms to solve a wide range of problems.

## Asymptotic Analysis

Asymptotic analysis is a method used to analyze the performance of an algorithm in terms of its input size. This allows us to compare the performance of different algorithms and choose the most efficient one for a given problem.

There are several types of complexity classes used in asymptotic analysis, including:

### Big-O notation

Big-O notation provides an upper bound on the growth rate of a function. It is used to express the worst-case time complexity of an algorithm. For example, if an algorithm has a time complexity of O(n), it means that the running time of the algorithm is proportional to the size of the input.

### Big-Omega notation

Big-Omega notation provides a lower bound on the growth rate of a function. It is used to express the best-case time complexity of an algorithm. For example, if an algorithm has a time complexity of Ω(n), it means that the running time of the algorithm is at least proportional to the size of the input.

### Big-Theta notation

Big-Theta notation provides both an upper and lower bound on the growth rate of a function. It is used to express the average-case time complexity of an algorithm. For example, if an algorithm has a time complexity of Θ(n), it means that the running time of the algorithm is proportional to the size of the input, within a constant factor.

Asymptotic analysis is important because it allows us to compare the performance of different algorithms without having to run them on actual data. By analyzing the worst-case, best-case, and average-case time complexity of an algorithm, we can determine its efficiency and choose the most appropriate algorithm for a given problem.

It is important to note that asymptotic analysis does not provide an exact measure of the running time of an algorithm. Instead, it provides an estimate of how the running time grows as the input size increases. Actual running times may vary depending on factors such as hardware, software, and the specific input data used.

## Data Structures

Data structures are used to store and organize data in a way that allows efficient access and modification. There are many different types of data structures, each with its own strengths and weaknesses. Some of the most common data structures used in computer programming include:

### Arrays

Arrays are a collection of elements, each identified by an index or a key. They are used to store homogeneous data types, such as integers or characters. Arrays are efficient for accessing individual elements, but less efficient for inserting or deleting elements.

### Linked Lists

Linked lists are a collection of elements, each containing a data value and a reference to the next element in the list. They are used to store heterogeneous data types and can be easily modified by adding or removing elements. However, they are less efficient for accessing individual elements.

### Stacks

Stacks are a collection of elements that follow a last-in, first-out (LIFO) order. They are used to store elements that need to be accessed in reverse order, such as the history of a web browser. Stacks are efficient for adding and removing elements, but less efficient for accessing individual elements.

### Queues

Queues are a collection of elements that follow a first-in, first-out (FIFO) order. They are used to store elements that need to be processed in the order they were received, such as messages in a chat application. Queues are efficient for adding and removing elements, but less efficient for accessing individual elements.

### Trees

Trees are a collection of elements, each containing a data value and references to its child elements. They are used to store hierarchical data, such as file systems or organization charts. Trees are efficient for searching and adding elements, but less efficient for deleting elements.

The choice of data structure depends on the problem at hand. By understanding the strengths and weaknesses of each data structure, you can choose the most appropriate one for a given problem.

It is also important to note that data structures can be combined to create more complex data structures. For example, a binary search tree is a combination of a tree and a sorting algorithm. By combining data structures, you can create solutions to even more complex problems.

## Algorithmic Paradigms

In addition to the algorithm design techniques discussed earlier, there are several algorithmic paradigms commonly used in computer programming. These paradigms provide a framework for solving problems and can be combined with the algorithm design techniques to create even more efficient algorithms. Some of the most common algorithmic paradigms are:

### Brute Force

Brute force is a straightforward approach to solving a problem by trying every possible solution. This approach is often used when the problem size is small or when the solution space is not well understood. While brute force can be effective for small problems, it quickly becomes impractical as the problem size increases.

### Backtracking

Backtracking is a technique used to find all possible solutions to a problem by exploring all possible paths. It involves choosing a path, exploring it to see if it leads to a solution, and backtracking if it does not. This approach is often used in combinatorial problems, such as finding all possible combinations of a set of items.

### Divide and Conquer

Divide and conquer is a technique used to break down a problem into smaller subproblems that can be solved independently. This approach is often used in sorting and searching algorithms, where the problem can be divided into smaller subproblems that can be solved more efficiently.

### Greedy Algorithms

Greedy algorithms make locally optimal choices at each step of the algorithm in the hope of finding a global optimum. This approach is often used in optimization problems, such as finding the shortest path between two points.

### Dynamic Programming

Dynamic programming is a technique used to solve problems that can be broken down into smaller subproblems. It involves solving each subproblem only once and storing the results to avoid redundant computations. This technique is often used in optimization problems, such as the knapsack problem.

### Branch and Bound

Branch and bound is a technique used to solve optimization problems by exploring all possible solutions and keeping track of the best solution found so far. It involves dividing the problem into smaller subproblems, exploring each subproblem, and pruning the search space based on the current best solution. This approach is often used in problems such as the traveling salesman problem.

Each of these algorithmic paradigms has its own strengths and weaknesses, and the choice of paradigm depends on the problem at hand. By understanding these paradigms and the algorithm design techniques, you can create efficient algorithms to solve a wide range of problems.

It is important to note that some problems may require the combination of multiple algorithmic paradigms and techniques to find an optimal solution. By understanding the strengths and weaknesses of each paradigm and technique, you can create powerful algorithms to solve even the most complex problems.