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Efficient Algorithm Design

Efficient Algorithm Design

By : Masoud Makrehchi
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Efficient Algorithm Design

Efficient Algorithm Design

By: Masoud Makrehchi

Overview of this book

Efficient Algorithm Design redefines algorithms, tracing the evolution of computer science as a discipline bridging natural science and mathematics. Author Masoud Makrehchi, PhD, with his extensive experience in delivering publications and presentations, explores the duality of computers as mortal hardware and immortal algorithms. The book guides you through essential aspects of algorithm design and analysis, including proving correctness and the importance of repetition and loops. This groundwork sets the stage for exploring algorithm complexity, with practical exercises in design and analysis using sorting and search as examples. Each chapter delves into critical topics such as recursion and dynamic programming, reinforced with practical examples and exercises that link theory with real-world applications. What sets this book apart is its focus on the practical application of algorithm design and analysis, equipping you to solve real programming challenges effectively. By the end of this book, you’ll have a deep understanding of algorithmic foundations and gain proficiency in designing efficient algorithms, empowering you to develop more robust and optimized software solutions.
Table of Contents (21 chapters)
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Part 1: Foundations of Algorithm Analysis
7
Part 2: Deep Dive in Algorithms
13
Part 3: Fundamental Data Structures
17
Part 4: Next Steps

The master theorem

In the analysis of algorithms, the master theorem plays a crucial role in solving recurrences for divide-and-conquer algorithms. Introduced in 1980, it has become a mainstream approach for estimating the complexity of a wide range of recurrence functions. The master theorem provides a straightforward framework for determining the asymptotic behavior of recurrences of the following form:

<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math" display="block"><mml:mi>T</mml:mi><mml:mfenced separators="|"><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:mfenced><mml:mo>=</mml:mo><mml:mi>a</mml:mi><mml:mi>T</mml:mi><mml:mfenced separators="|"><mml:mrow><mml:mfrac><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>b</mml:mi></mml:mrow></mml:mfrac></mml:mrow></mml:mfenced><mml:mo>+</mml:mo><mml:mi>f</mml:mi><mml:mfenced separators="|"><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:mfenced></mml:math>

Here, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mml:mi>a</mml:mi><mml:mo>≥</mml:mo><mml:mn>1</mml:mn></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mml:mi>b</mml:mi><mml:mo>></mml:mo><mml:mn>1</mml:mn></mml:math> are constants, and f(n), the driving function, is an asymptotically positive function bounded by polynomial functions. This means there exist two polynomial functions <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mml:mi>g</mml:mi><mml:mo>(</mml:mo><mml:mi>n</mml:mi><mml:mo>)</mml:mo></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mml:mi>h</mml:mi><mml:mo>(</mml:mo><mml:mi>n</mml:mi><mml:mo>)</mml:mo></mml:math> such that the following is the case:

<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math" display="block"><mml:mi>g</mml:mi><mml:mfenced separators="|"><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:mfenced><mml:mo>≤</mml:mo><mml:mi>f</mml:mi><mml:mfenced separators="|"><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:mfenced><mml:mo>≤</mml:mo><mml:mi>h</mml:mi><mml:mfenced separators="|"><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:mfenced></mml:math>

The importance of the master theorem lies in its ability to simplify the complexity analysis of many common algorithms, such as merge sort, quicksort, and binary search, among others. By categorizing the behavior of the recurrence based on the relationship between <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mml:mi>f</mml:mi><mml:mo>(</mml:mo><mml:mi>n</mml:mi><mml:mo>)</mml:mo></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math"><mml:msup><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant="normal">log</mml:mi></mml:mrow><mml:mrow><mml:mi>b</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>⁡</mml:mo><mml:mrow><mml:mi>a</mml:mi></mml:mrow></mml:mrow></mml:mrow></mml:msup></mml:math>, the master theorem allows for quick and accurate complexity estimation without...

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