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Proof of O(log*n) time complexity of union–find

In computer science, Union Find is an algorithm for doing certain operations on sets. This page is about proof of O amortized time of Union FindStatement: If m operations, either Union or Find, are applied to n elements, the total run time is O, where log* is the iterated logarithm. Lemma 1: As the find function follows the path along to the root, the rank of node it encounters is increasing. Proof: claim that as Find and Union operations are applied to the data set, this fact remains true over time; when each node is the root of its own tree, it's trivially true. The only case when the rank of a node might be changed is when the Union by Rank operation is applied. In this case, a tree with smaller rank will be attached to a tree with greater rank, rather than vice versa, and during the find operation, all nodes visited along the path will be attached to the root, which has larger rank than its children, so this operation won't change this fact either. Lemma 2: A node u, root of a subtree with rank r has at least 2r nodes.

Proof: Initially when each node is the root of its own tree, it's trivially true. Assume that a node u with rank r has at least 2r nodes; when two trees with rank r Union by Rank and form a tree with rank r + 1, the new node has at least 2r + 2r = 2r + 1 nodes. Lemma 3: The maximum number of nodes of rank r is at most n/2r. Proof: From lemma 2, we know that a node u, root of a subtree with rank r has at least 2r nodes. We will get the maximum number of nodes of rank r when each node with rank r is the root of a tree that has 2r nodes. In this case, the number of nodes of rank r is n/2rFor convenience, we define "bucket" here: a bucket is a set that contains vertices with particular ranks. We put vertices into the buckets according to their ranks inductively; that is, vertices with rank 0 go into the zeroth bucket, vertices with rank 1 go into the first bucket, vertices with ranks 2 and 3 go into the second bucket. If the Bth bucket contains vertices with ranks from interval = the st bucket will contain vertices with ranks from interval.

We can make two observations about the buckets. The total number of buckets is at most log*n Proof: When we go from one bucket to the next, we add one more two to the power, that is, the next bucket to will be The maximum number of elements in bucket is at most 2n/2BProof: The maximum number of elements in bucket is at most n/2B + n/2B+1 + n/2B+2 + … + n/22B – 1 ≤ 2n/2BLet F represent the list of "find" operations performed, let T 1 = ∑ F T 2 = ∑ F T 3 = ∑ F; the total cost of m finds is T = T1 + T2 + T3Since each find operation makes one traversal that leads to a root, we have T1 = O. Also, from the bound above on the number of buckets, we have T2 = O. For T3, suppose we are traversing an edge from u to v, where u and v have rank in the bucket and v is not the root. Fix u and consider the sequence v1,v2...vk that take the role of v in different find operations. Because of path compression and not accounting for the edge to a root, this sequence contains only different nodes and because of Lemma 1 we know that the ranks of the nodes in this sequence are increasing.

By both of the nodes being in the bucket we can conclude that the length k of the sequence is at most the number of ranks in the buckets B, i.e. at most 2B − 1 − B < 2B. Therefore, T 3 ≤ ∑ ∑ u 2 B. From Observations 1 and 2, we can conclude that T 3 ≤ ∑ B 2 B 2 n 2 B ≤ 2 n log ∗ ⁡ n. Therefore, T = T1 + T2 + T3 = O

Sandra Holden

Sandra Holden is an Australian softball pitcher. She resides in Ormeau, Queensland, she represents Queensland in national competitions and she is a member of the Australia women's national softball team, where she represented the country in a 2009 tour of Japan and in a January 2012 test against New Zealand. She is trying to earn a spot on the team that will allow her to compete at the 2012 ISF XIII Women's World Championships Holden is from Queensland, she is a high school science teacher at Livingstone Christian College, is married. Her maiden name was Avery, she resides in Queensland. Holden is a pitcher, she started playing softball as a ten-year-old in school. This led her to joining a club outside of school, she prays before she plays in games. She plays club softball for Peninsula Cougars. Holden represents Queensland in national competitions. However, in 2003, she represented Tasmania at the national championships, earning the spot because the Tasmanians were looking to fill in their squad as they were short on players.

Holden first represented Queensland on the junior U23 level in 2000 and played with them for four years. She was first named to the Queensland state senior team, Queensland Sunlanders, in 2004, first represented them at the 2005 national championships, she earned her spot after retirement of one player ranked above her. She represented her state in the Gilleys Shield in 2007, 2011 and 2012. In 2011, she picked up a bronze medal in the Shield; the Logan West Leader described her performance in 2012 as being in "top form". At the end of the fourth round of competition in 2012, her team was in fourth place. Two other years that she played for the team, they finished first in the Shield, she represented her state in the 2012 Edebone-Weber Shield, similar to the Gilleys Shield except it includes New Zealand as a competing side. Holden is a member of Australia women's national softball team, she rotates spots with teammate Jocelyn McCallum as one of the team's pitchers. She was a member of the 2009 national team.

She is a member of the 2012 Australia women's national softball team and is trying to earn a spot on the team that will allow her to compete at the 2012 ISF XIII Women's World Championships. In January 2012, she traveled with the team to play in a test series against New Zealand women's national softball team, she was chosen to represent the country in a March 2012 series against Japan, played in Canberra


Byutyay-Yurdya is a rural locality, the only inhabited locality, the administrative center of Betyunsky Rural Okrug of Namsky District in the Sakha Republic, located 12 kilometers from Namtsy, the administrative center of the district. Its population as of the 2010 Census was 608, up from 578 recorded during the 2002 Census. Official website of the Sakha Republic. Registry of the Administrative-Territorial Divisions of the Sakha Republic. Namsky District. Государственное Собрание Республики Саха. Закон №173-З №353-III от 30 ноября 2004 г. «Об установлении границ и о наделении статусом городского и сельского поселений муниципальных образований Республики Саха », в ред. Закона №1058-З №1007-IV от 25 апреля 2012 г. «О внесении изменений в Закон Республики Саха "Об установлении границ и о наделении статусом городского и сельского поселений муниципальных образований Республики Саха"». Вступил в силу со дня официального опубликования. Опубликован: "Якутия", №245, 31 декабря 2004 г


Fabletics is an American online subscription retailer that sells women's sportswear and accessories referred to as "athleisure." The company is best known for its e-commerce business approach and has 22 brick-and-mortar stores. It offers its members personalized outfits chosen for them based on their lifestyle and fashion preferences and claims to be "all inclusive" with a range of sizing, it is a subsidiary of JustFab, which re-branded to TechStyle Fashion Group in August 2016. Fabletics launched on October 1, 2013. In June 2015, Fabletics launched FL2, its men's activewear line with Kate Hudson's brother and actor, Oliver Hudson. In March 2016, the company expanded its inventory beyond athletic wear by adding dresses and swimsuits; the company has repeated gained press exposure for their advertising tactics. According to Forbes, Fabletics has increased its revenue by 35% each year, with a valuation of $250 million within its first three years of business; the company uses crowd marketing to increase membership.

In these locations, Fabletics stocks apparel based on analytics of its online trends. The company added collections for dresses and swimwear in the spring of 2017. In September 2017, Fabletics announced its first line of official footwear including slip-ons, lifestyle sneakers, workout shoes. Fabletic's VIP membership plan has raised some controversy due to the lack of clarity of the conditions and the difficulty to cancel the membership. Fabletic's VIP Membership plan is a monthly subscription of $49.95. VIP members have the option of paying for the monthly membership. Members can be spent whenever. Members who opt-out paying for the month must do so before the 5th of every month. If members choose to cancel, they can do it over chat, or online. In September and October 2015, Fabletics opened its first brick and mortar retail stores in malls owned and operated by Westfield and General Growth Properties Inc. In February 2016, Forbes reported that 75 to 100 Fabletics stores are scheduled to open over the next three to five years.

As of July 2018, the company has a presence in a total of 10 countries, with plans of opening up new stores in the Philippines for the first time by fall of the same year. As of 2020, Fabletics has opened 42 stores in North America. Hudson was named the ambassador for Fashion Targets Breast Cancer with Fabletics partnering on a pink capsule collection to support breast cancer awareness. In May 2017, the company announced a collaboration with singer Demi Lovato in support of the United Nations Foundation's Girl Up campaign. In 2019, Fabletics made collaboration lines with Kelly Rowland and Maddie Ziegler that are available to VIP and regular members; the company's subscription business model has been criticized. If subscribers forget to notify Fabletics parent JustFab within the first five days to "skip" the month, the customer is charged anyway. Members receive credit towards future purchases. A 2015 investigation of the parent company, JustFab, by Buzzfeed found 1,400 Better Business Bureau complaints against JustFab in just over three years.

Official website 21st Century Fashion Company Bets on Tech & Media Kate Hudson's athleisure brand is doubling down on a strategy it borrowed from Apple and Warby Parker Kate Hudson’s Fabletics Fiercely Giving Amazon A Run For Its Money

ISU Grand Prix of Figure Skating

The ISU Grand Prix of Figure Skating is a series of senior international figure skating competitions organized by the International Skating Union. The invitational series was inaugurated in 1995, incorporating several existing events. Medals are awarded in the disciplines of men's singles, ladies' singles, pair skating, ice dancing; the junior-level equivalent is the ISU Junior Grand Prix. The sanctioned competitions for the Grand Prix are: Skate America. First held in 1979 as Norton Skate, the event has been part of the series since 1995 and its location changes yearly. Skate Canada International. First held in 1973, the event has been part of the series since 1995 and its location changes yearly. Cup of China, it was created in 2003 and joined the Grand Prix series in the same year, replacing the German event. It has been held in Beijing, Harbin and Chongqing, it was replaced in 2018 by the Grand Prix of Helsinki. Internationaux de France. First held in 1987, the event has been part of the series since 1995.

From 1987 to 2014, it was always held in Paris, with the exception of 1991, 1994, 1995. Since 2014, it has been held in Bordeaux and Grenoble. Rostelecom Cup; the Prize of Moscow News having disappeared with the breakup of the Soviet Union, the Cup of Russia was established in 1996 and joined the series in the same year, adopting the name Rostelecom Cup in 2009. It is held in Moscow and, less in Saint Petersburg. NHK Trophy. First held in 1979, the event has been part of the series since 1995; the location changes yearly — it has been held in Tokyo, Kobe, Asahikawa, Chiba, Nagoya, Nagano, Kyoto and Sendai. Grand Prix of Figure Skating Final. Created in 1995 to serve as the concluding event, it features the top six qualifiers in each discipline from the six earlier competitions; the event adopted its current name in the 1998–99 season. Its location changes yearly. Bofrost Cup on Ice. First held in 1986, the event was part of the series from 1995 to 2002. Held in Gelsenkirchen, the event adopted the name Bofrost Cup on Ice in 2002.

Grand Prix of Helsinki. The event replaced Cup of China in 2018. Cup of China returned to the series from the 2019–20 season onwards. Fall international competitions such as Skate America, organized by the skating federations of their host countries, had been held for many years prior to being organized into a series as separate individual events. Following the Nancy Kerrigan attack in 1994, television coverage of skating was saturated with made-for-TV professional skating events, while the traditional "amateur" or "eligible" competitions were neglected. In order to remedy this situation, in 1995, the skating federations from the United States, Germany and Japan began to plan their events as a series with cooperative marketing of the television rights in those countries, with prize money funded by the sale of those rights. At this point, the International Skating Union stepped in and asserted its ownership of the international television rights to the series; when it was first created in the 1995–1996 skating season, the series was known as the ISU Champions Series.

It did not become known as the Grand Prix of Figure Skating until the 1998–1999 season, when the ISU gained the rights to use that name. It was composed of five events, held in the United States, Japan and France. Following the demise of the Prize of Moscow News, last held in 1990, the Russian federation created the Cup of Russia, which increased the number of events to six in 1996, the series' second year. In the fall of 2003, the event in Germany, the Bofrost Cup on Ice, was discontinued, was replaced with one in China, due to the ISU having negotiated a more favorable television contract in that country. In 1997, the ISU created a similar series of developmental events for junior age-eligible skaters. Known as the ISU Junior Series, these events are now named the ISU Junior Grand Prix; this season begins. Skaters are entered in the individual events either by invitation; the seeding of top skaters at Grand Prix events takes into account their placement from the previous World Championships, as well as their ISU international ranking.

Skaters who are not seeded can be invited by the hosting country and each country can invite up to three of their own skaters for each discipline. This is to give a balanced field throughout the series, as well as allowing the hosting country a chance to showcase their top competitors; the Grand Prix of Figure Skating uses a points-based system based on results from the selected international events. The top qualifying skaters from each discipline are eligible to compete in the Grand Prix Figure Skating Final; the entry and qualification rules for the individual events have varied from year to year, between the different disciplines. Seeded skaters can be entered in either two or three events, while other skaters may be entered in either one or two events. Starting with the 2003-04 season, the Interim Judging System was introduced for scoring events in the Grand Prix; this developed into the ISU Judging System called the Co