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Prim's algorithm

In computer science, Prim's algorithm is a greedy algorithm that finds a minimum spanning tree for a weighted undirected graph. This means it finds a subset of the edges that forms a tree that includes every vertex, where the total weight of all the edges in the tree is minimized; the algorithm operates by building this tree one vertex at a time, from an arbitrary starting vertex, at each step adding the cheapest possible connection from the tree to another vertex. The algorithm was developed in 1930 by Czech mathematician Vojtěch Jarník and rediscovered and republished by computer scientists Robert C. Prim in 1957 and Edsger W. Dijkstra in 1959. Therefore, it is sometimes called the Jarník's algorithm, Prim–Jarník algorithm, Prim–Dijkstra algorithm or the DJP algorithm. Other well-known algorithms for this problem include Borůvka's algorithm; these algorithms find the minimum spanning forest in a disconnected graph. However, running Prim's algorithm separately for each connected component of the graph, it can be used to find the minimum spanning forest.

In terms of their asymptotic time complexity, these three algorithms are fast for sparse graphs, but slower than other more sophisticated algorithms. However, for graphs that are sufficiently dense, Prim's algorithm can be made to run in linear time, meeting or improving the time bounds for other algorithms; the algorithm may informally be described as performing the following steps: In more detail, it may be implemented following the pseudocode below. As described above, the starting vertex for the algorithm will be chosen arbitrarily, because the first iteration of the main loop of the algorithm will have a set of vertices in Q that all have equal weights, the algorithm will automatically start a new tree in F when it completes a spanning tree of each connected component of the input graph; the algorithm may be modified to start with any particular vertex s by setting C to be a number smaller than the other values of C, it may be modified to only find a single spanning tree rather than an entire spanning forest by stopping whenever it encounters another vertex flagged as having no associated edge.

Different variations of the algorithm differ from each other in how the set Q is implemented: as a simple linked list or array of vertices, or as a more complicated priority queue data structure. This choice leads to differences in the time complexity of the algorithm. In general, a priority queue will be quicker at finding the vertex v with minimum cost, but will entail more expensive updates when the value of C changes; the time complexity of Prim's algorithm depends on the data structures used for the graph and for ordering the edges by weight, which can be done using a priority queue. The following table shows the typical choices: A simple implementation of Prim's, using an adjacency matrix or an adjacency list graph representation and linearly searching an array of weights to find the minimum weight edge to add, requires O running time. However, this running time can be improved further by using heaps to implement finding minimum weight edges in the algorithm's inner loop. A first improved version uses a heap to store all edges of the input graph, ordered by their weight.

This leads to an O worst-case running time. But storing vertices instead of edges can improve it still further; the heap should order the vertices by the smallest edge-weight that connects them to any vertex in the constructed minimum spanning tree. Every time a vertex v is chosen and added to the MST, a decrease-key operation is performed on all vertices w outside the partial MST such that v is connected to w, setting the key to the minimum of its previous value and the edge cost of. Using a simple binary heap data structure, Prim's algorithm can now be shown to run in time O where |E| is the number of edges and |V| is the number of vertices. Using a more sophisticated Fibonacci heap, this can be brought down to O, asymptotically faster when the graph is dense enough that |E| is ω, linear time when |E| is at least |V| log |V|. For graphs of greater density, Prim's algorithm can be made to run in linear time more by using a d-ary heap in place of a Fibonacci heap. Let P be a connected, weighted graph.

At every iteration of Prim's algorithm, an edge must be found that connects a vertex in a subgraph to a vertex outside the subgraph. Since P is connected, there will always be a path to every vertex; the output Y of Prim's algorithm is a tree, because the edge and vertex added to tree Y are connected. Let Y1 be a minimum spanning tree of graph P. If Y1=Y Y is a minimum spanning tree. Otherwise, let e be the first edge added during the construction of tree Y, not in tree Y1, V be the set of vertices connected by the edges added before edge e. One endpoint of edge e is in set V and the other is not. Since tree Y1 is a spanning tree of graph P, there is a path in tree Y1 joining the two endpoints; as one travels along the path, one must encounter an edge f joining a vertex in set V to one, not in set V. Now, at the iteration when edge e was added to tree Y, edge f could have been added and it would be added instead of edge e if its weight was less than e, since edge f was not added, we conclude that w ≥ w.

Let tree Y2 be the graph obtained b

Comrades Marathon

The Comrades Marathon is an ultramarathon of 89 kilometres, run annually in the KwaZulu-Natal province of South Africa between the cities of Durban and Pietermaritzburg. It is the world's oldest ultramarathon race; the direction of the race alternates each year between the "up" run starting from Durban and the "down" run starting from Pietermaritzburg. The 2019 field was capped at 25,000 runners, the entry process closed after one week. South African runners constitute the greater part of the field, but many entrants hail from the United Kingdom, India, the United States, Australia, Botswana and Swaziland. In all but three runnings since 1988, over 10,000 runners have reached the finish within the allowed 11 or 12 hours. With increased participation since the 1980s, the average finish times for both sexes, the average age of finishers have increased substantially. Runners over the age of 20 qualify when they are able to complete an recognised marathon in under five hours. During the event an athlete must reach five cut-off points in specified times to complete the race.

The spirit of the Comrades Marathon is said to be embodied by attributes of camaraderie, dedication and ubuntu. The race is run on the roads of KwaZulu-Natal province, marked by "The Big Five" set of hills. On the up run they appear in the following order: Cowies Hill, Fields Hill, Botha's Hill and Polly Shortts; the highest point of the race, at 2,850 feet above sea level, is situated near Umlaas Road interchange. 40 official refreshment stations along the route are stocked with soft drinks, water sachets, energy drink sachets, biscuits, energy bars and cooked potatoes. Eight physiotherapy and first aid stations are located at strategic points. Athletes have 12 hours to complete the course, extended from 11 hours in 2003; the original Comrades cut-off time from 1921 to 1927 was 12 hours, reduced to 11 hours in 1928. There are a number of cut-off points along the routes which runners must reach by a prescribed time or be forced to retire from the race. A runner who has completed nine marathons wears a yellow number, while those who have completed ten races wear a green number, permanently allocated to the runner for all future races.

Medals are awarded to all runners completing the course in under 12 hours. Medals are awarded as follows: Gold medal: the first 10 men and women. Wally Hayward medal: 11th position to sub 6hrs 00min. Isavel Roche-Kelly medal: women only, 11th position to sub 7hrs 30min. Silver medal: 6hrs 00min to sub 7hrs 30min. Bill Rowan medal: 7hrs 30min to sub 9hrs 00min. Robert Mtshali medal: 9hr 00min to sub 10hrs 00min. Bronze medal: 10hrs 00min to sub 11hrs 00min. Vic Clapham medal: 11hrs 00min to sub 12hrs 00min.- Prior to 2000, only gold and bronze medals were awarded. - The Bill Rowan medal was introduced in 2000 and named after the winner of the first Comrades Marathon in 1921. The time limit for this medal was inspired by Rowan's winning time in 1921 of 8hrs 59min. - A new copper medal, the Vic Clapham medal, was added in 2003. This medal coincided with the increase in the time allocation for completing the event from sub 11hrs to sub 12hrs. - The Wally Hayward medal, named after five-time winner Wally Hayward, was added in 2007 for runners finishing in under 6hrs, but outside the gold medals.

- In 2005 the back-to-back medal was created and henceforth was awarded to novice runners who complete an'up or down run' in succession. In terms of the implementation thereof, back-to-back medals were automatically awarded to 2005 Comrades Marathon finishers who had completed their first Comrades Marathon in 2004; as with any new innovation, the award was never intended to be retrospective, owing to administrative restrictions. However, in response to popular demand, the back-to-back medal is available for purchase to runners who have fulfilled the criteria of completing both an'up' and a'down' Comrades Marathon. - For 2019 the Isavel Roche-Kelly medal is being introduced for women finishing outside the gold medals, but under 7hrs 30min eliminating the silver medal for women. Twenty-year old Isavel Roche-Kelly was named the UCT Sportsperson of the Year for 1980. An unknown on the athletics scene, Roche-Kelly set the roads alight that year when she became the first woman to break the 7½-hour barrier and win the Comrades Marathon in 7:18:00.

Earlier that year she became only the third women in Africa to complete a marathon in under three hours. She went on to win the 1981 Comrades up run in a time of 6:44:35 the following year. Three years she died in a cycling accident in her native Northern Ireland at the age of 24. - Also in 2019, the titanium Robert Mtshali medal is being introduced for a time between 9hrs 00min and sub 10hrs 00min. Robert Mtshali was the first unofficial black runner in the 1935 Comrades Marathon, finishing his race in a time of 9 hours and 30 minutes, his efforts were not recorded as government and race rules of the time stipulated that, in order to compete in the Comrades Marathon, you had to be a white male. Friday, the 24th of May 1935, saw Mtshali participating in the 15th Comrades Marathon, a down run, joining the 48 official entrants on the starting line, he ran unofficially, but was warmly welcomed into the Durban finish venue on the Old Fort

Aguillon family

The Aguillon family, of French origin, were feudal landowners in England who held estates in several southern counties from before 1135 to 1312. Surviving records suggest various branches which all ended without male heirs, the lands going to daughters or sisters and their husbands; the family seems to have been associated as under-tenants and maybe through marriage, with the Marmion family, witnessing charters alongside them in Normandy in 1106 and occupying their land in England. The English branches may spring from William Aguillon, a descendant of the viscounts of Chaumont, seigneur of Trie near the French border with Normandy around 1119 and died on the Second Crusade, he married Margaret of Gisors and their son and heir was Enguerrand. Manser may have been a younger son. In England, family members can be found in four apparent groups but establishing definite connections between the four groups may be impossible. Manser, who before 1135 received two knight's fees in the honour of Arundel. and held a knight's fee from Robert Marmion of Tamworth in 1167.

In 1172 he was liable by knight-service for castle-guard at Falaise. He had one known son and may have had two others: Robert I inherited lands in West Sussex William I Richard IRobert I, son of Manser, in 1180 paid 15 marks to have seisin of Nutbourne in West Sussex and for leave to come to an agreement with his unnamed brother, who may have been William I. William I another son of Manser, in 1195 was claiming a knight's fee in Nutbourne against a Manser and a Richard, He married Mary, daughter of Eustace de Valle Pironis, an otherwise unknown family name, their sons were: Reginald John, it may be this John who in 1221 had land at Maltby in Lincolnshire, or else it was his contemporary, the son of Richard I and Margery. Reginald I, son of William I and Mary, from 1220 to 1226 was bailiff of the honour of Arundel, being ordered by King Henry III in 1225 to arrest all ships carrying corn in various Sussex ports. In 1223 he bought land in Offham and in 1227 was given the manor of Up Marden by his mother, who had inherited it from her father.

By 1240 his lands had been divided between his four daughters: Mary, who married William Covert Cecily, who married Peter Gatesden and gave her part to the Knights Hospitaller Godeheut, who married Ralph St Owen Alice, who married first William Russell and secondly Robert Hackett. Richard I another son of Manser and the Richard who asserted his right to a knight's fee in Nutbourne in 1206, married Margery, daughter of William Thorney, lord of Thorney, his wife Mabel, their sons were. William II, son of Richard I and Margery, before 1215 acquired the manor of Warblington which, together with lands in Emsworth, was confirmed to him in 1230. In 1242 he held three fees in Nutbourne, Up Marden, Burpham and one-third of West Thorney, his son was: Richard II. Richard II, son of William II, married Eleanor, who in 1308 held the three fees of Nutbourne, Up Marden and Burpham as his widow, she died before 1312, leaving them to her granddaughter Juliana, daughter of her deceased son Thomas, who herself died in 1312.

William III a childless younger son of William II, in 1259 was made Keeper of Guildford Castle and was Sheriff of Surrey in 1261. In 1265 he had one-third of the advowson of West Thorney and in 1278 he held half the manor of Nutbourne. William IV may be the William who in 1219 with his wife Joan claimed the manor of Greatham in Hampshire, he inherited the lands of his mother-in-law and of his wife's grandfather, including the manor of Addington, which carried the duty of making a special dish to be served at the king's coronation and entitled the holder to attend Parliament as a baron, with William said to have taken his seat in 1233 and his son to have followed him after 1244. Shortly before his death, he received a pardon for crimes of murder and robbery which he had committed in 1227, after which he had fled the country and been declared an outlaw. In 1212 he married Joan, widow of Ralph Parminter and younger daughter of Peter fitz Henry, the son of Henry fitz Ailwin and the husband of Isabel Cheney, heiress of Addington.

Their son was: Robert IISir Robert II, son of William II and Joan, had by 1248 inherited his father's manor of Perching in the West Sussex parish of Fulking and in 1260 acquired two-thirds of the neighbouring manor of Fulking, with the reversion of the other third. In 1264 he was licensed to enclose his manor house at Perching with a ditch and a stone wall, to crenellate it, so marking the origin of Perching Castle. In 1281 he acquired further land in Perching and in 1284 was reported as holding the whole settlement. In 1248 he obtained a grant of free warren in his demesne lands of Addington and in 1270 licence to embattle his house there. In 1267 he was Keeper of Guildford Castle. After 1265 he was granted lands at Berwick, East Sussex, taken from the rebel William Marmion, was Keeper of Arundel Castle during the minority of its heir in 1272. In 1274 he did service at the coronation of King Edward I. After his second marriage to a rich widow, he acquired more landholdings: Portslade in Sussex and Wendover in Buckinghamshire, both in 1270, together with Stapleford in Hertfordshire, where in 1285 he held both manor and advowson.

He first married by August 1256 Joan, widow of Sir John Mohun and daughter of William de Ferrers, 5th Earl of Derby and his first wife Sibyl Marshal. After her death before O

Challawa Gorge Dam

The Challawa Gorge Dam is in Karaye Local Government Area of Kano State in the Northwest of Nigeria, about 90 km southwest of Kano city. It is a major reservoir on the Challawa River, a tributary of the Kano River, the main tributary of the Hadejia River; the Challawa Gorge reservoir project was started by the Water Resources and Engineering Construction Agency of the Kano State Government, was handed over to the Federal Government who funded the project. The dam is owned and operated by the Hadejia-Jama’are River Basin Development Authority, a Federal agency; the dam was built by Julius Berger Nigeria in 1990 - 1992 using rock fill construction. It is 7.8 km in length. The dam has a full storage capacity of 904,000,000 m3; the direct catchment area is 3857 km2. The dam was designed with the potential for hydro power generation in mind, may have a capacity of 3MW on average - more in the rainy season and less in dry season. However, power would cost more to deliver than current retail prices, it is not clear how a project to install the generating equipment would be financed.

The soil in the immediate catchments of the dam has not been stabilized, so the reservoir may be silting up. Silt is being deposited in the Challawa River, affecting the intake structures of Kano City Water Supply; the dam has disrupted the natural balance along the river. Upstream areas are now subject to flooding while downstream riverine wetlands and croplands have dried out. A 2002 study noted that while the dam was intended to support irrigation projects, none had been started, although much farmland had been covered by the dam; the water was being used only to supply Kano city. The Challawa dam and the nearby Tiga Dam have had adverse effects on the downstream Hadejia-Nguru wetlands. Several studies have shown that these dams have delivered negative economic value when their effect on downstream communities was taken into account

GEM of Egypt

The GEM of Egypt was a power shovel used for strip mining. Built in 1966, the machine had worked in the Egypt Valley coalfield, GEM being an acronym for “Giant Earth Mover” or “Giant Excavating Machine”, it was one of only two Bucyrus-Erie 1950-B shovels built and one of two to use the knee action crowd, licensed from Marion Power Shovel in exchange to Marion's use of BE's cable crowd patent. The GEM had a 170' boom and a 130 cubic yard bucket which enabled it to dig 200 tons per'bite'; the machine began work in January, 1967 for Hanna Coal, was purchased by Consolidated Coal in "Little Egypt Valley" near Barnesville, Ohio. The area was where the GEM got its name; the GEM of Egypt was one of three in the service of the Hanna Coal Company, which by 1970, had been strip mining in Ohio for decades. The other two power shovels were The Tiger and The Mountaineer. Of the three, the GEM of Egypt was the largest of the power shovels in the Hanna Coal Company, of which it went into service via the official opening of the Egypt Valley mine in January 1967.

It was reported that an estimated 25,000 people traveled to the site, many from Ohio cities such as Cleveland and Canton, as well as those from neighboring states. The Gem of Egypt weighed 7,000 tons. Production there was expected to average 20,000 tons a day, which the company forecast would last for the next 30 to 40 years until the vein ran out; the machine was parked in 1988 and dismantled in 1991 off Ohio SR9 between New Athens and Fairpoint. Parts of the shovel were used to keep its twin, The Silver Spade, operating until it too was retired. <gallery>

Martin of Leon

Saint Martin of Leon was a priest and canon regular of the Augustinian Order. Born at León, along with his father Juan, withdrew from the world to the canonry of St. Marcellinus in León after the death of his mother. Martin was educated at this canonry, after the death of his father, Martin decided to undertake a major pilgrimage, visiting the cities of Rome and Constantinople. Returning to Spain, he took the religious habit at St. Marcellinus, but after seeing this monastery had been secularized by the bishops he entered the collegiate of church St. Isidore in the same city; this is a church he is where Saint Isidore was buried, hence its name. Martin distinguished himself by his zealous observance, his charity, his deep devotion to the Blessed Sacrament; the date of his death is given to us by the necrology preserved in the monastery. He died on January 1203 of natural causes; the religious of St. Isidore's dedicated a chapel to Martin early and celebrated his feast each year. Martin wrote commentaries on different Epistles and the Apocalypse, he left numerous discourses on the many varied subjects.

His complete works were published first by Espinosa, Migne in Patrologia Latina, LXXXI, 53-64, CCVIII, CCIX. Augustinian Canons: Blessed Martin of Leon Martin of Leon at the Catholic Encyclopedia