How much do you know about algorithms?

Summary Title: Understanding Algorithms and Heuristics

• An algorithm is a finite sequence of rigorous instructions used to solve a class of specific problems or to perform a computation.

• Algorithms are used as specifications for performing calculations and data processing.

• Advanced algorithms can use conditionals to divert the code execution through various routes (referred to as automated decision-making) and deduce valid inferences (referred to as automated reasoning), achieving automation eventually.

• A heuristic is an approach to problem-solving that may not be fully specified or may not guarantee correct or optimal results, especially in problem domains where there is no well-defined correct or optimal result.

• Step-by-step procedures for solving mathematical problems have been in existence since antiquity.

• The term algorithm was coined by Muhammad ibn Musa al-Khwarizmi in the 9th century.

• An algorithm can be considered to be any sequence of operations that can be simulated by a Turing-complete system.

• Algorithm design refers to a method or a mathematical process for problem-solving and engineering algorithms.

• The design of algorithms is part of many solution theories such as divide-and-conquer or dynamic programming within operation research.

• An algorithm's resource efficiency is important; the big O notation is used to describe its run-time growth as the size of its input increases.

• An algorithm can be expressed in many kinds of notation, including natural languages, pseudocode, flowcharts, drakon-charts, programming languages or control tables.

• Computers (and computors), models of computation, are a restricted type of machine, a "discrete deterministic mechanical device" that blindly follows its instructions.Summary Title: Understanding Algorithms and Algorithmic Analysis

• To effectively use an algorithm, the programmer must translate it into a language that the simulator/computer can execute, which requires knowledge of a "language" that is effective relative to the target computing agent.

• The choice of a model for simulation of algorithms is arbitrary, and the notion of simulation enters when measuring speed, where the instruction set matters.

• Structured programming offers a way to write effective algorithms using only four instruction types—conditional GOTO, unconditional GOTO, assignment, HALT—while avoiding "spaghetti code".

• Canonical flowchart symbols, including the directed arrow, rectangle, diamond, and dot, can be used to describe and document an algorithm.

• Euclid's algorithm is an efficient method for computing the greatest common divisor of two integers, and it can be used to reduce fractions to their simplest form.

• The compactness and speed of algorithms can be improved through trial and error, cleverness, insight, application of inductive reasoning, and detailed analysis.

• The analysis of algorithms involves determining how much of a particular resource (such as time or storage) is theoretically required for a given algorithm, and it is practiced abstractly without the use of a specific programming language or implementation.

• Empirical testing is useful to uncover unexpected interactions that affect performance, but it cannot replace formal analysis and is not trivial to perform in a fair manner.

• Scaling from small n to large n frequently exposes inefficient algorithms that are otherwise benign.

• Speed improvements in algorithms depend on special properties of the problem, which are common in practical applications.

• Recent significant innovations in FFT algorithms can decrease processing time up to 1,000 times for applications like medical imaging.

• Algorithmic analysis can be used to prove the validity of any algorithm, and a measure of the complexity of a program can be the length of its correctness proof.Classification of Algorithms: From Implementation to Complexity

• Algorithms can be classified by their implementation means and design methodology or paradigm.

• Some common paradigms include optimization problems, search algorithms, sorting algorithms, and machine learning.

• Fields of study that require efficient algorithms include graph algorithms, numerical algorithms, cryptography, and data compression algorithms.

• Algorithms can also be classified by the amount of time they need to complete compared to their input size, such as constant time, logarithmic time, and polynomial time.

• There are also continuous algorithms that deal with continuous variables and functions.

• Algorithms, by themselves, are not usually patentable, but practical applications of algorithms are sometimes patentable.

• The earliest evidence of algorithms is found in the Babylonian mathematics of ancient Mesopotamia and ancient Egyptian mathematics.

• The terms "algorithm" and "algorism" are derived from the name al-Khwārizmī, a Persian mathematician who wrote the Al-jabr in the 9th century.

• The first cryptographic algorithm for deciphering encrypted code was developed by Al-Kindi, a 9th-century Arab mathematician.

• The clock, Jacquard loom, and electromechanical relay were among the mechanical contrivances that led to the development of the first computers.

• Mathematicians such as George Boole, Gottlob Frege, and Giuseppe Peano reduced arithmetic to a sequence of symbols manipulated by rules in the 19th and early 20th centuries.

• In an effort to solve the Entscheidungsproblem defined precisely by Hilbert in 1928, mathematicians first set about to define what was meant by an "effective method" or "effective calculation" or "effective calculability." Emil Post and Alan Turing were among those who made significant contributions to this effort in the 1930s.The Church-Turing thesis and algorithm refinement

• Rosser's footnote No. 5 references the work of Church, Kleene, Herbrand, Gödel, Post, and Turing in their development of the Church-Turing thesis and mechanism-models of computation.

• Stephen C. Kleene defined "Thesis I" as the Church-Turing thesis, which has been subject to further refinement since 1950.

• Efforts to refine the definition of "algorithm" have been ongoing due to issues surrounding foundations of mathematics and philosophy of mind.

• The Church-Turing thesis is particularly relevant to the foundations of mathematics.

• The thesis is also relevant to arguments about artificial intelligence and the philosophy of mind.

• Algorithm characterizations are an ongoing area of research.

• The work of Church and Kleene focused on λ-definability and its use in solving unsolvable problems in elementary number theory.

• Herbrand and Gödel used recursion, with Gödel famously using it in his paper on undecidable propositions in Principia Mathematica.

• Post and Turing developed mechanism-models of computation.

• The Church-Turing thesis is sometimes referred to as the Church-Turing hypothesis.

• The thesis has been used to demonstrate the limits of what can be computed algorithmically.

• The thesis has also been used to argue against the possibility of certain types of artificial intelligence.