René Descartes (1596-1650), the French philosopher and mathematician, is renowned for his theory of mind-body dualism. He proposed that the mind and body are two fundamentally distinct substances: the mind as a non-extended, thinking entity and the body as an extended, non-thinking entity. Descartes believed in the interaction between these substances while maintaining their separateness and independent existence. This dualistic view emphasizes the separation of the mental (mind) and physical (body) aspects, positing that they differ in nature. His theory has had a significant influence on philosophical discussions about consciousness and the relationship between mind and body.

But why are we starting our discussion about algorithms with Descartes’ theory of dualism, despite serious criticisms of it from philosophers? The answer lies in the model’s ability to aid our understanding of the essence of algorithms and their...

In the last two decades, we have witnessed extraordinary advancements in advanced computational systems, particularly in fields such as Artificial Intelligence (AI), Machine Learning, Deep Learning, Robotics, and Computer Vision. This progress owes more to two significant revolutions in the tech world and information society than to improvements in algorithm design. The first revolution was the availability of vast amounts of data following the public release of the Internet in 1991. The second was the tremendous advancements in hardware, including the development of powerful yet affordable processors such as Graphical Processing Units (GPUs) and the creation of ultra-high-capacity memory and storage solutions.

Despite these remarkable improvements in computational resources, the study and analysis of algorithms remain critically important for several reasons:

• Algorithm correctness: Irrespective of the abundance of computational resources...

The objective of algorithm analysis is indeed twofold. Firstly, it seeks to ascertain the correctness of an algorithm. An algorithm is considered correct if it consistently solves the intended problem and produces the right output for all valid inputs. This correctness hinges on two critical criteria:

• Termination: An algorithm must always conclude after a finite number of steps. It should not fall into perpetual loops or continue indefinitely, regardless of the input provided.
• Validity: The algorithm must yield the expected outcome or a valid solution for every possible input. It needs to adhere to the problem’s specifications and requirements precisely.

Interestingly, demonstrating an algorithm’s correctness cannot be conclusively achieved merely by testing it with numerous positive examples. While many successful test cases may suggest correctness, it takes only a single...

This chapter introduced the fundamental nature of algorithms, emphasizing their role as structured and systematic tools for effective problem-solving across various fields. It began with a comprehensive definition and explored the distinct aspects of algorithms, including their dependence on the dual nature of hardware and software, influenced by physical laws and mathematical principles, respectively. The chapter focused on the critical importance of algorithm analysis, which revealed the necessity of evaluating algorithms, particularly as hardware becomes more accessible and cost-effective. It sets the stage for you to explore the complexities and applications of algorithms through guided and structured learning paths. This introductory chapter served as a foundational step toward mastering the art of algorithm analysis and application. The next chapter will establish the essential mathematical foundation that every software practitioner needs for effective algorithm analysis...