Zoho Office Suite is an online office suite developed by Zoho Corporation. == History == Zoho Office Suite was launched in 2005 with a web-based word processor. Additional products, such as those for spreadsheets and presentations, were incorporated later into the suite. The applications are distributed as software as a service (SaaS). == Products == Zoho uses an open API for its Writer, Sheet, Show, Creator, Meeting, and Planner products. It also has plugins into Microsoft Word and Excel, an OpenOffice.org plugin, and a plugin for Firefox. Zoho Office Suite is free for individuals but offers a plan for teams, which includes Zoho WorkDrive, Zoho Workplace and other Zoho apps. In October 2009, Zoho integrated some of their applications with the Google Apps online suite.
Synthesia (company)
Synthesia Limited is a British multinational artificial intelligence company based in London, United Kingdom. It is a synthetic media-generation software developer and creator of AI-generated video content, including audio-visual agents and cloned avatars. Britain's largest generative-AI firm, it is used by 70% of FTSE 100 and over 90% of Fortune 100 companies. == Overview == Synthesia is most often used by corporations for localized communication, orientation, employee training videos, advertising campaigns, reporting, product demonstrations, customer service, and to create chatbots. Its software algorithm mimics speech and facial movements based on video recordings of an individual’s speech and facial expressions. From this, a text-to-speech video is created to look and sound like the individual. Swiss bank UBS incorporated Synthesia AI-powered avatars of their human financial experts, for instance, in 2025. Users create content via the platform's pre-generated AI presenters or by creating digital representations of themselves, or personal avatars, using the platform's AI video editing tool. These avatars can be used to narrate videos generated from text. As of August 2021, Synthesia's voice database included multiple gender options in over 60 languages. Its free voice library doubled by 2025, to 140 languages and accents, and its Express-Voice technology can clone a user's own voice, or generate a synthetic one. === Deepfakes === The platform prohibits use of its software to create non-consensual clones, including of celebrities or political figures for satirical purposes. Explicit consent must be provided in addition to a strict pre-screening regimen for use of an individual's likeness to avoid “deepfaking”. While the company prohibits use of its technology for misinformation or "news-like content", an October 2023 Freedom House report stated that Synthesia tools had been used by governments in Venezuela, China, Burkina Faso, and Russia to create videos of fake TV news outlets with AI-generated avatars in order to spread propaganda. Actor Dan Dewhirst signed a contract with the company in 2021, becoming one of the first actors whose likeness would be made into an AI avatar, finding his likeness used in the Venezuelan generated-videos. The company stated, in February 2024, that it had improved its misuse detection systems, and, in April 2024, that new users of its technology are screened by the company, and content employing it is further vetted by Synthesia moderators. == History == Synthesia's software utilizes deep learning architecture developed by Lourdes Agapito and Matthias Niessner. The company was co-founded in 2017 by Agapito, Niessner, Victor Riparbelli, and Steffen Tjerrild. In 2018, the company first demonstrated the software's capabilities on the BBC programme Click when it presented a digitization of Matthew Amroliwala speaking Spanish, Mandarin, and Hindi. Through Synthesia's first two years of existence, it employed 10 people and struggled to make sales, leading to an expansion of the company's focus. It moved on from just targeting entertainment studios to a variety of businesses. In 2020, Synthesia users were reported to include Amazon, Tiffany & Co. and IHG Hotels & Resorts. In January 2024, the company introduced its AI video assistant, which turns text-to-video. That April, with a reported 55,000 customers, including half of the Fortune 100, Synthesia launched "expressive avatars". That September, an enhanced dubbing feature was launched, to translate video in 30 languages with naturalized lip-syncing. Peter Hill joined Synthesia as CTO in January 2025, following 25 years at Amazon, and two years as CEO and CPO of Wildfire Studios. That March, a million dollar base of shares was formed to furnish human actors, employed to generate digital avatars, with company stock, which all of its employees hold. By June of that year, 150,000 individuals from among Synthesia's 65,000 customers had created AI-generated avatars of themselves. In July 2025, the company's new global headquarters at Regent’s Place was opened by London mayor Sadiq Khan, who described Britain's largest generative-AI company, then valued at over $2 billion, as a "London success story". By that October, its technology was employed by 90% of the Fortune 100, and Synthesia 3.0 was launched, with hyper-realistic digital avatars equipped with AI-powered dubbing and translation, and a built-in video assistant. In January 2026, it reached a $4 billion valuation, with 70% of FTSE 100 companies noted among its customers. === Funding === The company raised $3.1 million in seed funding in 2019. In April 2021, the company raised $12.5 million in Series A funding. In December 2021, it raised $50 million in a Series B funding round led by Kleiner Perkins and GV (then Google Ventures). Synthesia gained a total valuation of $1 billion, and achieved unicorn status, when it raised $90 million from Accel and Nvidia partnership NVentures, in June 2023, during its Series C funding round. Counting 60,000 customers by January 2025, including over 60% of Fortune 100 companies; the company raised $180 million in a Series D round led by NEA, with new investors World Innovation Lab (WiL), Atlassian Ventures and PSP Growth, as well as existing investors GV, MMC Ventures and FirstMark, doubling Synthesia's valuation to $2.1 billion. Capital raised by 2025 had reached $330 million, with investments slated to further product innovation, talent growth, and company expansion in North America, Europe, Japan and Australia. In April 2025, Adobe Inc. invested £10 million in the company for a strategic partnership. Synthesia subsequently rejected a $3 billion acquisition offer from Adobe, choosing to remain independent. With a revenue stream then exceeding $100 million annually; GV led a Series E funding round in October 2025, resulting in Synthesia's $4 billion valuation, raising $200 million from GV, Nvidia and Accel to develop, in 2026, interactive audio-visual avatar "agents" that converse on topic, for automated sales training and corporate communications, such as recruiting. == Recognition == In 2021, Synthesia partnered with Lay's to create the Messi Messages campaign featuring Argentine footballer Lionel Messi. Users created personalized messages with Synthesia's software and sent custom artificial reality video messages from Messi based on their text input. The campaign received a Cannes Lion Award under the Bronze category. In February 2025, UK Science and Technology Minister Peter Kyle commended Synthesia's "pioneering generative AI innovations."
LPBoost
Linear Programming Boosting (LPBoost) is a supervised classifier from the boosting family of classifiers. LPBoost maximizes a margin between training samples of different classes, and thus also belongs to the class of margin classifier algorithms. Consider a classification function f : X → { − 1 , 1 } , {\displaystyle f:{\mathcal {X}}\to \{-1,1\},} which classifies samples from a space X {\displaystyle {\mathcal {X}}} into one of two classes, labelled 1 and -1, respectively. LPBoost is an algorithm for learning such a classification function, given a set of training examples with known class labels. LPBoost is a machine learning technique especially suited for joint classification and feature selection in structured domains. == LPBoost overview == As in all boosting classifiers, the final classification function is of the form f ( x ) = ∑ j = 1 J α j h j ( x ) , {\displaystyle f({\boldsymbol {x}})=\sum _{j=1}^{J}\alpha _{j}h_{j}({\boldsymbol {x}}),} where α j {\displaystyle \alpha _{j}} are non-negative weightings for weak classifiers h j : X → { − 1 , 1 } {\displaystyle h_{j}:{\mathcal {X}}\to \{-1,1\}} . Each individual weak classifier h j {\displaystyle h_{j}} may be just a little bit better than random, but the resulting linear combination of many weak classifiers can perform very well. LPBoost constructs f {\displaystyle f} by starting with an empty set of weak classifiers. Iteratively, a single weak classifier to add to the set of considered weak classifiers is selected, added and all the weights α {\displaystyle {\boldsymbol {\alpha }}} for the current set of weak classifiers are adjusted. This is repeated until no weak classifiers to add remain. The property that all classifier weights are adjusted in each iteration is known as totally-corrective property. Early boosting methods, such as AdaBoost do not have this property and converge slower. == Linear program == More generally, let H = { h ( ⋅ ; ω ) | ω ∈ Ω } {\displaystyle {\mathcal {H}}=\{h(\cdot ;\omega )|\omega \in \Omega \}} be the possibly infinite set of weak classifiers, also termed hypotheses. One way to write down the problem LPBoost solves is as a linear program with infinitely many variables. The primal linear program of LPBoost, optimizing over the non-negative weight vector α {\displaystyle {\boldsymbol {\alpha }}} , the non-negative vector ξ {\displaystyle {\boldsymbol {\xi }}} of slack variables and the margin ρ {\displaystyle \rho } is the following. min α , ξ , ρ − ρ + D ∑ n = 1 ℓ ξ n sb.t. ∑ ω ∈ Ω y n α ω h ( x n ; ω ) + ξ n ≥ ρ , n = 1 , … , ℓ , ∑ ω ∈ Ω α ω = 1 , ξ n ≥ 0 , n = 1 , … , ℓ , α ω ≥ 0 , ω ∈ Ω , ρ ∈ R . {\displaystyle {\begin{array}{cl}{\underset {{\boldsymbol {\alpha }},{\boldsymbol {\xi }},\rho }{\min }}&-\rho +D\sum _{n=1}^{\ell }\xi _{n}\\{\textrm {sb.t.}}&\sum _{\omega \in \Omega }y_{n}\alpha _{\omega }h({\boldsymbol {x}}_{n};\omega )+\xi _{n}\geq \rho ,\qquad n=1,\dots ,\ell ,\\&\sum _{\omega \in \Omega }\alpha _{\omega }=1,\\&\xi _{n}\geq 0,\qquad n=1,\dots ,\ell ,\\&\alpha _{\omega }\geq 0,\qquad \omega \in \Omega ,\\&\rho \in {\mathbb {R} }.\end{array}}} Note the effects of slack variables ξ ≥ 0 {\displaystyle {\boldsymbol {\xi }}\geq 0} : their one-norm is penalized in the objective function by a constant factor D {\displaystyle D} , which—if small enough—always leads to a primal feasible linear program. Here we adopted the notation of a parameter space Ω {\displaystyle \Omega } , such that for a choice ω ∈ Ω {\displaystyle \omega \in \Omega } the weak classifier h ( ⋅ ; ω ) : X → { − 1 , 1 } {\displaystyle h(\cdot ;\omega ):{\mathcal {X}}\to \{-1,1\}} is uniquely defined. When the above linear program was first written down in early publications about boosting methods it was disregarded as intractable due to the large number of variables α {\displaystyle {\boldsymbol {\alpha }}} . Only later it was discovered that such linear programs can indeed be solved efficiently using the classic technique of column generation. === Column generation for LPBoost === In a linear program a column corresponds to a primal variable. Column generation is a technique to solve large linear programs. It typically works in a restricted problem, dealing only with a subset of variables. By generating primal variables iteratively and on-demand, eventually the original unrestricted problem with all variables is recovered. By cleverly choosing the columns to generate the problem can be solved such that while still guaranteeing the obtained solution to be optimal for the original full problem, only a small fraction of columns has to be created. ==== LPBoost dual problem ==== Columns in the primal linear program corresponds to rows in the dual linear program. The equivalent dual linear program of LPBoost is the following linear program. max λ , γ γ sb.t. ∑ n = 1 ℓ y n h ( x n ; ω ) λ n + γ ≤ 0 , ω ∈ Ω , 0 ≤ λ n ≤ D , n = 1 , … , ℓ , ∑ n = 1 ℓ λ n = 1 , γ ∈ R . {\displaystyle {\begin{array}{cl}{\underset {{\boldsymbol {\lambda }},\gamma }{\max }}&\gamma \\{\textrm {sb.t.}}&\sum _{n=1}^{\ell }y_{n}h({\boldsymbol {x}}_{n};\omega )\lambda _{n}+\gamma \leq 0,\qquad \omega \in \Omega ,\\&0\leq \lambda _{n}\leq D,\qquad n=1,\dots ,\ell ,\\&\sum _{n=1}^{\ell }\lambda _{n}=1,\\&\gamma \in \mathbb {R} .\end{array}}} For linear programs the optimal value of the primal and dual problem are equal. For the above primal and dual problems, the optimal value is equal to the negative 'soft margin'. The soft margin is the size of the margin separating positive from negative training instances minus positive slack variables that carry penalties for margin-violating samples. Thus, the soft margin may be positive although not all samples are linearly separated by the classification function. The latter is called the 'hard margin' or 'realized margin'. ==== Convergence criterion ==== Consider a subset of the satisfied constraints in the dual problem. For any finite subset we can solve the linear program and thus satisfy all constraints. If we could prove that of all the constraints which we did not add to the dual problem no single constraint is violated, we would have proven that solving our restricted problem is equivalent to solving the original problem. More formally, let γ ∗ {\displaystyle \gamma ^{}} be the optimal objective function value for any restricted instance. Then, we can formulate a search problem for the 'most violated constraint' in the original problem space, namely finding ω ∗ ∈ Ω {\displaystyle \omega ^{}\in \Omega } as ω ∗ = argmax ω ∈ Ω ∑ n = 1 ℓ y n h ( x n ; ω ) λ n . {\displaystyle \omega ^{}={\underset {\omega \in \Omega }{\textrm {argmax}}}\sum _{n=1}^{\ell }y_{n}h({\boldsymbol {x}}_{n};\omega )\lambda _{n}.} That is, we search the space H {\displaystyle {\mathcal {H}}} for a single decision stump h ( ⋅ ; ω ∗ ) {\displaystyle h(\cdot ;\omega ^{})} maximizing the left hand side of the dual constraint. If the constraint cannot be violated by any choice of decision stump, none of the corresponding constraint can be active in the original problem and the restricted problem is equivalent. ==== Penalization constant ==== D {\displaystyle D} The positive value of penalization constant D {\displaystyle D} has to be found using model selection techniques. However, if we choose D = 1 ℓ ν {\displaystyle D={\frac {1}{\ell \nu }}} , where ℓ {\displaystyle \ell } is the number of training samples and 0 < ν < 1 {\displaystyle 0<\nu <1} , then the new parameter ν {\displaystyle \nu } has the following properties. ν {\displaystyle \nu } is an upper bound on the fraction of training errors; that is, if k {\displaystyle k} denotes the number of misclassified training samples, then k ℓ ≤ ν {\displaystyle {\frac {k}{\ell }}\leq \nu } . ν {\displaystyle \nu } is a lower bound on the fraction of training samples outside or on the margin. == Algorithm == Input: Training set X = { x 1 , … , x ℓ } {\displaystyle X=\{{\boldsymbol {x}}_{1},\dots ,{\boldsymbol {x}}_{\ell }\}} , x i ∈ X {\displaystyle {\boldsymbol {x}}_{i}\in {\mathcal {X}}} Training labels Y = { y 1 , … , y ℓ } {\displaystyle Y=\{y_{1},\dots ,y_{\ell }\}} , y i ∈ { − 1 , 1 } {\displaystyle y_{i}\in \{-1,1\}} Convergence threshold θ ≥ 0 {\displaystyle \theta \geq 0} Output: Classification function f : X → { − 1 , 1 } {\displaystyle f:{\mathcal {X}}\to \{-1,1\}} Initialization Weights, uniform λ n ← 1 ℓ , n = 1 , … , ℓ {\displaystyle \lambda _{n}\leftarrow {\frac {1}{\ell }},\quad n=1,\dots ,\ell } Edge γ ← 0 {\displaystyle \gamma \leftarrow 0} Hypothesis count J ← 1 {\displaystyle J\leftarrow 1} Iterate h ^ ← argmax ω ∈ Ω ∑ n = 1 ℓ y n h ( x n ; ω ) λ n {\displaystyle {\hat {h}}\leftarrow {\underset {\omega \in \Omega }{\textrm {argmax}}}\sum _{n=1}^{\ell }y_{n}h({\boldsymbol {x}}_{n};\omega )\lambda _{n}} if ∑ n = 1 ℓ y n h ^ ( x n ) λ n + γ ≤ θ {\displaystyle \sum _{n=1}^{\ell }y_{n}{\hat {h}}({\boldsymbol {x}}_{n})\lambda _{n}+\gamma \leq \theta } then break h J ← h ^ {\displaystyle h_{J}\leftarrow {\hat {h}}} J
Detrended correspondence analysis
Detrended correspondence analysis (DCA) is a multivariate statistical technique widely used by ecologists to find the main factors or gradients in large, species-rich but usually sparse data matrices that typify ecological community data. DCA is frequently used to suppress artifacts inherent in most other multivariate analyses when applied to gradient data. == History == DCA was created in 1979 by Mark Hill of the United Kingdom's Institute for Terrestrial Ecology (now merged into Centre for Ecology and Hydrology) and implemented in FORTRAN code package called DECORANA (Detrended Correspondence Analysis), a correspondence analysis method. DCA is sometimes erroneously referred to as DECORANA; however, DCA is the underlying algorithm, while DECORANA is a tool implementing it. == Issues addressed == According to Hill and Gauch, DCA suppresses two artifacts inherent in most other multivariate analyses when applied to gradient data. An example is a time-series of plant species colonising a new habitat; early successional species are replaced by mid-successional species, then by late successional ones (see example below). When such data are analysed by a standard ordination such as a correspondence analysis: the ordination scores of the samples will exhibit the 'edge effect', i.e. the variance of the scores at the beginning and the end of a regular succession of species will be considerably smaller than that in the middle, when presented as a graph the points will be seen to follow a horseshoe shaped curve rather than a straight line ('arch effect'), even though the process under analysis is a steady and continuous change that human intuition would prefer to see as a linear trend. Outside ecology, the same artifacts occur when gradient data are analysed (e.g. soil properties along a transect running between 2 different geologies, or behavioural data over the lifespan of an individual) because the curved projection is an accurate representation of the shape of the data in multivariate space. Ter Braak and Prentice (1987, p. 121) cite a simulation study analysing two-dimensional species packing models resulting in a better performance of DCA compared to CA. == Method == DCA is an iterative algorithm that has shown itself to be a highly reliable and useful tool for data exploration and summary in community ecology (Shaw 2003). It starts by running a standard ordination (CA or reciprocal averaging) on the data, to produce the initial horse-shoe curve in which the 1st ordination axis distorts into the 2nd axis. It then divides the first axis into segments (default = 26), and rescales each segment to have mean value of zero on the 2nd axis - this effectively squashes the curve flat. It also rescales the axis so that the ends are no longer compressed relative to the middle, so that 1 DCA unit approximates to the same rate of turnover all the way through the data: the rule of thumb is that 4 DCA units mean that there has been a total turnover in the community. Ter Braak and Prentice (1987, p. 122) warn against the non-linear rescaling of the axes due to robustness issues and recommend using detrending-by-polynomials only. == Drawbacks == No significance tests are available with DCA, although there is a constrained (canonical) version called DCCA in which the axes are forced by Multiple linear regression to correlate optimally with a linear combination of other (usually environmental) variables; this allows testing of a null model by Monte-Carlo permutation analysis. == Example == The example shows an ideal data set: The species data is in rows, samples in columns. For each sample along the gradient, a new species is introduced but another species is no longer present. The result is a sparse matrix. Ones indicate the presence of a species in a sample. Except at the edges each sample contains five species. The plot of the first two axes of the correspondence analysis result on the right hand side clearly shows the disadvantages of this procedure: the edge effect, i.e. the points are clustered at the edges of the first axis, and the arch effect. == Software == An open source implementation of DCA, based on the original FORTRAN code, is available in the vegan R-package.
Farthest-first traversal
In computational geometry, the farthest-first traversal of a compact metric space is a sequence of points in the space, where the first point is selected arbitrarily and each successive point is as far as possible from the set of previously-selected points. The same concept can also be applied to a finite set of geometric points, by restricting the selected points to belong to the set or equivalently by considering the finite metric space generated by these points. For a finite metric space or finite set of geometric points, the resulting sequence forms a permutation of the points, also known as the greedy permutation. Every prefix of a farthest-first traversal provides a set of points that is widely spaced and close to all remaining points. More precisely, no other set of equally many points can be spaced more than twice as widely, and no other set of equally many points can be less than half as far to its farthest remaining point. In part because of these properties, farthest-point traversals have many applications, including the approximation of the traveling salesman problem and the metric k-center problem. They may be constructed in polynomial time, or (for low-dimensional Euclidean spaces) approximated in near-linear time. == Definition and properties == A farthest-first traversal is a sequence of points in a compact metric space, with each point appearing at most once. If the space is finite, each point appears exactly once, and the traversal is a permutation of all of the points in the space. The first point of the sequence may be any point in the space. Each point p after the first must have the maximum possible distance to the set of points earlier than p in the sequence, where the distance from a point to a set is defined as the minimum of the pairwise distances to points in the set. A given space may have many different farthest-first traversals, depending both on the choice of the first point in the sequence (which may be any point in the space) and on ties for the maximum distance among later choices. Farthest-point traversals may be characterized by the following properties. Fix a number k, and consider the prefix formed by the first k points of the farthest-first traversal of any metric space. Let r be the distance between the final point of the prefix and the other points in the prefix. Then this subset has the following two properties: All pairs of the selected points are at distance at least r from each other, and All points of the metric space are at distance at most r from the subset. Conversely any sequence having these properties, for all choices of k, must be a farthest-first traversal. These are the two defining properties of a Delone set, so each prefix of the farthest-first traversal forms a Delone set. == Applications == Rosenkrantz, Stearns & Lewis (1977) used the farthest-first traversal to define the farthest-insertion heuristic for the travelling salesman problem. This heuristic finds approximate solutions to the travelling salesman problem by building up a tour on a subset of points, adding one point at a time to the tour in the ordering given by a farthest-first traversal. To add each point to the tour, one edge of the previous tour is broken and replaced by a pair of edges through the added point, in the cheapest possible way. Although Rosenkrantz et al. prove only a logarithmic approximation ratio for this method, they show that in practice it often works better than other insertion methods with better provable approximation ratios. Later, the same sequence of points was popularized by Gonzalez (1985), who used it as part of greedy approximation algorithms for two problems in clustering, in which the goal is to partition a set of points into k clusters. One of the two problems that Gonzalez solve in this way seeks to minimize the maximum diameter of a cluster, while the other, known as the metric k-center problem, seeks to minimize the maximum radius, the distance from a chosen central point of a cluster to the farthest point from it in the same cluster. For instance, the k-center problem can be used to model the placement of fire stations within a city, in order to ensure that every address within the city can be reached quickly by a fire truck. For both clustering problems, Gonzalez chooses a set of k cluster centers by selecting the first k points of a farthest-first traversal, and then creates clusters by assigning each input point to the nearest cluster center. If r is the distance from the set of k selected centers to the next point at position k + 1 in the traversal, then with this clustering every point is within distance r of its center and every cluster has diameter at most 2r. However, the subset of k centers together with the next point are all at distance at least r from each other, and any k-clustering would put some two of these points into a single cluster, with one of them at distance at least r/2 from its center and with diameter at least r. Thus, Gonzalez's heuristic gives an approximation ratio of 2 for both clustering problems. Gonzalez's heuristic was independently rediscovered for the metric k-center problem by Dyer & Frieze (1985), who applied it more generally to weighted k-center problems. Another paper on the k-center problem from the same time, Hochbaum & Shmoys (1985), achieves the same approximation ratio of 2, but its techniques are different. Nevertheless, Gonzalez's heuristic, and the name "farthest-first traversal", are often incorrectly attributed to Hochbaum and Shmoys. For both the min-max diameter clustering problem and the metric k-center problem, these approximations are optimal: the existence of a polynomial-time heuristic with any constant approximation ratio less than 2 would imply that P = NP. As well as for clustering, the farthest-first traversal can also be used in another type of facility location problem, the max-min facility dispersion problem, in which the goal is to choose the locations of k different facilities so that they are as far apart from each other as possible. More precisely, the goal in this problem is to choose k points from a given metric space or a given set of candidate points, in such a way as to maximize the minimum pairwise distance between the selected points. Again, this can be approximated by choosing the first k points of a farthest-first traversal. If r denotes the distance of the kth point from all previous points, then every point of the metric space or the candidate set is within distance r of the first k − 1 points. By the pigeonhole principle, some two points of the optimal solution (whatever it is) must both be within distance r of the same point among these first k − 1 chosen points, and (by the triangle inequality) within distance 2r of each other. Therefore, the heuristic solution given by the farthest-first traversal is within a factor of two of optimal. Other applications of the farthest-first traversal include color quantization (clustering the colors in an image to a smaller set of representative colors), progressive scanning of images (choosing an order to display the pixels of an image so that prefixes of the ordering produce good lower-resolution versions of the whole image rather than filling in the image from top to bottom), point selection in the probabilistic roadmap method for motion planning, simplification of point clouds, generating masks for halftone images, hierarchical clustering, finding the similarities between polygon meshes of similar surfaces, choosing diverse and high-value observation targets for underwater robot exploration, fault detection in sensor networks, modeling phylogenetic diversity, matching vehicles in a heterogenous fleet to customer delivery requests, uniform distribution of geodetic observatories on the Earth's surface or of other types of sensor network, generation of virtual point lights in the instant radiosity computer graphics rendering method, and geometric range searching data structures. == Algorithms == === Greedy exact algorithm === The farthest-first traversal of a finite point set may be computed by a greedy algorithm that maintains the distance of each point from the previously selected points, performing the following steps: Initialize the sequence of selected points to the empty sequence, and the distances of each point to the selected points to infinity. While not all points have been selected, repeat the following steps: Scan the list of not-yet-selected points to find a point p that has the maximum distance from the selected points. Remove p from the not-yet-selected points and add it to the end of the sequence of selected points. For each remaining not-yet-selected point q, replace the distance stored for q by the minimum of its old value and the distance from p to q. For a set of n points, this algorithm takes O(n2) steps and O(n2) distance computations. === Approximations === A faster approximation algorithm, given by Har-Peled & Mendel (2006), applie
Vinted
Vinted Group UAB is a Lithuanian technology company best known for its online marketplace Vinted. Vinted is the leading second-hand fashion marketplace in Europe and a go-to destination for all kinds of second-hand items. According to the company, its mission is to make second-hand the first choice worldwide. The company operates as an ecosystem of businesses, including the Vinted Marketplace (its peer-to-peer resale platform), Vinted Go (logistics and shipping services), Vinted Pay (in-app payment solutions), and Vinted Ventures (an investment arm supporting the circular economy). Headquartered in Vilnius, Lithuania, it also has offices in Germany and the Netherlands and employs more than 2,200 people. == History == Vinted was co-founded in 2008 by Milda Mitkute and Justas Janauskas in Vilnius, Lithuania. The idea originated when Mitkute was moving house and wanted a way to sell clothes she no longer needed. Janauskas helped her create a website where users could trade clothing items. In 2016, Dutch entrepreneur Thomas Plantenga joined Vinted as a strategy consultant and later became Chief Executive Officer, leading the company through a period of international growth. In 2019, Vinted became Lithuania’s first technology unicorn after raising €128 million at a €1 billion valuation in a funding round led by Lightspeed Venture Partners. In October 2020, it acquired United Wardrobe, a Dutch competitor, and in November 2020 German Kleiderkreisel and Mamikreisel were officially merged into the Vinted platform. In 2024 it acquired Trendsales, a Danish resale platform. According to Vogue Business, Vinted’s revenue grew 61% between 2022 and 2023 and the company posted a net profit of €17.8 million in 2023. Usage of Vinted in the UK has grown from 1.2 million users in 2021, to 8 million in 2023. In 2024, the group reported consolidated revenue of €813.4 million (up 36% from 2023) and a net profit of €76.7 million, up 330% from 2023. As of 2024, Vinted was valued at approximately €5 billion, operating in more than 26 markets worldwide and announcing plans to launch in Ireland, Greece, Latvia, Slovenia, and Estonia in 2025. As of 2025 the company employed more than 2,200 people. In April 2026, Vinted completed a secondary share transaction of €880m, valuing the company at €8bn. == Products and operations == Vinted primarily resells clothing but now supports multiple categories including homeware, kidswear, electronics, books, collectibles, and high-value fashion. Vinted has worked with public figures such as Paul Mescal and Alexa Chung on exclusive wardrobe sales and has also partnered directly with charities including Oxfam on initiatives which promote the social and environmental value of second-hand fashion, such as the Style for Change fashion show at London Fashion Week. In 2025, Vinted produced its first television format, the second-hand fashion competition series RE/Style, hosted by Emma Willis. The show features emerging fashion designers from across Europe creating runway-ready looks from second-hand garments and aired on Prime Video UK. In 2025, Vinted was reported as France’s top clothing retailer by sales volume. == Criticism == Vinted has faced scrutiny from European data protection authorities in France, Lithuania, and Poland following complaints regarding GDPR compliance and account blocking practices. In July 2024, the Lithuanian authority fined the company €2,375,276. The case was coordinated by a dedicated Vinted Working Group under the European Data Protection Board. In early 2024, Swedish police reported around 300 fraud cases linked to the platform, in which users’ bank accounts were targeted by scammers. In October 2024, Channel 4 in the United Kingdom aired a documentary examining safety and privacy concerns related to the platform, including the sexualisation of underage users’ images and risks associated with second-hand baby products lacking safety certification. In November 2025, BBC News reported that Vinted’s update to its sizing system in the United Kingdom led to widespread user criticism. Vinted said the update was intended to standardise sizing across international brands.
Lattice Miner
Lattice Miner is a formal concept analysis software tool for the construction, visualization and manipulation of concept lattices. It allows the generation of formal concepts and association rules as well as the transformation of formal contexts via apposition, subposition, reduction and object/attribute generalization, and the manipulation of concept lattices via approximation, projection and selection. Lattice Miner allows also the drawing of nested line diagrams. == Introduction == Formal concept analysis (FCA) is a branch of applied mathematics based on the formalization of concept and concept hierarchy and mainly used as a framework for conceptual clustering and rule mining. Over the last two decades, a collection of tools have emerged to help FCA users visualize and analyze concept lattices. They range from the earliest DOS-based implementations (e.g., ConImp and GLAD) to more recent implementations in Java like ToscanaJ, Galicia, ConExp and Coron. A main issue in the development of FCA tools is to visualize large concept lattices and provide efficient mechanisms to highlight patterns (e.g., concepts, associations) that could be relevant to the user. The initial objective of the FCA tool called Lattice Miner was to focus on visualization mechanisms for the representation of concept lattices, including nested line diagrams. Later on, many other interesting features were integrated into the tool. == Functional architecture of Lattice Miner == Lattice Miner is a Java-based platform whose functions are articulated around a core. The Lattice Miner core provides all low-level operations and structures for the representation and manipulation of contexts, lattices and association rules. Mainly, the core of Lattice Miner consists of three modules: context, concept and association rule modules. The user interface offers a context editor and concept lattice manipulator to assist the user in a set of tasks. The architecture of Lattice Miner is open and modular enough to allow the integration of new features and facilities in each one of its components. === Context module === The context module offers all the basic operations and structures to manipulate binary and valued contexts as well as context decomposition to produce nested line diagrams. Basic context operations include apposition, subposition, generalization, clarification, reduction as well as the complementary context computation. The module provides also the arrow relations (for context reduction and decomposition) [2]. The tool has an input LMB format and recognizes the binary format SLF found in Galicia and the format CEX produced by ConExp. === Concept module === The main function of the concept module is to generate the concepts of the current binary context and construct the corresponding lattice and nested structure (see Figures 2 and 3). It provides the user with basic operators such as projection, selection, and exact search as well as advanced features like pair approximation. Some known algorithms are included in this module such as Bordat’s procedure, Godin’s algorithm and NextClosure algorithm. The approximation feature implemented in Lattice Miner is based on the following idea: given a pair (X,Y) where X ⊆ G, and Y ⊆ M, is there a set of formal concepts (Ai,Bi) which are “close to” (X,Y)? To answer this question, The tool starts to identify the type of couple that the pair (X,Y) represents. It can be a formal concept, a protoconcept, a semiconcept or a preconcept. In the last case, the approximation is given by the interval [(X",X′),(Y′,Y")] and highlighted in the line diagram. === Association rule module === This module includes procedures for computing the (stem) Guigues–Duquenne base using NextClosure algorithm [3], as well as the generic and informative bases. Implications with negation can be obtained using the apposition of a context and its complementary. This module embeds also procedures for the computation of a non-redundant family C of implications and the closure of a set Y of attributes for the given implication set C. === User interface === The initial objective of Lattice Miner was to focus on lattice drawing and visualization either as a flat or nested structure by taking into account the cognitive process of human beings and known principles for lattice drawing (e.g., reducing the number of edge intersections, ensuring diagram symmetry). Some well-known visualization techniques were implemented such as focus & context and fisheye view. The basic idea behind focus & context visualization paradigm is to allow a viewer to see key (important) objects in full detail in the foreground (focus) while at the same time an overview of all the surrounding information (context) remains available in the background. Lattice Miner translates the focus & context paradigm into clear and blurred elements while the size of nodes and the intensity of their color were used to indicate their importance. Various forms of highlighting, labelling and animation are also provided. In order to better handle the display of large lattices, nested line diagrams are offered in the tool. Figure 3 shows the third level of the nested line diagram corresponding to the binary context of Figure 1 where three levels of nesting are defined. Each one of the inner nodes of this diagram represents a combination of attributes from the previous two (outer) levels. Real inner concepts (see the node on the left hand-side of the diagram) are identified by colored nodes while void elements are in grey color. Each node of levels 1 and 2 can be expanded to exhibit its internal line diagram. Both flat and nested diagrams can be saved as an image. Simple (flat) lattices can also be saved as an XML format file.