Scroll (web service)

Scroll was a subscription-based web service developed by Scroll Labs Inc., offering ad-free access to websites in exchange for a fee. Scroll was not an ad blocker; instead, it partnered directly with internet publishers who voluntarily removed ads from their sites for Scroll users in exchange for a portion of the subscription fee. In May 2021, Scroll was acquired by Twitter. In October 2021, Scroll sent out an email announcing its integration into Twitter Blue within 30 days. == Functionality == Scroll enabled users to browse websites that partnered with Scroll without encountering online advertising, in exchange for a subscription fee. Unlike ad blocker, which disable advertisements without compensating the publisher, Scroll sent a browser cookie indicating that the user was a subscriber. The Scroll software integrated into the website detected this cookie and served an ad-free version of the site. In exchange for disabling advertisements, partner websites received a portion of the subscription fee. As of January 2020, Scroll retained 30% of the subscription fee, with the remaining 70% distributed among publisher sites. Payments to sites were made individually by users based on their 'engagement and loyalty,' rather than from a single pool of all subscription revenue. Scroll did not grant subscribers access to partner sites behind a paywall; it only removed ads from the site if the user also paid the publication's subscription fee. == History == Scroll was founded in 2016 by former Chartbeat Chief Executive Tony Haile. Scroll raised US$3 million in its first round of funding in 2016, including investments from The New York Times, Uncork Capital, and Axel Springer SE. By October 2018, Scroll had raised US$10 million in funding. In 2018, Scroll signed its first partner websites, which included The Atlantic, Fusion Media Group, Business Insider, Slate, MSNBC, The Philadelphia Inquirer, and Talking Points Memo. In February 2019, Scroll acquired the social media curation app Nuzzel. The same month, Mozilla and Scroll announced a partnership to run a "test pilot" together, but did not go into details. Scroll entered beta testing in 2019 and launched to the general public on January 28, 2020. In March 2020, Mozilla started offering Scroll as part of its "Firefox Better Web" service bundle. In May 2021, Scroll was acquired by Twitter, with the future of Scroll cited as being uncertain. An email to customers announcing the change said, "Later this year, Scroll will become part of a wider Twitter subscription that will expand on and adapt our services and functionality".

Law practice management software

Law practice management software is software designed to manage the business operations of a law firm. This can include software that manages cases, client intake, court communications, electronic discovery, time tracking, trust accounting, and billing. == Features of law practice management software == Common features of practice management software include: Case management Time tracking Document assembly Contact management Calendaring Docket management Client portal Contract Management Court Case Status Tracker Trust accounting == Examples of law practice management software == Smokeball LEAP Legal Software PracticeEvolve Dye & Durham

Informetrics

Informetrics is the study of quantitative aspects of information, it is an extension and evolution of traditional bibliometrics and scientometrics. Informetrics uses bibliometrics and scientometrics methods to study mainly the problems of literature information management and evaluation of science and technology. Informetrics is an independent discipline that uses quantitative methods from mathematics and statistics to study the process, phenomena, and law of informetrics. Informetrics has gained more attention as it is a common scientific method for academic evaluation, research hotspots in discipline, and trend analysis. Informetrics includes the production, dissemination, and use of all forms of information, regardless of its form or origin. Informetrics encompasses the following fields: Scientometrics, which studies quantitative aspects of science Webometrics, which studies quantitative aspects of the World Wide Web Bibliometrics, which studies quantitative aspects of recorded information Cybermetrics, which is similar to webometrics, but broadens its definition to include electronic resources == Origin and Development == The term informetrics (French: informétrie) was coined by German scholar Otto Nacke in 1979, and came from the German word 'informetrie’. The corresponding English terminology soon appeared in the subsequent literature. In September 1980, Professor Otto Nacke introduced the term 'informetrics' at the first seminar on Informetrics in Frankfurt, Germany. Later, Committee on Informetrics has established through The International Federation for Information and Documentation (FID). In 1987, informetrics started to be officially recognized by the international information community and several foreign information scientists. In 1988, at First International Conference on Bibliometrics and Theoretical Aspects of Information Retrieval Archived 2022-05-23 at the Wayback Machine, Brooks suggested bibliometrics and scientometrics can be included in the field of informetrics. In 1990, Leo Egghe and Ronald Rousseau proposed the formation of the discipline of informetrics: statistical bibliography (1923) to bibliometrics and scientometrics (1969) and then to informetrics (1979). In 1993, the International Society for Scientometrics and Informetrics (ISSI) Archived 2023-11-05 at the Wayback Machine was founded at the International Conference on Bibliometrics, Informetrics and Scientometrics in Berlin, and the first one was held in Belgium and organized by Leo Egghe and Ronald Rousseau. The society was formally incorporated in 1994 in the Netherlands and plays a significant role in the development of informetrics. The ISSI aims to promote the "exchange and communication of professional information in the fields of scientometrics and informetrics, including improve standards, theory and practice, as well as promote research, education and training". In addition, to "engage in relevant public conversation and policy discussions". In the western world, 20th century's Informetrics is mostly based on Lotka's law, named after Alfred J. Lotka, Zipf's law, named after George Kingsley Zipf, Bradford's law named after Samuel C. Bradford and on the work of Derek J. de Solla Price, Gerard Salton, Leo Egghe, Ronald Rousseau, Tibor Braun, Olle Persson, Peter Ingwersen, Manfred Bonitz, and Eugene Garfield. == Difference Between Informetrics, Bibliometrics and Scientometrics == Since the 1960s, three similar terms have emerged in the fields of library science, philology and science of science, they are bibliometrics, scientometrics and informetrics, representing three very similar quantitative sub-disciplines. The three metrics terms can be confusing and often misused. Informetrics and bibliometrics interpenetrate each other but have different aspects in research object, research scope, and measuring unit. Informetrics and scientometrics are very different in their research purpose and research object, as well as the research scope and application. Bibliometrics is categorised under the field of library science, it uses mathematical and statistical methods to describe, evaluate, and predict the current status and trends of science and technology. Also to study the "distribution structure, quantitative relationship, change law and quantitative management of literature information, quantitative relationships, patterns and quantitative management of literature and information". The term was first used by Alan Pritchard in 1969 in his paper Statistical Bibliography or Bibliometrics?. Scientometrics is a branch of science that quantitatively evaluates and predicts the process and management of scientific activities in order to reveal their development patterns and trends. The definition of scientometrics was described by Derek De Solla Price in his book Science to Science as the “quantitative study of science, communication in science, and science policy”. === Links between the three metrics terms === The most prominent connection between the three metrics terms is in their research objects. Since all three disciplines use literature information as their research object, therefore, they have some similarities and overlaps in their research methods and fields. Moreover, they all use mathematical methods as the basic research methods and they all apply the three basic laws, Bradford's law, Lotka's law and Zipf's law. === Distinctions between the three metrics terms === The distinction between the three metrics terms can tell from their research object and research purpose. The research of bibliometrics focuses on the analysis of "scientific output in the form of articles, publications, citations, and others". Scientometrics is to measure the basic characteristics and laws of scientific activities. Where informetrics is to investigate information sources and information distribution process. == Concept and System Structure == === Purpose of Informetrics Research === The main purpose of informetrics is to use its theocratical research to solve the methodological issues in the research process, and to discover and reveal the basic laws of information distribution through the study of information process and phenomenon. In this way, makes information management more scientific and provides a quantitative basis for information services and information management decisions. For informetrics, it is necessary to bring quantitative analysis methods to further reveal the structure of information units and the "quantitative change law of literature information”. Further to this, to improve the scientific accuracy of information science from a theoretical point of view. At the same time, to better solve the basic contradictions in the information service, overcome the information crisis, and make the information management work more effective to serve science and technology, economic and social development. Quantitative analysis of bibliographic data was pioneered by Robert K. Merton in an article called Science, Technology, and Society in Seventeenth Century England and originally published by Merton in 1938. === The Significance of Informetrics Research === The significance of informetrics research is to summarize various empirical laws from the theoretical point of view, at the same time test and modify the various empirical laws in the new information unit conditions, and explore its new applicability, therefore, the scientific nature of information science can be improved, but also to provide theoretical guidance for practical work. === The Objects of Informetrics Research === The object of informetrics is broader than the field of bibliometrics and scientometrics, including "messages, data, events, objects, text, and documents”. Informetrics is often used to inform policies and decisions across a broad range of fields, such as economy, politics, technology and social spheres that "influence the flow and use patterns of information". Tague-Sutcliffe describes the following uses of informetrics: Citation analysis; Characteristics of authors; Use of recorded information; Obsolescence of the literature; Concomitant growth of new concepts; Characteristics of publication sources; Definition and measurement o information; Growth of subject literature, databases, libraries; Types and characteristics of retrieval performance measures; Statistical aspects of language, word, and phrase frequencies. == Basic Laws == In the field of informetrics research, there are many outstanding contributors in the discipline with a solid knowledge of quantitative research methods. In the early 20th century, several scientists contributed empirical applications that have become the three basic laws of informetrics, Bradford's law, Lotka's law, and Zipf's law, which promote the development of informetrics. === Bradford's Law === The British documentalist and librarian Samuel C. Bradford first discovered the law of concentration and scattering of literature, and in 1934, it has be

Flajolet–Martin algorithm

The Flajolet–Martin algorithm is an algorithm for approximating the number of distinct elements in a stream with a single pass and space-consumption logarithmic in the maximal number of possible distinct elements in the stream (the count-distinct problem). The algorithm was introduced by Philippe Flajolet and G. Nigel Martin in their 1984 article "Probabilistic Counting Algorithms for Data Base Applications". Later it has been refined in "LogLog counting of large cardinalities" by Marianne Durand and Philippe Flajolet, and "HyperLogLog: The analysis of a near-optimal cardinality estimation algorithm" by Philippe Flajolet et al. In their 2010 article "An optimal algorithm for the distinct elements problem", Daniel M. Kane, Jelani Nelson and David P. Woodruff give an improved algorithm, which uses nearly optimal space and has optimal O(1) update and reporting times. == The algorithm == Assume that we are given a hash function h a s h ( x ) {\displaystyle \mathrm {hash} (x)} that maps input x {\displaystyle x} to integers in the range [ 0 ; 2 L − 1 ] {\displaystyle [0;2^{L}-1]} , and where the outputs are sufficiently uniformly distributed. Note that the set of integers from 0 to 2 L − 1 {\displaystyle 2^{L}-1} corresponds to the set of binary strings of length L {\displaystyle L} . For any non-negative integer y {\displaystyle y} , define b i t ( y , k ) {\displaystyle \mathrm {bit} (y,k)} to be the k {\displaystyle k} -th bit in the binary representation of y {\displaystyle y} , such that: y = ∑ k ≥ 0 b i t ( y , k ) 2 k . {\displaystyle y=\sum _{k\geq 0}\mathrm {bit} (y,k)2^{k}.} We then define a function ρ ( y ) {\displaystyle \rho (y)} that outputs the position of the least-significant set bit in the binary representation of y {\displaystyle y} , and L {\displaystyle L} if no such set bit can be found as all bits are zero: ρ ( y ) = { min { k ≥ 0 ∣ b i t ( y , k ) ≠ 0 } y > 0 L y = 0 {\displaystyle \rho (y)={\begin{cases}\min\{k\geq 0\mid \mathrm {bit} (y,k)\neq 0\}&y>0\\L&y=0\end{cases}}} Note that with the above definition we are using 0-indexing for the positions, starting from the least significant bit. For example, ρ ( 13 ) = ρ ( 1101 2 ) = 0 {\displaystyle \rho (13)=\rho (1101_{2})=0} , since the least significant bit is a 1 (0th position), and ρ ( 8 ) = ρ ( 1000 2 ) = 3 {\displaystyle \rho (8)=\rho (1000_{2})=3} , since the least significant set bit is at the 3rd position. At this point, note that under the assumption that the output of our hash function is uniformly distributed, then the probability of observing a hash output ending with 2 k {\displaystyle 2^{k}} (a one, followed by k {\displaystyle k} zeroes) is 2 − ( k + 1 ) {\displaystyle 2^{-(k+1)}} , since this corresponds to flipping k {\displaystyle k} heads and then a tail with a fair coin. Now the Flajolet–Martin algorithm for estimating the cardinality of a multiset M {\displaystyle M} is as follows: Initialize a bit-vector BITMAP to be of length L {\displaystyle L} and contain all 0s. For each element x {\displaystyle x} in M {\displaystyle M} : Calculate the index i = ρ ( h a s h ( x ) ) {\displaystyle i=\rho (\mathrm {hash} (x))} . Set B I T M A P [ i ] = 1 {\displaystyle \mathrm {BITMAP} [i]=1} . Let R {\displaystyle R} denote the smallest index i {\displaystyle i} such that B I T M A P [ i ] = 0 {\displaystyle \mathrm {BITMAP} [i]=0} . Estimate the cardinality of M {\displaystyle M} as 2 R / ϕ {\displaystyle 2^{R}/\phi } , where ϕ ≈ 0.77351 {\displaystyle \phi \approx 0.77351} . The idea is that if n {\displaystyle n} is the number of distinct elements in the multiset M {\displaystyle M} , then B I T M A P [ 0 ] {\displaystyle \mathrm {BITMAP} [0]} is accessed approximately n / 2 {\displaystyle n/2} times, B I T M A P [ 1 ] {\displaystyle \mathrm {BITMAP} [1]} is accessed approximately n / 4 {\displaystyle n/4} times and so on. Consequently, if i ≫ log 2 ⁡ n {\displaystyle i\gg \log _{2}n} , then B I T M A P [ i ] {\displaystyle \mathrm {BITMAP} [i]} is almost certainly 0, and if i ≪ log 2 ⁡ n {\displaystyle i\ll \log _{2}n} , then B I T M A P [ i ] {\displaystyle \mathrm {BITMAP} [i]} is almost certainly 1. If i ≈ log 2 ⁡ n {\displaystyle i\approx \log _{2}n} , then B I T M A P [ i ] {\displaystyle \mathrm {BITMAP} [i]} can be expected to be either 1 or 0. The correction factor ϕ ≈ 0.77351 {\displaystyle \phi \approx 0.77351} (OEIS: A244256) is found by calculations, which can be found in the original article. == Improving accuracy == A problem with the Flajolet–Martin algorithm in the above form is that the results vary significantly. A common solution has been to run the algorithm multiple times with k {\displaystyle k} different hash functions and combine the results from the different runs. One idea is to take the mean of the k {\displaystyle k} results together from each hash function, obtaining a single estimate of the cardinality. The problem with this is that averaging is very susceptible to outliers (which are likely here). A different idea is to use the median, which is less prone to be influences by outliers. The problem with this is that the results can only take form 2 R / ϕ {\displaystyle 2^{R}/\phi } , where R {\displaystyle R} is integer. A common solution is to combine both the mean and the median: Create k ⋅ l {\displaystyle k\cdot l} hash functions and split them into k {\displaystyle k} distinct groups (each of size l {\displaystyle l} ). Within each group use the mean for aggregating together the l {\displaystyle l} results, and finally take the median of the k {\displaystyle k} group estimates as the final estimate. The 2007 HyperLogLog algorithm splits the multiset into subsets and estimates their cardinalities, then it uses the harmonic mean to combine them into an estimate for the original cardinality.

Algorithmic Puzzles

Algorithmic Puzzles is a book of puzzles based on computational thinking. It was written by computer scientists Anany and Maria Levitin, and published in 2011 by Oxford University Press. == Topics == The book begins with a "tutorial" introducing classical algorithm design techniques including backtracking, divide-and-conquer algorithms, and dynamic programming, methods for the analysis of algorithms, and their application in example puzzles. The puzzles themselves are grouped into three sets of 50 puzzles, in increasing order of difficulty. A final two chapters provide brief hints and more detailed solutions to the puzzles, with the solutions forming the majority of pages of the book. Some of the puzzles are well known classics, some are variations of known puzzles making them more algorithmic, and some are new. They include: Puzzles involving chessboards, including the eight queens puzzle, knight's tours, and the mutilated chessboard problem Balance puzzles River crossing puzzles The Tower of Hanoi Finding the missing element in a data stream The geometric median problem for Manhattan distance == Audience and reception == The puzzles in the book cover a wide range of difficulty, and in general do not require more than a high school level of mathematical background. William Gasarch notes that grouping the puzzles only by their difficulty and not by their themes is actually an advantage, as it provides readers with fewer clues about their solutions. Reviewer Narayanan Narayanan recommends the book to any puzzle aficionado, or to anyone who wants to develop their powers of algorithmic thinking. Reviewer Martin Griffiths suggests another group of readers, schoolteachers and university instructors in search of examples to illustrate the power of algorithmic thinking. Gasarch recommends the book to any computer scientist, evaluating it as "a delight".

Question answering

Question answering (QA) is a computer science discipline within the fields of information retrieval and natural language processing (NLP) that is concerned with building systems that automatically answer questions that are posed by humans in a natural language. A question-answering implementation, usually a computer program, may construct its answers by querying a structured database of knowledge or information, usually a knowledge base. More commonly, question-answering systems can pull answers from an unstructured collection of natural language documents. Some examples of natural language document collections used for question answering systems include reference texts, compiled newswire reports, Wikipedia pages and other World Wide Web pages. == History == Two early question answering systems were BASEBALL and LUNAR. BASEBALL answered questions about Major League Baseball over a period of one year. LUNAR answered questions about the geological analysis of rocks returned by the Apollo Moon missions. Both question answering systems were very effective in their chosen domains. LUNAR was demonstrated at a lunar science convention in 1971 and it was able to answer 90% of the questions in its domain that were posed by people untrained on the system. Further restricted-domain question answering systems were developed in the following years. The common feature of all these systems is that they had a core database or knowledge system that was hand-written by experts of the chosen domain. The language abilities of BASEBALL and LUNAR used techniques similar to ELIZA and DOCTOR, the first chatterbot programs. SHRDLU was a successful question-answering program developed by Terry Winograd in the late 1960s and early 1970s. It simulated the operation of a robot in a toy world (the "blocks world"), and it offered the possibility of asking the robot questions about the state of the world. The strength of this system was the choice of a very specific domain and a very simple world with rules of physics that were easy to encode in a computer program. In the 1970s, knowledge bases were developed that targeted narrower domains of knowledge. The question answering systems developed to interface with these expert systems produced more repeatable and valid responses to questions within an area of knowledge. These expert systems closely resembled modern question answering systems except in their internal architecture. Expert systems rely heavily on expert-constructed and organized knowledge bases, whereas many modern question answering systems rely on statistical processing of a large, unstructured, natural language text corpus. The 1970s and 1980s saw the development of comprehensive theories in computational linguistics, which led to the development of ambitious projects in text comprehension and question answering. One example was the Unix Consultant (UC), developed by Robert Wilensky at U.C. Berkeley in the late 1980s. The system answered questions pertaining to the Unix operating system. It had a comprehensive, hand-crafted knowledge base of its domain, and it aimed at phrasing the answer to accommodate various types of users. Another project was LILOG, a text-understanding system that operated on the domain of tourism information in a German city. The systems developed in the UC and LILOG projects never went past the stage of simple demonstrations, but they helped the development of theories on computational linguistics and reasoning. Specialized natural-language question answering systems have been developed, such as EAGLi for health and life scientists. Question answering systems have been extended in recent years to encompass additional domains of knowledge For example, systems have been developed to automatically answer temporal and geospatial questions, questions of definition and terminology, biographical questions, multilingual questions, and questions about the content of audio, images, and video. Current question answering research topics include: interactivity—clarification of questions or answers answer reuse or caching semantic parsing answer presentation knowledge representation and semantic entailment social media analysis with question answering systems sentiment analysis utilization of thematic roles Image captioning for visual question answering Embodied question answering In 2011, Watson, a question answering computer system developed by IBM, competed in two exhibition matches of Jeopardy! against Brad Rutter and Ken Jennings, winning by a significant margin. Facebook Research made their DrQA system available under an open source license. This system uses Wikipedia as knowledge source. The open source framework Haystack by deepset combines open-domain question answering with generative question answering and supports the domain adaptation of the underlying language models for industry use cases. Large Language Models (LLMs)[36] like GPT-4[37], Gemini[38] are examples of successful QA systems that are enabling more sophisticated understanding and generation of text. When coupled with Multimodal[39] QA Systems, which can process and understand information from various modalities like text, images, and audio, LLMs significantly improve the capabilities of QA systems. == Types == Question-answering research attempts to develop ways of answering a wide range of question types, including fact, list, definition, how, why, hypothetical, semantically constrained, and cross-lingual questions. Answering questions related to an article in order to evaluate reading comprehension is one of the simpler form of question answering, since a given article is relatively short compared to the domains of other types of question-answering problems. An example of such a question is "What did Albert Einstein win the Nobel Prize for?" after an article about this subject is given to the system. Closed-book question answering is when a system has memorized some facts during training and can answer questions without explicitly being given a context. This is similar to humans taking closed-book exams. Closed-domain question answering deals with questions under a specific domain (for example, medicine or automotive maintenance) and can exploit domain-specific knowledge frequently formalized in ontologies. Alternatively, "closed-domain" might refer to a situation where only a limited type of questions are accepted, such as questions asking for descriptive rather than procedural information. Question answering systems in the context of machine reading applications have also been constructed in the medical domain, for instance related to Alzheimer's disease. Open-domain question answering deals with questions about nearly anything and can only rely on general ontologies and world knowledge. Systems designed for open-domain question answering usually have much more data available from which to extract the answer. An example of an open-domain question is "What did Albert Einstein win the Nobel Prize for?" while no article about this subject is given to the system. Another way to categorize question-answering systems is by the technical approach used. There are a number of different types of QA systems, including: rule-based systems, statistical systems, and hybrid systems. Rule-based systems use a set of rules to determine the correct answer to a question. Statistical systems use statistical methods to find the most likely answer to a question. Hybrid systems use a combination of rule-based and statistical methods. == Architecture == As of 2001, question-answering systems typically included a question classifier module that determined the type of question and the type of answer. Different types of question-answering systems employ different architectures. For example, modern open-domain question answering systems may use a retriever-reader architecture. The retriever is aimed at retrieving relevant documents related to a given question, while the reader is used to infer the answer from the retrieved documents. Systems such as GPT-3, T5, and BART use an end-to-end architecture in which a transformer-based architecture stores large-scale textual data in the underlying parameters. Such models can answer questions without accessing any external knowledge sources. == Methods == Question answering is dependent on a good search corpus; without documents containing the answer, there is little any question answering system can do. Larger collections generally mean better question answering performance, unless the question domain is orthogonal to the collection. Data redundancy in massive collections, such as the web, means that nuggets of information are likely to be phrased in many different ways in differing contexts and documents, leading to two benefits: If the right information appears in many forms, the question answering system needs to perform fewer complex NLP techniques to understand the text. Correct answers can be filtered from false positives because the syst

Generalized distributive law

The generalized distributive law (GDL) is a generalization of the distributive property which gives rise to a general message passing algorithm. It is a synthesis of the work of many authors in the information theory, digital communications, signal processing, statistics, and artificial intelligence communities. The law and algorithm were introduced in a semi-tutorial by Srinivas M. Aji and Robert J. McEliece with the same title. == Introduction == "The distributive law in mathematics is the law relating the operations of multiplication and addition, stated symbolically, a ∗ ( b + c ) = a ∗ b + a ∗ c {\displaystyle a(b+c)=ab+ac} ; that is, the monomial factor a {\displaystyle a} is distributed, or separately applied, to each term of the binomial factor b + c {\displaystyle b+c} , resulting in the product a ∗ b + a ∗ c {\displaystyle ab+ac} " – Britannica. As it can be observed from the definition, application of distributive law to an arithmetic expression reduces the number of operations in it. In the previous example the total number of operations reduced from three (two multiplications and an addition in a ∗ b + a ∗ c {\displaystyle ab+ac} ) to two (one multiplication and one addition in a ∗ ( b + c ) {\displaystyle a(b+c)} ). Generalization of distributive law leads to a large family of fast algorithms. This includes the FFT and Viterbi algorithm. This is explained in a more formal way in the example below: α ( a , b ) = d e f ∑ c , d , e ∈ A f ( a , c , b ) g ( a , d , e ) {\displaystyle \alpha (a,\,b){\stackrel {\mathrm {def} }{=}}\displaystyle \sum \limits _{c,d,e\in A}f(a,\,c,\,b)\,g(a,\,d,\,e)} where f ( ⋅ ) {\displaystyle f(\cdot )} and g ( ⋅ ) {\displaystyle g(\cdot )} are real-valued functions, a , b , c , d , e ∈ A {\displaystyle a,b,c,d,e\in A} and | A | = q {\displaystyle |A|=q} (say) Here we are "marginalizing out" the independent variables ( c {\displaystyle c} , d {\displaystyle d} , and e {\displaystyle e} ) to obtain the result. When we are calculating the computational complexity, we can see that for each q 2 {\displaystyle q^{2}} pairs of ( a , b ) {\displaystyle (a,b)} , there are q 3 {\displaystyle q^{3}} terms due to the triplet ( c , d , e ) {\displaystyle (c,d,e)} which needs to take part in the evaluation of α ( a , b ) {\displaystyle \alpha (a,\,b)} with each step having one addition and one multiplication. Therefore, the total number of computations needed is 2 ⋅ q 2 ⋅ q 3 = 2 q 5 {\displaystyle 2\cdot q^{2}\cdot q^{3}=2q^{5}} . Hence the asymptotic complexity of the above function is O ( n 5 ) {\displaystyle O(n^{5})} . If we apply the distributive law to the RHS of the equation, we get the following: α ( a , b ) = d e f ∑ c ∈ A f ( a , c , b ) ⋅ ∑ d , e ∈ A g ( a , d , e ) {\displaystyle \alpha (a,\,b){\stackrel {\mathrm {def} }{=}}\displaystyle \sum \limits _{c\in A}f(a,\,c,\,b)\cdot \sum _{d,\,e\in A}g(a,\,d,\,e)} This implies that α ( a , b ) {\displaystyle \alpha (a,\,b)} can be described as a product α 1 ( a , b ) ⋅ α 2 ( a ) {\displaystyle \alpha _{1}(a,\,b)\cdot \alpha _{2}(a)} where α 1 ( a , b ) = d e f ∑ c ∈ A f ( a , c , b ) {\displaystyle \alpha _{1}(a,b){\stackrel {\mathrm {def} }{=}}\displaystyle \sum \limits _{c\in A}f(a,\,c,\,b)} and α 2 ( a ) = d e f ∑ d , e ∈ A g ( a , d , e ) {\displaystyle \alpha _{2}(a){\stackrel {\mathrm {def} }{=}}\displaystyle \sum \limits _{d,\,e\in A}g(a,\,d,\,e)} Now, when we are calculating the computational complexity, we can see that there are q 3 {\displaystyle q^{3}} additions in α 1 ( a , b ) {\displaystyle \alpha _{1}(a,\,b)} and α 2 ( a ) {\displaystyle \alpha _{2}(a)} each and there are q 2 {\displaystyle q^{2}} multiplications when we are using the product α 1 ( a , b ) ⋅ α 2 ( a ) {\displaystyle \alpha _{1}(a,\,b)\cdot \alpha _{2}(a)} to evaluate α ( a , b ) {\displaystyle \alpha (a,\,b)} . Therefore, the total number of computations needed is q 3 + q 3 + q 2 = 2 q 3 + q 2 {\displaystyle q^{3}+q^{3}+q^{2}=2q^{3}+q^{2}} . Hence the asymptotic complexity of calculating α ( a , b ) {\displaystyle \alpha (a,b)} reduces to O ( n 3 ) {\displaystyle O(n^{3})} from O ( n 5 ) {\displaystyle O(n^{5})} . This shows by an example that applying distributive law reduces the computational complexity which is one of the good features of a "fast algorithm". == History == Some of the problems that used distributive law to solve can be grouped as follows: Decoding algorithms: A GDL like algorithm was used by Gallager's for decoding low density parity-check codes. Based on Gallager's work Tanner introduced the Tanner graph and expressed Gallagers work in message passing form. The tanners graph also helped explain the Viterbi algorithm. It is observed by Forney that Viterbi's maximum likelihood decoding of convolutional codes also used algorithms of GDL-like generality. Forward–backward algorithm: The forward backward algorithm helped as an algorithm for tracking the states in the Markov chain. And this also was used the algorithm of GDL like generality Artificial intelligence: The notion of junction trees has been used to solve many problems in AI. Also the concept of bucket elimination used many of the concepts. == The MPF problem == MPF or marginalize a product function is a general computational problem which as special case includes many classical problems such as computation of discrete Hadamard transform, maximum likelihood decoding of a linear code over a memory-less channel, and matrix chain multiplication. The power of the GDL lies in the fact that it applies to situations in which additions and multiplications are generalized. A commutative semiring is a good framework for explaining this behavior. It is defined over a set K {\displaystyle K} with operators " + {\displaystyle +} " and " . {\displaystyle .} " where ( K , + ) {\displaystyle (K,\,+)} and ( K , . ) {\displaystyle (K,\,.)} are a commutative monoids and the distributive law holds. Let p 1 , … , p n {\displaystyle p_{1},\ldots ,p_{n}} be variables such that p 1 ∈ A 1 , … , p n ∈ A n {\displaystyle p_{1}\in A_{1},\ldots ,p_{n}\in A_{n}} where A {\displaystyle A} is a finite set and | A i | = q i {\displaystyle |A_{i}|=q_{i}} . Here i = 1 , … , n {\displaystyle i=1,\ldots ,n} . If S = { i 1 , … , i r } {\displaystyle S=\{i_{1},\ldots ,i_{r}\}} and S ⊂ { 1 , … , n } {\displaystyle S\,\subset \{1,\ldots ,n\}} , let A S = A i 1 × ⋯ × A i r {\displaystyle A_{S}=A_{i_{1}}\times \cdots \times A_{i_{r}}} , p S = ( p i 1 , … , p i r ) {\displaystyle p_{S}=(p_{i_{1}},\ldots ,p_{i_{r}})} , q S = | A S | {\displaystyle q_{S}=|A_{S}|} , A = A 1 × ⋯ × A n {\displaystyle \mathbf {A} =A_{1}\times \cdots \times A_{n}} , and p = { p 1 , … , p n } {\displaystyle \mathbf {p} =\{p_{1},\ldots ,p_{n}\}} Let S = { S j } j = 1 M {\displaystyle S=\{S_{j}\}_{j=1}^{M}} where S j ⊂ { 1 , . . . , n } {\displaystyle S_{j}\subset \{1,...\,,n\}} . Suppose a function is defined as α i : A S i → R {\displaystyle \alpha _{i}:A_{S_{i}}\rightarrow R} , where R {\displaystyle R} is a commutative semiring. Also, p S i {\displaystyle p_{S_{i}}} are named the local domains and α i {\displaystyle \alpha _{i}} as the local kernels. Now the global kernel β : A → R {\displaystyle \beta :\mathbf {A} \rightarrow R} is defined as: β ( p 1 , . . . , p n ) = ∏ i = 1 M α ( p S i ) {\displaystyle \beta (p_{1},...\,,p_{n})=\prod _{i=1}^{M}\alpha (p_{S_{i}})} Definition of MPF problem: For one or more indices i = 1 , . . . , M {\displaystyle i=1,...\,,M} , compute a table of the values of S i {\displaystyle S_{i}} -marginalization of the global kernel β {\displaystyle \beta } , which is the function β i : A S i → R {\displaystyle \beta _{i}:A_{S_{i}}\rightarrow R} defined as β i ( p S i ) = ∑ p S i c ∈ A S i c β ( p ) {\displaystyle \beta _{i}(p_{S_{i}})\,=\displaystyle \sum \limits _{p_{S_{i}^{c}}\in A_{S_{i}^{c}}}\beta (p)} Here S i c {\displaystyle S_{i}^{c}} is the complement of S i {\displaystyle S_{i}} with respect to { 1 , . . . , n } {\displaystyle \mathbf {\{} 1,...\,,n\}} and the β i ( p S i ) {\displaystyle \beta _{i}(p_{S_{i}})} is called the i t h {\displaystyle i^{th}} objective function, or the objective function at S i {\displaystyle S_{i}} . It can observed that the computation of the i t h {\displaystyle i^{th}} objective function in the obvious way needs M q 1 q 2 q 3 ⋯ q n {\displaystyle Mq_{1}q_{2}q_{3}\cdots q_{n}} operations. This is because there are q 1 q 2 ⋯ q n {\displaystyle q_{1}q_{2}\cdots q_{n}} additions and ( M − 1 ) q 1 q 2 . . . q n {\displaystyle (M-1)q_{1}q_{2}...q_{n}} multiplications needed in the computation of the i th {\displaystyle i^{\text{th}}} objective function. The GDL algorithm which is explained in the next section can reduce this computational complexity. The following is an example of the MPF problem. Let p 1 , p 2 , p 3 , p 4 , {\displaystyle p_{1},\,p_{2},\,p_{3},\,p_{4},} and p 5 {\displaystyle p_{5}} be variables such that p 1 ∈ A 1 , p 2 ∈ A 2 , p 3 ∈ A 3 , p 4 ∈ A 4 , {\displaystyle p_{1}\in