Small data is data that is 'small' enough for human comprehension. It is data in a volume and format that makes it accessible, informative and actionable. The term "big data" is about machines and "small data" is about people. This is to say that eyewitness observations or five pieces of related data could be small data. Small data is what we used to think of as data. The only way to comprehend Big data is to reduce the data into small, visually-appealing objects representing various aspects of large data sets (such as histogram, charts, and scatter plots). Big Data is all about finding correlations, but Small Data is all about finding the causation, the reason why. A formal definition of small data has been proposed by Allen Bonde, former vice-president of Innovation at Actuate - now part of OpenText: "Small data connects people with timely, meaningful insights (derived from big data and/or “local” sources), organized and packaged – often visually – to be accessible, understandable, and actionable for everyday tasks." Another definition of small data is: The small set of specific attributes produced by the Internet of Things. These are typically a small set of sensor data such as temperature, wind speed, vibration and status. It was estimated (2016) that “If one takes the top 100 biggest innovations of our time, perhaps around 60% to 65% percent are really based on Small Data.” as Martin Lindstrom puts it. Small data includes everything from Snapchat to simple objects such as the post-it note. Lindstrom believes we become so focused on Big-Data that we tend to forget about more basic concepts and creativity. Lindstrom defines Small Data "as seemingly insignificant observations you identify in consumers’ homes, is everything from how you place your shoes on how you hang your paintings". He thus considers that one should perfectly master the basic (Small Data) in order to mine and find correlations. == Academic Recognition and Methodology == The growing significance of "small data" as a distinct field of inquiry was highlighted by the 2024 Thematic Einstein Semester (TES) on Small Data Analysis, hosted by the Berlin Mathematics Research Center MATH+. A central focus of this semester was the transition from theoretical analysis to practical decision-making. Because small data sets are primarily used to drive specific actions, the presentation of results becomes an essential methodological step. The semester’s findings emphasized that while small data may lack volume, it often contains a high density of "many possible interpretations." Consequently, the final conference of the TES was structured around the pillars of interpretation, explanation, and knowledge gain. Participants sought to develop new mathematical and methodical representations that could accurately depict this wealth of interpretative possibilities. This work underscores that analyzing small data is not purely a computational task; it requires a robust interface between mathematics and diverse disciplines to ensure that insights are both contextually grounded and scientifically rigorous. == Uses in business == === Marketing === Bonde has written about the topic for Forbes, Direct Marketing News, CMO.com and other publications. According to Martin Lindstrom, in his book, Small Data: "{In customer research, small data is} Seemingly insignificant behavioural observations containing very specific attributes pointing towards an unmet customer need. Small data is the foundation for breakthrough ideas or completely new ways to turnaround brands." His approach is based on the combination of the observation of small samples with intuition. Marketers can obtain market insights from gathering Small Data by engaging with and observing people in their own environments. In comparison to Big Data, Small Data has the power to trigger emotions and to provide insights into the reasons behind the behaviours of customers. It may uncover detailed information on a person's extroversion or introversion, self-confidence, whether one is having problems in his/her relationship, etc. According to Lindstrom, relationships among people and customer segments are organized around four criteria: Climate: It reveals for example how a person's environment affects their diet. Rulership: The power or government in charge Religion: The prevalence of religion in a country, depending on its influence, indicates whether a person's decision making process is impacted by their belief system. Tradition: Cultural norms influence people's behaviors and interactions. Many companies underestimate the power of Small Data, using samples of millions of consumers instead of recognizing the value of closely observing small samples in their market research. In his book, Lindstrom defines "7Cs", which companies should consider in the attempt to derive meaningful customer insights and market trends through small data from their customers: Collecting: Understanding the manner in which observations are translated inside a home. Clues: Uncovering other distinctive emotional reflections that can be observed. Connecting: Identifying the consequences of emotional behaviour. Causation: Understanding what emotions are being evoked. Correlation: Identifying the initial date of appearance of the behaviour or emotion. Compensation: Identifying the unmet or unfulfilled desire. Concept: Defining the “big idea” compensation for the identified consumer need. Some of Lindstrom's clients such as Lowes Foods looked at data in a different way and actually chose to live with the customer. “As you enter their store, they have now created an amazing community where every staff member acts in a character mood, based on Small Data”. The supermarket made everything it can to make the customer feel at home. All the behaviours of employees are inspired by customer feedbacks gathered from interviews directly done at customer’s home. === Healthcare === Researchers at Cornell University started developing applications to monitor health problems in patients, based on small data. This is an initiative of Cornell's Small Data Lab, in close cooperation with Weill Cornell Medicine College, led by Deborah Estrin. The Small Data Lab developed a series of apps, focusing not only on gathering data from patients' pain but also tracking habits in areas such as grocery shopping. In the case of patients with rheumatoid arthritis for example, which has flares and remissions that do not follow a particular cycle, the app gathers information passively, thus allowing to forecast when a flare might be coming up based on small changes in behaviour. Other apps developed also include monitoring online grocery shopping, to use this information from every user to adapt their groceries to the recommendations of nutritionists, or monitoring email language to identify patterns that might indicate "fluctuations in cognitive performance, fatigue, side effects of medication or poor sleep, and other conditions and treatments that are typically self-reported and self-medicated". === Postal Service === The United States Postal Service (USPS) used optical character recognition (OCR) to automatically read and process 98% of all hand-addressed mail and 99.5% of machine-printed mail. By combining this technology with its small data sample of US zip codes, the USPS can now process more than 36,000 pieces of mail per hour. === Aerospace === In 2015, Boeing established the analytics lab for aerospace data in cooperation with the Carnegie Mellon University to leverage the university's leadership in machine learning, language technologies and data analytics. One of the initiatives projects aims to by standardize maintenance logs using AI to dramatically reduce costs. Currently, there is no standardized procedure to document maintenance logs leading to small but highly unstructured data sets. As a result, it becomes highly difficult for maintenance workers to translate these variations in maintenance logs within a short period of time. However, with AI and a narrow data set of common aircraft maintenance terminology, it becomes possible to dynamically translate these logs in real time. By using AI to enhance the speed and accuracy of the airline maintenance workflow, airlines stand to save billions according to the Harvard Business Review.
FedRAMP
The Federal Risk and Authorization Management Program (FedRAMP) is a United States federal government-wide compliance program that provides a standardized approach to security assessment, authorization, and continuous monitoring for cloud products and services. The US government describes FedRAMP as FISMA for the cloud. == Overview == The FedRAMP PMO mission is to promote the adoption of secure cloud services across the federal government by providing a standardized approach to security and risk assessment. Per the OMB memorandum, any cloud services that hold federal data must be FedRAMP authorized. FedRAMP prescribes the security requirements and processes that cloud service providers must follow in order for the government to use their service. There are two ways to authorize a cloud service through FedRAMP: a Joint Authorization Board (JAB) provisional authorization (P-ATO), and through individual agencies. FedRAMP provides accreditation for cloud services for the various cloud offering models which are Infrastructure as a Service (IaaS), Platform as a Service (PaaS), and Software as a Service, (SaaS). == History == In 2011, the Office of Management and Budget (OMB) released a memorandum establishing FedRAMP "to provide a cost-effective, risk-based approach for the adoption and use of cloud services to Executive departments and agencies." The General Services Administration (GSA) established the FedRAMP Program Management Office (PMO) in June 2012. Before the introduction of FedRAMP, individual federal agencies managed their own assessment methodologies following guidance set by the Federal Information Security Management Act of 2002. == Governance and applicable laws == FedRAMP is governed by different Executive Branch entities that collaborate to develop, manage, and operate the program. These entities include: The Office of Management and Budget (OMB): The governing body that issued the FedRAMP policy memo, which defines the key requirements and capabilities of the program The Joint Authorization Board (JAB): The primary governance and decision-making body for FedRAMP comprises the chief information officers (CIOs) from the Department of Homeland Security (DHS), General Services Administration (GSA), and Department of Defense (DOD) The National Institute of Standards and Technology (NIST): Advises FedRAMP on FISMA compliance requirements and assists in developing the standards for the accreditation of independent 3PAOs The Department of Homeland Security (DHS): Manages the FedRAMP continuous monitoring strategy including data feed criteria, reporting structure, threat notification coordination, and incident response The Federal Chief Information Officers (CIO) Council: Disseminates FedRAMP information to Federal CIOs and other representatives through cross-agency communications and events The FedRAMP PMO: Established within GSA and responsible for the development of the FedRAMP program, including the management of day-to-day operations There are several laws, mandates, and policies that are foundational to FedRAMP. FISMA–the Federal Information Security Modernization Act–requires that agencies authorize the information systems that they use. The US government describes FedRAMP as FISMA for the cloud. The FedRAMP Policy Memo requires federal agencies to use FedRAMP when assessing, authorizing, and continuously monitoring cloud services in order to aid agencies in the authorization process as well as save government resources and eliminate duplicative efforts. FedRAMP's security baselines are derived from NIST SP 800-53 (as revised) with a set of control enhancements that pertain to the unique security requirements of cloud computing. == Third-party assessment organizations == Third-party assessment organizations (3PAOs) play a critical role in the FedRAMP security assessment process, as they are the independent assessment organizations that verify cloud providers' security implementations and provide the overall risk posture of a cloud environment for a security authorization decision. Accredited by the American Association for Laboratory Accreditation (A2LA), these assessment organizations must demonstrate independence and the technical competence required to test security implementations and collect representative evidence. == FedRAMP Marketplace == The FedRAMP Marketplace provides a searchable, sortable database of Cloud Service Offerings (CSOs) that have achieved a FedRAMP designation. 3PAOs, accredited auditors that can perform the FedRAMP assessment, are listed within the Marketplace. The FedRAMP Marketplace is maintained by the FedRAMP Program Management Office (PMO). == Security and authorization concerns == A 2026 ProPublica investigation found that FedRAMP entered into a partnership with Microsoft despite considerable concerns about the security of its cloud technology.
Influencer
An influencer is an individual who has the capacity to shape the attitudes, behavior, or decisions of others through authority, knowledge, position, or the nature of the relationship with the audience. The term is used in various fields such as media, business, politics, religion, and communication, referring to influencers such as social media influencers, podcasters, public speakers, religious influencers, writers, and newsletter writers etc who have dedicated followings in various areas. One writer defines influencers as "a range of third parties who exercise influence over the organization and its potential customers." Another writer defines an influencer as a "third party who significantly shapes the customer's purchasing decision but may never be accountable for it." According to another writer, influencers are "well-connected, create an impact, have active minds, and are trendsetters". Just because a person has many followers does not necessarily mean they have much influence over those people. In contemporary usage, the term frequently refers to a social media influencer, (also known as an online influencer or simply influencer) a person who builds a grassroots online presence through engaging content such as photos, videos, and updates. This is done by using direct audience interaction to establish authenticity, expertise, and appeal, and by standing apart from traditional celebrities by growing their platform through social media rather than pre-existing fame. The modern referent of the term is commonly a paid role in which a business entity pays for the social media influence-for-hire activity to promote its products and services, known as influencer marketing. A 1% increase in spending on influencer marketing can lead to a 0.5% increase in audience engagement. As such, an influencer effectively acts as a modern salesperson or a marketer. Types of influencers include fashion influencer, travel influencer, and virtual influencer, and they involve content creators and streamers. Some influencers are associated primarily with specific social media apps such as TikTok, Instagram, or Pinterest; many influencers are also considered internet celebrities. As of 2023, Instagram is the social media platform businesses spend the most advertising money towards marketing with influencers. However, influencers can have an impact on any social media network. == History == === Origins === The word influencer in its general sense of a person or thing that exerts influence, is attested in historical sources at least since the 17th century. The Oxford English Dictionary (OED) gives 1664 as the earliest example of usage and cites a sentence from Henry More's A Modest Enquiry into the Mystery of Iniquity: "The head and influencer of the whole Church". The origins of online influencing can be traced back to the emergence of digital blogs and platforms in the early 2000s. Nevertheless, recent studies demonstrate that Instagram, an application with more than one billion users, harbors the majority of the influencer demographic. These individuals are sometimes referred to as "Instagrammers" or "Instafamous". A crucial aspect of influencing is their association with sponsors. The 2015 debut of Vamp, a company that links influencers with sponsorships, transformed the landscape of influencing. There is much debate about whether social media influencers can be considered celebrities, as their path to fame is often less traditional and arguably easier. Melody Nouri addressed the differences between the two types in her article "The Power of Influence: Traditional Celebrities vs Social Media Influencer". Nouri asserts that social media platforms have a greater negative impact on young, impressionable audiences in comparison with traditional media such as magazines, billboards, advertisements, and tabloids featuring celebrities. Online, it is thought to be simpler to manipulate an image and lifestyle in such a way that viewers are more susceptible to believing it. One theory considers the former American First Lady Eleanor Roosevelt (1884–1962) to be the "original media influencer." While she achieved celebrity in her role as First Lady, she built a global personal brand as a wise, informative, trustworthy American woman. Her voice was her own, unrestricted by political advisors and powerful men, and with it, Roosevelt exerted unprecedented social and cultural influence in radio, print, public speaking, film, and television until she died. In one notable example, it may have been Roosevelt's television support of John F. Kennedy which nudged his "hairline victory" during the 1960 Presidential campaign. In another example, David Ogilvy paid Roosevelt more than a quarter of a million dollars in today's currency to make a TV commercial for Good Luck margarine (1959), in which Roosevelt also managed to mention world hunger. As a content creator, she wrote My Day, a popular daily newspaper column that ran nationwide for twenty-six years. Like a social media post, My Day covered all aspects of her life, and in it Roosevelt often recommended movies, books, and products that she admired. Roosevelt also had a hand in designing all three of her public affairs television shows. Unlike contemporary influencers, she was less motivated by a pay-to-play situation than by a desire to educate and inspire; but she did use her influence to benefit the entertainment industry careers of her children, and she welcomed the revenue that her influence bought, most of which was donated to charity. === 2000s === The early 2000s showed corporate endeavors to leverage the internet for influence, with some companies participating in forums for promotions or providing bloggers with complimentary products in return for favorable reviews. A few of these practices were viewed as unethical for taking advantage of the labor of young individuals without providing remuneration. In 2004, The Blogstar Network was established by Ted Murphy of MindComet. Bloggers were encouraged to join an email list and receive remunerated offers from corporations in exchange for creating specific posts. For instance, bloggers were compensated for writing reviews of fast-food meals on their blogs. Blogstar is widely regarded as the first influencer marketing network. Murphy succeeded Blogstar with PayPerPost, which was introduced in 2006. This platform compensated significant posters on prominent forums and social media platforms for every post made about a corporate product. Payment rates were determined by the influencer's status. Though very popular, PayPerPost, received a great deal of criticism as these influencers were not required to disclose their involvement with PayPerPost as traditional journalism would have. With the success of PayPerPost, the public became aware that there was a drive for corporate interests to influence what some people were posting to these sites. The platform also incentivized other firms to establish comparable programs. Despite concerns, marketing networks with influencers continued to grow throughout the 2000s and into the 2010s. The influencer marketing industry was worth as much as $8 billion in 2019, according to estimates from Business Insider Intelligence, which are based on Mediakix data. Evan Asano, the Former CEO and founder of the agency Mediakix, previously spoke with Business Insider and said he believed influencer marketing on Instagram would continue to grow despite likes being hidden. === 2010s === By the 2010s, the term "influencer" described digital content creators with a large following, distinctive brand persona, and a patterned relationship with commercial sponsors. By this period, influencer marketing had become a widely researched field globally, with systematic reviews drawing on hundreds of studies that documented the growing role of authenticity, audience engagement, and parasocial relationships in shaping how consumers responded to influencer content across different markets. During this period, influencer culture also developed through distinct channels outside Western markets. In South Korea, the global spread of Korean pop culture, also called K-Pop, through platforms such as YouTube, Facebook, and Twitter gave rise to what scholars have called 'Hallyu 2.0' or the 'New Korean Wave', where fans throughout Southeast Asia, North America, Latin America, and Europe shared, subtitled, and redistributed Korean music and film content on a large scale. This helped Korean entertainers to build substantial followings internationally. Consumers often mistakenly view celebrities as reliable, leading to trust and confidence in the products being promoted. A 2001 study from Rutgers University discovered that individuals were using "internet forums as influential sources of consumer information." The study proposes that consumers preferred internet forums and social media when making purchasing decisions over conventional advertising and print sources. An in
Cryptosystem
In cryptography, a cryptosystem is a suite of cryptographic algorithms needed to implement a particular security service, such as confidentiality (encryption). Typically, a cryptosystem consists of three algorithms: one for key generation, one for encryption, and one for decryption. The term cipher (sometimes cypher) is often used to refer to a pair of algorithms, one for encryption and one for decryption. Therefore, the term cryptosystem is most often used when the key generation algorithm is important. For this reason, the term cryptosystem is commonly used to refer to public key techniques; however both "cipher" and "cryptosystem" are used for symmetric key techniques. == Formal definition == Mathematically, a cryptosystem or encryption scheme can be defined as a tuple ( P , C , K , E , D ) {\displaystyle ({\mathcal {P}},{\mathcal {C}},{\mathcal {K}},{\mathcal {E}},{\mathcal {D}})} with the following properties. P {\displaystyle {\mathcal {P}}} is a set called the "plaintext space". Its elements are called plaintexts. C {\displaystyle {\mathcal {C}}} is a set called the "ciphertext space". Its elements are called ciphertexts. K {\displaystyle {\mathcal {K}}} is a set called the "key space". Its elements are called keys. E = { E k : k ∈ K } {\displaystyle {\mathcal {E}}=\{E_{k}:k\in {\mathcal {K}}\}} is a set of functions E k : P → C {\displaystyle E_{k}:{\mathcal {P}}\rightarrow {\mathcal {C}}} . Its elements are called "encryption functions". D = { D k : k ∈ K } {\displaystyle {\mathcal {D}}=\{D_{k}:k\in {\mathcal {K}}\}} is a set of functions D k : C → P {\displaystyle D_{k}:{\mathcal {C}}\rightarrow {\mathcal {P}}} . Its elements are called "decryption functions". For each e ∈ K {\displaystyle e\in {\mathcal {K}}} , there is d ∈ K {\displaystyle d\in {\mathcal {K}}} such that D d ( E e ( p ) ) = p {\displaystyle D_{d}(E_{e}(p))=p} for all p ∈ P {\displaystyle p\in {\mathcal {P}}} . Note; typically this definition is modified in order to distinguish an encryption scheme as being either a symmetric-key or public-key type of cryptosystem. == Examples == A classical example of a cryptosystem is the Caesar cipher. A more contemporary example is the RSA cryptosystem. Another example of a cryptosystem is the Advanced Encryption Standard (AES). AES is a widely used symmetric encryption algorithm that has become the standard for securing data in various applications. Paillier cryptosystem is another example used to preserve and maintain privacy and sensitive information. It is featured in electronic voting, electronic lotteries and electronic auctions.
Ultra (cryptography)
Ultra was the designation adopted by British military intelligence in June 1941 for wartime signals intelligence obtained by breaking high-level encrypted enemy radio and teleprinter communications at the Government Code and Cypher School (GC&CS) at Bletchley Park. Ultra eventually became the standard designation among the western Allies for all such intelligence. The name arose because the intelligence obtained was considered more important than that designated by the highest British security classification then used (Most Secret) and so was regarded as being Ultra Secret. Several other cryptonyms had been used for such intelligence. The code name "Boniface" was used as a cover name for Ultra. In order to ensure that the successful code-breaking did not become apparent to the Germans, British intelligence created a fictional MI6 master spy, Boniface, who controlled a fictional series of agents throughout Germany. Information obtained through code-breaking was often attributed to the human intelligence from the Boniface network. The U.S. used the codename Magic for its decrypts from Japanese sources, including the "Purple" cipher. Much of the German cipher traffic was encrypted on the Enigma machine. Used properly, the German military Enigma would have been virtually unbreakable; in practice, shortcomings in operation allowed it to be broken. The term "Ultra" has often been used almost synonymously with "Enigma decrypts". However, Ultra also encompassed decrypts of the German Lorenz SZ 40/42 machines that were used by the German High Command, and the Hagelin machine. Many observers, at the time and later, regarded Ultra as immensely valuable to the Allies. Winston Churchill was reported to have told King George VI, when presenting to him Stewart Menzies (head of the Secret Intelligence Service and the person who controlled distribution of Ultra decrypts to the government): "It is thanks to the secret weapon of General Menzies, put into use on all the fronts, that we won the war!" F. W. Winterbotham quoted the western Supreme Allied Commander, Dwight D. Eisenhower, at war's end describing Ultra as having been "decisive" to Allied victory. Sir Harry Hinsley, Bletchley Park veteran and official historian of British Intelligence in World War II, made a similar assessment of Ultra, saying that while the Allies would have won the war without it, "the war would have been something like two years longer, perhaps three years longer, possibly four years longer than it was." However, Hinsley and others have emphasized the difficulties of counterfactual history in attempting such conclusions, and some historians, such as John Keegan, have said the shortening might have been as little as the three months it took the United States to deploy the atomic bomb. == Sources of intelligence == Most Ultra intelligence was derived from reading radio messages that had been encrypted with cipher machines, complemented by material from radio communications using traffic analysis and direction finding. In the early phases of the war, particularly during the eight-month Phoney War, the Germans could transmit most of their messages using land lines and so had no need to use radio. This meant that those at Bletchley Park had some time to build up experience of collecting and starting to decrypt messages on the various radio networks. German Enigma messages were the main source, with those of the German air force (the Luftwaffe) predominating, as they used radio more and their operators were particularly ill-disciplined. === German === ==== Enigma ==== "Enigma" refers to a family of electro-mechanical rotor cipher machines. These produced a polyalphabetic substitution cipher and were widely thought to be unbreakable in the 1920s, when a variant of the commercial Model D was first used by the Reichswehr. The German Army (Heer), Navy, Air Force, Nazi party, Gestapo and German diplomats used Enigma machines in several variants. Abwehr (German military intelligence) used a four-rotor machine without a plugboard and Naval Enigma used different key management from that of the army or air force, making its traffic far more difficult to cryptanalyse; each variant required different cryptanalytic treatment. The commercial versions were not as secure and Dilly Knox of GC&CS is said to have broken one before the war. German military Enigma was first broken in December 1932 by Marian Rejewski and the Polish Cipher Bureau, using a combination of brilliant mathematics, the services of a spy in the German office responsible for administering encrypted communications, and good luck. The Poles read Enigma to the outbreak of World War II and beyond, in France. At the turn of 1939, the Germans made the systems ten times more complex, which required a tenfold increase in Polish decryption equipment, which they could not meet. On 25 July 1939, the Polish Cipher Bureau handed reconstructed Enigma machines and their techniques for decrypting ciphers to the French and British. Gordon Welchman wrote, Ultra would never have got off the ground if we had not learned from the Poles, in the nick of time, the details both of the German military Enigma machine, and of the operating procedures that were in use. At Bletchley Park, some of the key people responsible for success against Enigma included mathematicians Alan Turing and Hugh Alexander and, at the British Tabulating Machine Company, chief engineer Harold Keen. After the war, interrogation of German cryptographic personnel led to the conclusion that German cryptanalysts understood that cryptanalytic attacks against Enigma were possible but were thought to require impracticable amounts of effort and investment. The Poles' early start at breaking Enigma and the continuity of their success gave the Allies an advantage when World War II began. ==== Lorenz cipher ==== In June 1941, the Germans started to introduce on-line stream cipher teleprinter systems for strategic point-to-point radio links, to which the British gave the code-name Fish. Several systems were used, principally the Lorenz SZ 40/42 (codenamed "Tunny" by the British) and Geheimfernschreiber ("Sturgeon"). These cipher systems were cryptanalysed, particularly Tunny, which the British thoroughly penetrated. It was eventually attacked using Colossus machines, which were the first digital programme-controlled electronic computers. In many respects the Tunny work was more difficult than for the Enigma, since the British codebreakers had no knowledge of the machine producing it and no head-start such as that the Poles had given them against Enigma. Although the volume of intelligence derived from this system was much smaller than that from Enigma, its importance was often far higher because it produced primarily high-level, strategic intelligence that was sent between Wehrmacht high command (Oberkommando der Wehrmacht, OKW). The eventual bulk decryption of Lorenz-enciphered messages contributed significantly, and perhaps decisively, to the defeat of Nazi Germany. Nevertheless, the Tunny story has become much less well known among the public than the Enigma one. At Bletchley Park, some of the key people responsible for success in the Tunny effort included mathematicians W. T. "Bill" Tutte and Max Newman and electrical engineer Tommy Flowers. === Italian === In June 1940, the Italians were using book codes for most of their military messages, except for the Italian Navy, which in early 1941 had started using a version of the Hagelin rotor-based cipher machine C-38. This was broken from June 1941 onwards by the Italian subsection of GC&CS at Bletchley Park. === Japanese === In the Pacific theatre, a Japanese cipher machine, called "Purple" by the Americans, was used for highest-level Japanese diplomatic traffic. It produced a polyalphabetic substitution cipher, but unlike Enigma, was not a rotor machine, being built around electrical stepping switches. It was broken by the US Army Signal Intelligence Service and disseminated as Magic. Detailed reports by the Japanese ambassador to Germany were encrypted on the Purple machine. His reports included reviews of German assessments of the military situation, reviews of strategy and intentions, reports on direct inspections by the ambassador (in one case, of Normandy beach defences), and reports of long interviews with Hitler. The Japanese are said to have obtained an Enigma machine in 1937, although it is debated whether they were given it by the Germans or bought a commercial version, which, apart from the plugboard and internal wiring, was the German Heer/Luftwaffe machine. Having developed a similar machine, the Japanese did not use the Enigma machine for their most secret communications. The chief fleet communications code system used by the Imperial Japanese Navy was called JN-25 by the Americans, and by early 1942 the US Navy had made considerable progress in decrypting Japanese naval messages. The US Army also made progress on the
Algorithm selection
Algorithm selection (sometimes also called per-instance algorithm selection or offline algorithm selection) is a meta-algorithmic technique to choose an algorithm from a portfolio on an instance-by-instance basis. It is motivated by the observation that on many practical problems, different algorithms have different performance characteristics. That is, while one algorithm performs well in some scenarios, it performs poorly in others and vice versa for another algorithm. If we can identify when to use which algorithm, we can optimize for each scenario and improve overall performance. This is what algorithm selection aims to do. The only prerequisite for applying algorithm selection techniques is that there exists (or that there can be constructed) a set of complementary algorithms. == Definition == Given a portfolio P {\displaystyle {\mathcal {P}}} of algorithms A ∈ P {\displaystyle {\mathcal {A}}\in {\mathcal {P}}} , a set of instances i ∈ I {\displaystyle i\in {\mathcal {I}}} and a cost metric m : P × I → R {\displaystyle m:{\mathcal {P}}\times {\mathcal {I}}\to \mathbb {R} } , the algorithm selection problem consists of finding a mapping s : I → P {\displaystyle s:{\mathcal {I}}\to {\mathcal {P}}} from instances I {\displaystyle {\mathcal {I}}} to algorithms P {\displaystyle {\mathcal {P}}} such that the cost ∑ i ∈ I m ( s ( i ) , i ) {\displaystyle \sum _{i\in {\mathcal {I}}}m(s(i),i)} across all instances is optimized. == Examples == === Boolean satisfiability problem (and other hard combinatorial problems) === A well-known application of algorithm selection is the Boolean satisfiability problem. Here, the portfolio of algorithms is a set of (complementary) SAT solvers, the instances are Boolean formulas, the cost metric is for example average runtime or number of unsolved instances. So, the goal is to select a well-performing SAT solver for each individual instance. In the same way, algorithm selection can be applied to many other N P {\displaystyle {\mathcal {NP}}} -hard problems (such as mixed integer programming, CSP, AI planning, TSP, MAXSAT, QBF and answer set programming). Competition-winning systems in SAT are SATzilla, 3S and CSHC === Machine learning === In machine learning, algorithm selection is better known as meta-learning. The portfolio of algorithms consists of machine learning algorithms (e.g., Random Forest, SVM, DNN), the instances are data sets and the cost metric is for example the error rate. So, the goal is to predict which machine learning algorithm will have a small error on each data set. == Instance features == The algorithm selection problem is mainly solved with machine learning techniques. By representing the problem instances by numerical features f {\displaystyle f} , algorithm selection can be seen as a multi-class classification problem by learning a mapping f i ↦ A {\displaystyle f_{i}\mapsto {\mathcal {A}}} for a given instance i {\displaystyle i} . Instance features are numerical representations of instances. For example, we can count the number of variables, clauses, average clause length for Boolean formulas, or number of samples, features, class balance for ML data sets to get an impression about their characteristics. === Static vs. probing features === We distinguish between two kinds of features: Static features are in most cases some counts and statistics (e.g., clauses-to-variables ratio in SAT). These features ranges from very cheap features (e.g. number of variables) to very complex features (e.g., statistics about variable-clause graphs). Probing features (sometimes also called landmarking features) are computed by running some analysis of algorithm behavior on an instance (e.g., accuracy of a cheap decision tree algorithm on an ML data set, or running for a short time a stochastic local search solver on a Boolean formula). These feature often cost more than simple static features. === Feature costs === Depending on the used performance metric m {\displaystyle m} , feature computation can be associated with costs. For example, if we use running time as performance metric, we include the time to compute our instance features into the performance of an algorithm selection system. SAT solving is a concrete example, where such feature costs cannot be neglected, since instance features for CNF formulas can be either very cheap (e.g., to get the number of variables can be done in constant time for CNFs in the DIMACs format) or very expensive (e.g., graph features which can cost tens or hundreds of seconds). It is important to take the overhead of feature computation into account in practice in such scenarios; otherwise a misleading impression of the performance of the algorithm selection approach is created. For example, if the decision which algorithm to choose can be made with perfect accuracy, but the features are the running time of the portfolio algorithms, there is no benefit to the portfolio approach. This would not be obvious if feature costs were omitted. == Approaches == === Regression approach === One of the first successful algorithm selection approaches predicted the performance of each algorithm m ^ A : I → R {\displaystyle {\hat {m}}_{\mathcal {A}}:{\mathcal {I}}\to \mathbb {R} } and selected the algorithm with the best predicted performance a r g min A ∈ P m ^ A ( i ) {\displaystyle arg\min _{{\mathcal {A}}\in {\mathcal {P}}}{\hat {m}}_{\mathcal {A}}(i)} for an instance i {\displaystyle i} . === Clustering approach === A common assumption is that the given set of instances I {\displaystyle {\mathcal {I}}} can be clustered into homogeneous subsets and for each of these subsets, there is one well-performing algorithm for all instances in there. So, the training consists of identifying the homogeneous clusters via an unsupervised clustering approach and associating an algorithm with each cluster. A new instance is assigned to a cluster and the associated algorithm selected. A more modern approach is cost-sensitive hierarchical clustering using supervised learning to identify the homogeneous instance subsets. === Pairwise cost-sensitive classification approach === A common approach for multi-class classification is to learn pairwise models between every pair of classes (here algorithms) and choose the class that was predicted most often by the pairwise models. We can weight the instances of the pairwise prediction problem by the performance difference between the two algorithms. This is motivated by the fact that we care most about getting predictions with large differences correct, but the penalty for an incorrect prediction is small if there is almost no performance difference. Therefore, each instance i {\displaystyle i} for training a classification model A 1 {\displaystyle {\mathcal {A}}_{1}} vs A 2 {\displaystyle {\mathcal {A}}_{2}} is associated with a cost | m ( A 1 , i ) − m ( A 2 , i ) | {\displaystyle |m({\mathcal {A}}_{1},i)-m({\mathcal {A}}_{2},i)|} . == Requirements == The algorithm selection problem can be effectively applied under the following assumptions: The portfolio P {\displaystyle {\mathcal {P}}} of algorithms is complementary with respect to the instance set I {\displaystyle {\mathcal {I}}} , i.e., there is no single algorithm A ∈ P {\displaystyle {\mathcal {A}}\in {\mathcal {P}}} that dominates the performance of all other algorithms over I {\displaystyle {\mathcal {I}}} (see figures to the right for examples on complementary analysis). In some application, the computation of instance features is associated with a cost. For example, if the cost metric is running time, we have also to consider the time to compute the instance features. In such cases, the cost to compute features should not be larger than the performance gain through algorithm selection. == Application domains == Algorithm selection is not limited to single domains but can be applied to any kind of algorithm if the above requirements are satisfied. Application domains include: hard combinatorial problems: SAT, Mixed Integer Programming, CSP, AI Planning, TSP, MAXSAT, QBF and Answer Set Programming combinatorial auctions in machine learning, the problem is known as meta-learning software design black-box optimization multi-agent systems numerical optimization linear algebra, differential equations evolutionary algorithms vehicle routing problem power systems For an extensive list of literature about algorithm selection, we refer to a literature overview. == Variants of algorithm selection == === Online selection === Online algorithm selection refers to switching between different algorithms during the solving process. This is useful as a hyper-heuristic. In contrast, offline algorithm selection selects an algorithm for a given instance only once and before the solving process. === Computation of schedules === An extension of algorithm selection is the per-instance algorithm scheduling problem, in which we do not select only one solver, but we select a time budget for each algorithm
Superincreasing sequence
In mathematics, a sequence of positive real numbers ( s 1 , s 2 , . . . ) {\displaystyle (s_{1},s_{2},...)} is called superincreasing if every element of the sequence is greater than the sum of all previous elements in the sequence. Formally, this condition can be written as s n + 1 > ∑ j = 1 n s j {\displaystyle s_{n+1}>\sum _{j=1}^{n}s_{j}} for all n ≥ 1. == Program == The following Python source code tests a sequence of numbers to determine if it is superincreasing: This produces the following output: Sum: 0 Element: 1 Sum: 1 Element: 3 Sum: 4 Element: 6 Sum: 10 Element: 13 Sum: 23 Element: 27 Sum: 50 Element: 52 Is it a superincreasing sequence? True == Examples == (1, 3, 6, 13, 27, 52) is a superincreasing sequence, but (1, 3, 4, 9, 15, 25) is not. The series a^x for a>=2 == Properties == Multiplying a superincreasing sequence by a positive real constant keeps it superincreasing.