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" In the mathematical theory of directed graphs, a graph is said to be strongly connected or diconnected if every vertex is reachable from every other vertex. The strongly connected components or diconnected components of an arbitrary directed graph form a partition into subgraphs that are themselves strongly connected. It is possible to test the strong connectivity of a graph, or to find its strongly connected components, in linear time (that is, Θ(V+E)).
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" A directed graph is called strongly connected if there is a path in each direction between each pair of vertices of the graph. That is, a path exists from the first vertex in the pair to the second, and another path exists from the second vertex to the first.
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" In a directed graph ""G"" that may not itself be strongly connected, a pair of vertices ""u"" and ""v"" are said to be strongly connected to each other if there is a path in each direction between them.
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" The binary relation of being strongly connected is an equivalence relation, and the induced subgraphs of its equivalence classes are called strongly connected components.
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" Equivalently, a strongly connected component of a directed graph ""G"" is a subgraph that is strongly connected, and is maximal with this property: no additional edges or vertices from ""G"" can be included in the subgraph without breaking its property of being strongly connected. The collection of strongly connected components forms a partition of the set of vertices of ""G"".
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" If each strongly connected component is contracted to a single vertex, the resulting graph is a directed acyclic graph, the condensation of ""G"". A directed graph is acyclic if and only if it has no strongly connected subgraphs with more than one vertex, because a directed cycle is strongly connected and every nontrivial strongly connected component contains at least one directed cycle.
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" Several algorithms based on depth first search compute strongly connected components in linear time.
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" Although Kosaraju's algorithm is conceptually simple, Tarjan's and the path-based algorithm require only one depth-first search rather than two.
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" Previous linear-time algorithms are based on depth-first search which is generally considered hard to parallelize. Fleischer et al. in 2000 proposed a divide-and-conquer approach based on reachability queries, and such algorithms are usually called reachability-based SCC algorithms. The idea of this approach is to pick a random pivot vertex and apply forward and backward reachability queries from this vertex. The two queries partition the vertex set into 4 subsets: vertices reached by both, either one, or none of the searches. One can show that a strongly connected component has to be contained in one of the subsets. The vertex subset reached by both searches forms a strongly connected components, and the algorithm then recurses on the other 3 subsets.
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" The expected sequential running time of this algorithm is shown to be O(""n"" log ""n""), a factor of O(log ""n"") more than the classic algorithms. The parallelism comes from: (1) the reachability queries can be parallelized more easily (e.g. by a BFS, and it can be fast if the diameter of the graph is small); and (2) the independence between the subtasks in the divide-and-conquer process.
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" This algorithm performs well on real-world graphs, but does not have theoretical guarantee on the parallelism (consider if a graph has no edges, the algorithm requires O(""n"") levels of recursions).
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" Blelloch et al. in 2016 shows that if the reachability queries are applied in a random order, the cost bound of O(""n"" log ""n"") still holds. Furthermore, the queries then can be batched in a prefix-doubling manner (i.e. 1, 2, 4, 8 queries) and run simultaneously in one round. The overall span of this algorithm is log ""n"" reachability queries, which is probably the optimal parallelism that can be achieved using the reachability-based approach.
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" Peter M. Maurer describes an algorithm for generating random strongly-connected graphs, based on a modification of Tarjan's algorithm to create a spanning tree and adding a minimum of edges such that the result becomes strongly-connected. When used in conjunction with the Gilbert or Erdős-Rényi models with node relabelling, the algorithm is capable of generating any strongly-connected graph on ""n"" nodes, without restriction on the kinds of structures that can be generated.
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" Algorithms for finding strongly connected components may be used to solve 2-satisfiability problems (systems of Boolean variables with constraints on the values of pairs of variables): as showed, a 2-satisfiability instance is unsatisfiable if and only if there is a variable ""v"" such that ""v"" and its complement are both contained in the same strongly connected component of the implication graph of the instance.
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" Strongly connected components are also used to compute the Dulmage–Mendelsohn decomposition, a classification of the edges of a bipartite graph, according to whether or not they can be part of a perfect matching in the graph.
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" A directed graph is strongly connected if and only if it has an ear decomposition, a partition of the edges into a sequence of directed paths and cycles such that the first subgraph in the sequence is a cycle, and each subsequent subgraph is either a cycle sharing one vertex with previous subgraphs, or a path sharing its two endpoints with previous subgraphs.
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" According to Robbins' theorem, an undirected graph may be oriented in such a way that it becomes strongly connected, if and only if it is 2-edge-connected. One way to prove this result is to find an ear decomposition of the underlying undirected graph and then orient each ear consistently.
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"= = = Royal Academy of Dance = = =
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" The Royal Academy of Dance (RAD) is a UK-based examination board specialising in dance education and training, with an emphasis on classical ballet. The RAD was founded in London, England in 1920 as the Association of Teachers of Operatic Dancing, and was granted a Royal Charter in 1935. Queen Elizabeth II is patron of the RAD, and Darcey Bussell was elected to serve as President in 2012, succeeding Antoinette Sibley who served as President for 21 years.
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" The RAD was created with the objective to improve the standard of ballet teaching in the UK and, in pursuit of that goal, a new teaching method and dance technique was devised for the Academy by a group of eminent European dancers. The RAD is one of the largest dance organisations in the world with over 14,000 members in 79 countries, including about 7,500 who hold Registered Teacher Status. There are currently about 1,000 students in full-time or part-time teacher training programmes with the RAD, and each year about 250,000 candidates enter RAD examinations worldwide.
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" RAD exams are recognised by the national qualifications regulators of all for UK nations (England, Scotland, Wales and Northern Ireland) with selected exams also carrying a UCAS tariff towards university admission. The RAD is also a validated awarding body of the Council for Dance Education and Training. The RAD works in partnership with the International Dance Teachers' Association. Royal Academy of Dance is charity 312826 registered in England and Wales. Royal Academy of Dance Enterprises Ltd is a wholly owned subsidiary of the RAD, which sells Academy branded merchandise, with all profits being gift-aided to the RAD.
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" On 18 July 1920, Phillip J. S. Richardson, then Editor of ""Dancing Times"" magazine, organised a dinner for eminent dance professionals at the former Trocadero Restaurant in Piccadilly. The diners included five special guests, representing the principal methods of ballet training in use at that time.
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" Other guests included Ninette de Valois, the founder of England's Royal Ballet company, and Anton Dolin, the co-founder of English National Ballet. The purpose of the meeting was to discuss the poor quality of dance training in Britain at that time and following further meetings, the Association of Teachers of Operatic Dancing of Great Britain was formed, with Adeline Genée as its first President. The Association would be the predecessor of the RAD.
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" In 1921, the Association established headquarters at the offices of ""Dancing Times"" magazine in London and the first Elementary, Intermediate and Advanced level examinations were held in the years that followed, with the first Children's syllabus being published and then examined in 1923 and 1924. In 1927 a scholarship scheme was introduced for students studying dance with an Association member and in 1928, Queen Mary consented to become Patron of the Association of Teachers of Operatic Dancing.
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" In November 1930, the first issue of ""Dance Gazette"" was published containing a historical review of the Association's first ten years. 'Dance Gazette' is still the official magazine of the RAD, being distributed quarterly to all fully paid members. Since 2011, it is also available to buy in selected branches of WHSmith in the UK. In 1931, the first Genée International Ballet Competition for female dancers was held in London, with awards for male dancers being introduced later in 1939. The competition is the RAD's flagship annual event, named after Dame Adeline Genée DBE and now one of the most prestigious dance competitions in the world. The competition was introduced as an additional incentive for candidates who had passed the Solo Seal examination, and has taken place almost every year since 1931, even during the Second World War. In 1934, the Royal Opera House, Covent Garden agreed to an alliance with the Association and a Grand Council of prestigious men & women was formed to act as a governing body to advise the executive committee.
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" In 1935 King George V approved the granting of a Royal Charter for the Association of Teachers of Operatic Dancing. The charter was eventually affixed with his Great Seal in 1936, and the Association of Teachers of Operatic Dancing officially became the 'Royal Academy of Dancing' (RAD). The following year in 1937, a coat of arms was designed by the College of Arms in London.
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" The RAD coat of arts was granted by the College of Arms in 1937, designed by the Hon. George Bellew, Somerset Herald of
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" Arms and Registrar. The crest at the top of the coat of arms is a figure of Terpsichore, one of the Muses from Greek mythology, representing dance. The supporter at either side of the shield is a winged doe, symbolising lightness and grace of movement. The Escutcheon shows a pentagram symbolising health, with a wavy and zigzag line conveying the movement of dancing. The motto 'Salus et Felicitas’ translates to mean health and happiness.
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" In 1950, founder President of the RAD Adeline Genée was appointed a Dame Commander of the Order of the British Empire in the New Year Honours List. Further honours were awarded to members of the Academy in 1951, with Ninette de Valois being appointed a Dame, Philip Richardson being awarded an Order of the British Empire (OBE) and Margot Fonteyn awarded a Commander of the Order of the British Empire (CBE). Following the death of Queen Mary in 1953, Queen Elizabeth II consented to become the Royal Patron of the Royal Academy of Dancing and Dame Adeline Genée instituted the Queen Elizabeth II Coronation Award to mark the occasion. The award is presented annually in recognition of outstanding services to the art of ballet. The first recipient was Dame Ninette de Valois, founder of The Royal Ballet Company. The following is a list of selected recipients of the award.
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" Dame Adeline Genée retired as President of the RAD in 1954 and Dame Margot Fonteyn was elected as her successor. The leading Principal dancer of the Royal Ballet at that time and considered the greatest Ballerina of her generation, Fonteyn was later appointed Prima Ballerina Assoluta of the Royal Ballet by Queen Elizabeth II.
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" In 1963, the Academy was granted charitable status. In 1965, the first RAD summer school was held at Elmhurst Ballet School in Camberley, Surrey. Teachers and dancers attended the summer school from the United States, Canada, New Zealand as well as the UK. In 1968, Dame Margot Fonteyn devised a new children’s syllabus, which would become the basis for lower graded examinations.
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" On 23 April 1970, Founder President of the Academy, Dame Adeline Genée died at the age of 92 and in 1972, the RAD moved into its current headquarters in Battersea Square, London. The building was fully refurbished to provide high quality dance facilities and was officially opened by Queen Elizabeth II in 1974. The following year in 1975, the Academy established the Professional Dancer's Teaching Diploma, an intensive course for professional ballet dancers and other dance professionals to gain registered teacher status with the RAD and in 1976, the RAD's teacher training schemes were organised into what is now officially the Faculty of Education of the RAD.
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" In 1979, a new bursary was founded and named in honour of Phyllis Bedells, a founder member and former vice-president of the RAD. The bursary of £1,000 is held annually, awarded for further training in the Academy’s method and is awarded to young dancers who are under 17 years of age and have passed RAD Advanced 1 with Distinction but have not entered for the Advanced 2 examination. Past winners have gone on to win medals in very high-profile competitions, such as the Genée International Ballet Competition, and to dance with world-renowned companies including the Australian Ballet, Dutch National Ballet, English National Ballet, London City Ballet, Maurice Béjart and The Royal Ballet. Payment of the Bursary can be offset against existing tuition fees where the winner may be studying, or against material / equipment needed for the winner to continue their training.
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" In 1983, the RAD opened its first International headquarters in Darlinghurst, New South Wales, Australia. Since that time, the RAD has opened offices worldwide, including Brazil, China, Germany, Greece, Israel, Italy, Jamaica, Japan, Mexico, New Zealand, Singapore, South Africa, Spain, United Arab Emirates and the United States of America,
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" In 1990, a new studio complex was opened at RAD headquarters. Titled the 'Fonteyn Centre', it was opened by Queen Elizabeth II. The following year in 1991, Dame Margot Fonteyn died on the 21 February in Panama City and Antoinette Sibley was elected as the RAD's third President. Also in 1991, the 'Graded Examination Syllabus' was introduced, with Grades 6, 7 and 8 being introduced later in 1992. The three-year teacher training course was also amended in 1992, being replaced by the RAD's first full-time Degree qualification, the BA (Hons) Art & Teaching of Ballet (now the BA in Ballet Education).
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" The RAD celebrated its 75th Anniversary in 1995, the event being marked by events throughout the world. In 1996, RAD President Antoinette Sibley was appointed a Dame in the Queen's New Year Honours List and the Academy's first 12 degree students attended their graduation at Durham Castle. In 1997, the Benesh Institute was incorporated into the RAD and in 1999, The 'Faculty of Education' was established with further teaching qualifications being launched, validated by the University of Durham.
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" On 20 December 2000, the RAD officially adopted its current title, ""Royal Academy of Dance"" and in the same year, qualifications were established for students studying 'Benesh Movement Notation'.
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" In 2010 the RAD became an Accredited Institute of the University of Surrey.
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" On 4 January 2011 in London and 17 January in Sydney, the RAD launched its new Intermediate Foundation & Intermediate syllabi. RAD Grades 1-8 and the first five levels of the Vocational Grades were accredited by the Office of Qualifications and Examinations Regulation (Ofqual) in England with the RAD as an Awarding Organisation, and the Academy also achieved SQA Accreditation (Scottish Qualifications Authority Accreditation) for RAD exams. UCAS tariff points for Graded and Vocational Graded Examinations in Dance at Level 3 were confirmed in August, meaning those candidates who have achieved a Pass or higher at Grades 6-8, Intermediate or Advanced Foundation can use their exam result as contributing towards their points tariff when applying to a university or Higher Education institution in the UK. The Genée International Ballet Competition 2011 was held for the first time in Cape Town, South Africa, at Artscape Theatre Centre, looking to present the competition in New Zealand in 2012. The first ever Dance Proms took place on Sunday 13 November at the Royal Albert Hall, London, as a partnership project between the International Dance Teachers Association, the Imperial Society of Teachers of Dancing, RAD and the Royal Albert Hall working together in celebration of dance and hosting more than 450 young dancers on the stage. UK and Republic of Ireland members from the three dance organisations were invited to video and submit a short dance piece online, and a variety of acts were then chosen both by an esteemed panel of professionals and through a public vote to perform live at the Royal Albert Hall.
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" The RAD offers two examination syllabi, which are suitable for both male and female students and form the basis of the RAD's work in the UK and overseas. Each programme consists of a series of progressively more difficult practical examinations.
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" Grades 1-8 and the first five levels of the Vocational Grades are accredited by the Office of Qualifications and Examinations Regulation (Ofqual) in England, the Department for Children, Education, Lifelong Learning and Skills (DCELLS) in Wales and the Council for the Curriculum, Examinations & Assessment (CCEA) in Northern Ireland. These organisations are the Government approved qualifications regulators for their respective countries. Candidates who successfully pass an RAD examination are awarded credit on the Qualifications and Credit Framework. The Academy also achieves SQA Accreditation (Scottish Qualifications Authority Accreditation) for RAD exams.
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" In 2009, the RAD launched a new syllabus at Pre-Primary and Primary level in dance. These two new grades replace the existing Pre-Primary and Primary grades in ballet. The Academy has since launched new work for grades 1-5
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" The Graded Examination Syllabus consists of Pre-Primary, Primary, and numbered Grades 1-8. Each grade incorporates classical ballet, free movement and character dance. The syllabus is devised to progress in difficulty from one grade to the next and a student studying the grades in sequence would be expected to develop a greater degree of dance technique at each level.
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" Students who have studied a complete grade from the syllabus will normally take an examination organised by their dance teacher and held at the teachers studio or a suitable venue hired for the occasion. In the examination, the students perform the relevant grade work for an RAD examiner, who assesses the artistic and technical execution of the grade. A successful candidate receives a certificate printed with their name and the grade passed. They also receive a report detailing what marks were awarded and highlighting which aspects of their performance need further improvement. Unsuccessful candidates receive a report and a certificate of participation. The Pre-Primary grade is devised for younger students who are not old enough to enter the Primary examination. There is no examination offered at this level, however students can take part in a Presentation Class.
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" Presentation Classes were introduced for students who a dance teacher feels may not benefit from, or be suitable for taking the series of examinations. In a Presentation Class, in contrast to an examination, the teacher and a small audience of spectators are present in the room. The students perform for an examiner but are not assessed. They dance a condensed selection of exercises from the appropriate grade and receive a certificate of participation, which is normally presented by the examiner at the end of the class.
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" As a syllabus devised specifically with children in mind, examinations are not compulsory. The entry level of a new student is decided by the dance teacher based upon the child’s age, natural ability for dance or previous examination passes with another dance school or examination board, such as the IDTA or ISTD.
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" The Grades are: