Algorithms

1. Algorithms: The Unseen Architects of Our Digital World

Without algorithms, computers would be useless, and none of modern technology would exist.

Algorithms Everywhere. Algorithms are the backbone of all computer programs, providing the step-by-step instructions that enable computers to solve problems. They are not just abstract concepts but practical tools that drive everything from web searches and social media feeds to medical diagnoses and financial trading.

• Algorithms are cognitive tools, like numbers and arithmetic, that enhance our ability to understand and interact with the world.
• The word "algorithm" comes from the name of Muḥammad ibn Mūsā al-Khwārizmī (ca. 780–ca. 850), a Persian scholar who worked on mathematics, astronomy, and geography.

Beyond Computers. Algorithms existed long before computers and are not inherently tied to them. They are simply a set of well-defined instructions for solving a problem, which can be executed by humans or machines.

• Euclid's algorithm for finding the greatest common divisor (GCD) is an example of an algorithm that predates computers by centuries.
• The process of long division is also an algorithm that we learn in school.

Algorithmic Thinking. Understanding algorithms fosters a new way of reasoning, focusing on practical and efficient problem-solving. This "algorithmic thinking" is a valuable mental tool, even for those who are not professional programmers.

• Algorithmic thinking involves breaking down complex problems into smaller, manageable steps.
• It emphasizes the importance of designing processes that are both practical and efficient.

2. Graphs: Modeling Relationships and Solving Complex Problems

The definition of a graph is wide so that it can encompass everything that can be represented as things connected to other things.

Graphs as Models. Graphs are mathematical structures that consist of nodes (vertices) and edges (links) connecting them. They provide a powerful way to model relationships between objects and solve a wide range of problems.

• Social networks, road networks, and the World Wide Web can all be represented as graphs.
• The Königsberg bridge problem, solved by Euler, is considered the foundation of graph theory.

Applications of Graphs. Graphs are used in various fields, including computer science, biology, and social sciences.

• DNA sequencing: Finding the composition of an unknown DNA piece.
• Tournament scheduling: Scheduling the matches so that each contestant plays only one match per day.
• Shortest path finding: Determining the most efficient route between two points on a map.

Algorithms on Graphs. Many algorithms are designed to solve problems related to graphs, such as finding the shortest path, identifying communities, and scheduling tasks.

• Dijkstra's algorithm finds the shortest path between two nodes in a graph.
• Hierholzer's algorithm finds Eulerian circuits on graphs.

3. Searching: Finding Needles in Ever-Growing Haystacks

A good search algorithm can result in dramatic improvements in speed.

Ubiquity of Searching. Searching is a fundamental operation in computer science, used in almost every application to locate specific items within a collection of data. Efficient search algorithms are crucial for performance, especially as data volumes continue to grow.

• Searching is used in databases, web search engines, and even simple programs like text editors.
• The efficiency of a search algorithm can significantly impact the overall performance of an application.

Types of Search Algorithms. Different search algorithms are suited for different types of data and search requirements.

• Linear search: Examining each item in a list sequentially until the target item is found.
• Self-organizing search: Rearranging the list as we go with our searches and will reflect the popularity of the searched items.
• Binary search: Efficiently searching sorted data by repeatedly dividing the search interval in half.

Binary Search Efficiency. Binary search is incredibly efficient, with a time complexity of O(log n). This means that the number of steps required to find an item increases logarithmically with the size of the data set.

• It will not take more than 20 probes to search among a million items.
• With a hundred probes we are able to search and find any item among , which is more than one nonillion.

4. Sorting: Ordering Chaos for Efficiency and Insight

Although all of them have the same objective, particular screws require particular drivers. . . . The same with sorting. While all sorting algorithms put things in order, each is more suitable for particular uses.

Importance of Sorting. Sorting is a fundamental operation in computer science, used to arrange data in a specific order for efficient retrieval and analysis. It is a ubiquitous task, from organizing files on a computer to ranking search results.

• Sorting algorithms are used in databases, operating systems, and various applications.
• The efficiency of a sorting algorithm can significantly impact the performance of an application.

Types of Sorting Algorithms. Many different sorting algorithms exist, each with its own strengths and weaknesses.

• Selection sort: Repeatedly finding the minimum element and placing it in its correct position.
• Insertion sort: Inserting each element into its correct position within an already sorted portion of the list.
• Radix sort: Sorting items by processing individual digits or characters, without direct comparisons.
• Quicksort: A divide-and-conquer algorithm that partitions the data around a pivot element.
• Merge sort: A divide-and-conquer algorithm that repeatedly merges sorted sublists.

Algorithm Selection. The choice of sorting algorithm depends on the specific characteristics of the data and the performance requirements of the application.

• Quicksort is generally the fastest sorting algorithm in practice, but its performance can vary depending on the pivot selection.
• Merge sort has a guaranteed O(n log n) time complexity and is well-suited for parallel processing.
• Radix sort can be very efficient for sorting data with a limited range of values.

5. PageRank: Determining Importance in a Hyperlinked World

The more links that a web page accrues, the more authors judged it important enough to link to it from their own page, and thus the more important it becomes overall.

PageRank's Insight. PageRank is an algorithm that assigns a numerical value to each web page, representing its importance based on the link structure of the web. It was a key factor in the success of the Google search engine.

• PageRank is based on the idea that a web page is important if other important web pages link to it.
• The algorithm takes into account both the number and quality of backlinks to a page.

Random Surfer Model. PageRank can be understood through the metaphor of a random surfer who clicks on links to navigate the web. The pagerank of a page is the probability that the random surfer will visit that page.

• The random surfer may also jump to a random page, even if there are no links to follow.
• This helps to prevent the algorithm from getting stuck in loops or dead ends.

Mathematical Formulation. PageRank can be expressed mathematically using linear algebra. The pageranks of all web pages can be represented as a vector, and the algorithm involves repeatedly multiplying this vector by a matrix representing the link structure of the web.

• The power method is used to calculate the pagerank vector.
• The Google matrix is a modified version of the hyperlink matrix that ensures the algorithm converges to a stable solution.

6. Deep Learning: Mimicking the Brain for Intelligent Systems

The goal is to exhibit some useful behavior, which we often associate with intelligence.

Neural Networks. Deep learning is a subfield of machine learning that uses artificial neural networks to learn complex patterns from data. These networks are inspired by the structure and function of the human brain.

• Artificial neurons are the basic building blocks of neural networks.
• Neural networks consist of layers of interconnected neurons.

Learning Process. Neural networks learn by adjusting the weights and biases of their connections. This process is called training and involves feeding the network with a large amount of data and using an algorithm called backpropagation to minimize the difference between the network's output and the desired output.

• Supervised learning: The algorithm is provided with both the input and the desired output.
• Backpropagation: The algorithm for training neural networks.

Deep Architectures. Deep learning networks have many layers, allowing them to learn hierarchical representations of data. This enables them to extract complex features and patterns that would be difficult to identify with traditional machine learning techniques.

• The first layer of a multilayer network may learn to recognize small local patterns, such as edges in the image.
• The second layer may learn to recognize patterns that are built from the patterns recognized by the first layer, such as eyes, noses, and ears.

7. The Limits of Computation: Understanding What Algorithms Cannot Do

Our computational limits are given by Turing machines. Anything a computer can do, we could really do with a pen and paper. . . . Everything you see executed on any digital device is . . . a series of such elementary symbol manipulations.

Turing Machines. The Turing machine is a theoretical model of computation that provides a fundamental understanding of what can be computed. It consists of a tape, a head, a finite control, and a finite instructions table.

• The Church-Turing thesis states that any algorithm that can be computed by a computer can also be computed by a Turing machine.
• Quantum computers, while offering potential speed advantages, cannot solve problems that are fundamentally unsolvable by Turing machines.

The Essence of Algorithms. Algorithms, at their core, are a series of elementary symbol manipulations that can be performed by a Turing machine. This means that anything a computer can do, we could theoretically do with a pen and paper, given enough time and patience.

• The steps are so elementary that the head of the Turing machine scampers around a lot in order to perform the operation.
• You do not need any advanced qualifications to perform the steps of a Turing machine; you only need to look up a table, move around on a tape, read and write one symbol at a time, and keep track of what your state is.

Creativity and Algorithms. Despite the limitations of computation, human creativity continues to drive the development of new and innovative algorithms. The ability to model problems effectively and design efficient algorithms remains a key area of focus in computer science.

• The success of an algorithm does not hinge only on its elegance and efficiency.
• It also has to do with the mapping of the algorithm to a problem.

Last updated: April 15, 2025