Transkribus is a platform for the text recognition, image analysis and structure recognition of historical documents. The platform was created in the context of the two EU projects "tranScriptorium" (2013–2015) and "READ" (Recognition and Enrichment of Archival Documents – 2016–2019). It was developed by the University of Innsbruck. Since July 1, 2019 the platform has been directed and further developed by the READ-COOP, a non-profit cooperative. The platform integrates tools developed by research groups throughout Europe, including the Pattern Recognition and Human Language Technology (PRHLT) group of the Technical University of Valencia and the Computational Intelligence Technology Lab (CITlab) group of University of Rostock. Comparable programs that offer similar functions are eScriptorium and OCR4All.
Image warping
Image warping is the process of digitally manipulating an image such that any shapes portrayed in the image have been significantly distorted. Warping may be used for correcting image distortion as well as for creative purposes (e.g., morphing). The same techniques are equally applicable to video. While an image can be transformed in various ways, pure warping means that points are mapped to points without changing the colors. This can be based mathematically on any function from (part of) the plane to the plane. If the function is injective the original can be reconstructed. If the function is a bijection any image can be inversely transformed. Some methods are: Images may be distorted through simulation of optical aberrations. Images may be viewed as if they had been projected onto a curved or mirrored surface. (This is often seen in ray traced images.) Images can be partitioned into image polygons and each polygon distorted. Images can be distorted using morphing. The most obvious approach to transforming a digital image is the forward mapping. This applies the transform directly to the source image, typically generating unevenly-spaced points that will then be interpolated to generate the required regularly-spaced pixels. However, for injective transforms reverse mapping is also available. This applies the inverse transform to the target pixels to find the unevenly-spaced locations in the source image that contribute to them. Estimating them from source image pixels will require interpolation of the source image. To work out what kind of warping has taken place between consecutive images, one can use optical flow estimation techniques. == Image warping toolbox == ImWIP is an open-source, image warping tool for modeling deformation and motion in digital images, which contains differentiable image warping operators, together with their exact adjoints and derivatives.
Emotion Markup Language
An Emotion Markup Language (EML or EmotionML) has first been defined by the W3C Emotion Incubator Group (EmoXG) as a general-purpose emotion annotation and representation language, which should be usable in a large variety of technological contexts where emotions need to be represented. Emotion-oriented computing (or "affective computing") is gaining importance as interactive technological systems become more sophisticated. Representing the emotional states of a user or the emotional states to be simulated by a user interface requires a suitable representation format; in this case a markup language is used. EmotionML version 1.0 was published by the group in May 2014. == Example == Here is an example of an EmotionML document describing emotions expressed in a video recording of the interaction between a teacher, Alice, and a student, Bob. == History == In 2006, a first W3C Incubator Group, the Emotion Incubator Group (EmoXG), was set up "to investigate a language to represent the emotional states of users and the emotional states simulated by user interfaces" with the final Report published on 10 July 2007. In 2007, the Emotion Markup Language Incubator Group (EmotionML XG) was set up as a follow-up to the Emotion Incubator Group, "to propose a specification draft for an Emotion Markup Language, to document it in a way accessible to non-experts, and to illustrate its use in conjunction with a number of existing markups." The final report of the Emotion Markup Language Incubator Group, Elements of an EmotionML 1.0, was published on 20 November 2008. The work then was continued in 2009 in the frame of the W3C's Multimodal Interaction Activity, with the First Public Working Draft of "Emotion Markup Language (EmotionML) 1.0" being published on 29 October 2009. The Last Call Working Draft of "Emotion Markup Language 1.0", was published on 7 April 2011. The Last Call Working Draft addressed all open issues that arose from feedback of the community on the First Call Working Draft as well as results of a workshop held in Paris in October 2010. Along with the Last Call Working Draft, a list of vocabularies for EmotionML has been published to aid developers using common vocabularies for annotating or representing emotions. Annual draft updates were published until the 1.0 version was finished in 2014. == Reasons for defining an emotion markup language == A standard for an emotion markup language would be useful for the following purposes: To enhance computer-mediated human-human or human-machine communication. Emotions are a basic part of human communication and should therefore be taken into account, e.g. in emotional Chat systems or emphatic voice boxes. This involves specification, analysis and display of emotion related states. To enhance systems' processing efficiency. Emotion and intelligence are strongly interconnected. The modeling of human emotions in computer processing can help to build more efficient systems, e.g. using emotional models for time-critical decision enforcement. To allow the analysis of non-verbal behavior, emotion, mental states that can be provided using web services to enable data collection, analysis, and reporting. Concrete examples of existing technology that could apply EmotionML include: Opinion mining / sentiment analysis in Web 2.0, to automatically track customer's attitude regarding a product across blogs; Affective monitoring, such as ambient assisted living applications, fear detection for surveillance purposes, or using wearable sensors to test customer satisfaction; Wellness technologies that provide assistance according to a person's emotional state with the goal to improve the person's well-being; Character design and control for games and virtual worlds; Building web services to capture, analysis, and report data of non-verbal behavior, emotion and mental states of an individual or group across the internet using standard web technologies such as HTML5 and JSON. Social robots, such as guide robots engaging with visitors; Expressive speech synthesis, generating synthetic speech with different emotions, such as happy or sad, friendly or apologetic; expressive synthetic speech would for example make more information available to blind and partially sighted people, and enrich their experience of the content; Emotion recognition (e.g., for spotting angry customers in speech dialog systems, to improve computer games or e-Learning applications); Support for people with disabilities, such as educational programs for people with autism. EmotionML can be used to make the emotional intent of content explicit. This would enable people with learning disabilities (such as Asperger syndrome) to realise the emotional context of the content; EmotionML can be used for media transcripts and captions. Where emotions are marked up to help deaf or hearing impaired people who cannot hear the soundtrack, more information is made available to enrich their experience of the content. The Emotion Incubator Group has listed 39 individual use cases for an Emotion markup language. A standardised way to mark up the data needed by such "emotion-oriented systems" has the potential to boost development primarily because data that was annotated in a standardised way can be interchanged between systems more easily, thereby simplifying a market for emotional databases, and the standard can be used to ease a market of providers for sub-modules of emotion processing systems, e.g. a web service for the recognition of emotion from text, speech or multi-modal input. == The challenge of defining a generally usable emotion markup language == Any attempt to standardize the description of emotions using a finite set of fixed descriptors is doomed to failure, as there is no consensus on the number of relevant emotions, on the names that should be given to them or how else best to describe them. For example, the difference between ":)" and "(:" is small, but using a standardized markup it would make one invalid. Even more basically, the list of emotion-related states that should be distinguished varies depending on the application domain and the aspect of emotions to be focused. Basically, the vocabulary needed depends on the context of use. On the other hand, the basic structure of concepts is less controversial: it is generally agreed that emotions involve triggers, appraisals, feelings, expressive behavior including physiological changes, and action tendencies; emotions in their entirety can be described in terms of categories or a small number of dimensions; emotions have an intensity, and so on. For details, see the Scientific Descriptions of Emotions in the Final Report of the Emotion Incubator Group. Given this lack of agreement on descriptors in the field, the only practical way of defining an emotion markup language is the definition of possible structural elements and to allow users to "plug in" vocabularies that they consider appropriate for their work. An additional challenge lies in the aim to provide a markup language that is generally usable. The requirements that arise from different use cases are rather different. Whereas manual annotation tends to require all the fine-grained distinctions considered in the scientific literature, automatic recognition systems can usually distinguish only a very small number of different states and affective avatars need yet another level of detail for expressing emotions in an appropriate way. For the reasons outlined here, it is clear that there is an inevitable tension between flexibility and interoperability, which need to be weighed in the formulation of an EmotionML. The guiding principle in the following specification has been to provide a choice only where it is needed, and to propose reasonable default options for every choice. == Applications and web services benefiting from an emotion markup language == There are a range of existing projects and applications to which an emotion markup language will enable the building of webservices to measure capture data of individuals non-verbal behavior, mental states, and emotions and allowing results to be reported and rendered in a standardized format using standard web technologies such as JSON and HTML5. One such project is measuring affect data across the Internet using EyesWeb.
Ubiquitous computing
Ubiquitous computing (or "ubicomp") is a concept in software engineering, hardware engineering and computer science where computing is made to appear seamlessly anytime and everywhere. In contrast to desktop computing, ubiquitous computing implies use on any device, in any location, and in any format. A user interacts with the computer, which can exist in many different forms, including laptop computers, tablets, smart phones and terminals in everyday objects such as a refrigerator or a pair of glasses. The underlying technologies to support ubiquitous computing include the Internet, advanced middleware, kernels, operating systems, mobile codes, sensors, microprocessors, new I/Os and user interfaces, computer networks, mobile protocols, global navigational systems, and new materials. This paradigm is also described as pervasive computing, ambient intelligence, or "everyware". Each term emphasizes slightly different aspects. When primarily concerning the objects involved, it is also known as physical computing, the Internet of Things, haptic computing, and "things that think". Rather than propose a single definition for ubiquitous computing and for these related terms, a taxonomy of properties for ubiquitous computing has been proposed, from which different kinds or flavors of ubiquitous systems and applications can be described. Ubiquitous computing themes include: distributed computing, mobile computing, location computing, mobile networking, sensor networks, human–computer interaction, context-aware smart home technologies, and artificial intelligence. == Core concepts == Ubiquitous computing is the concept of using small internet connected and inexpensive computers to help with everyday functions in an automated fashion. Mark Weiser proposed three basic forms for ubiquitous computing devices: Tabs: a wearable device that is approximately a centimeter in size Pads: a hand-held device that is approximately a decimeter in size Boards: an interactive larger display device that is approximately a meter in size Ubiquitous computing devices proposed by Mark Weiser are all based around flat devices of different sizes with a visual display. These conceptual device categories were later implemented at Xerox PARC in experimental systems including the PARCTab, PARCPad, and LiveBoard, which served as early prototypes of handheld, tablet-style, and large interactive display computing environments. Expanding beyond those concepts there is a large array of other ubiquitous computing devices that could exist. == History == Mark Weiser coined the phrase "ubiquitous computing" around 1988, during his tenure as Chief Technologist of the Xerox Palo Alto Research Center (PARC). Both alone and with PARC Director and Chief Scientist John Seely Brown, Weiser wrote some of the earliest papers on the subject, largely defining it and sketching out its major concerns. == Recognizing the effects of extending processing power == Recognizing that the extension of processing power into everyday scenarios would necessitate understandings of social, cultural and psychological phenomena beyond its proper ambit, Weiser was influenced by many fields outside computer science, including "philosophy, phenomenology, anthropology, psychology, post-Modernism, sociology of science and feminist criticism". He was explicit about "the humanistic origins of the 'invisible ideal in post-modernist thought'", referencing as well the ironically dystopian Philip K. Dick novel Ubik. Andy Hopper from Cambridge University UK proposed and demonstrated the concept of "Teleporting" – where applications follow the user wherever he/she moves. Roy Want (now at Google), while at Olivetti Research Ltd, designed the first "Active Badge System", which is an advanced location computing system where personal mobility is merged with computing. Later at Xerox PARC, he designed and built the "PARCTab" or simply "Tab", widely recognized as the world's first Context-Aware computer, which has great similarity to the modern smartphone. Bill Schilit (now at Google) also did some earlier work in this topic, and participated in the early Mobile Computing workshop held in Santa Cruz in 1996. Ken Sakamura of the University of Tokyo, Japan leads the Ubiquitous Networking Laboratory (UNL), Tokyo as well as the T-Engine Forum. The joint goal of Sakamura's Ubiquitous Networking specification and the T-Engine forum, is to enable any everyday device to broadcast and receive information. MIT has also contributed significant research in this field, notably Things That Think consortium (directed by Hiroshi Ishii, Joseph A. Paradiso and Rosalind Picard) at the Media Lab and the CSAIL effort known as Project Oxygen. Other major contributors include University of Washington (Shwetak Patel, Anind Dey and James Landay), Dartmouth College's HealthX Lab (directed by Andrew Campbell), Georgia Tech's College of Computing (Gregory Abowd and Thad Starner), Cornell Tech's People Aware Computing Lab (directed by Tanzeem Choudhury), NYU's Interactive Telecommunications Program, UC Irvine's Department of Informatics, Microsoft Research, Intel Research and Equator, Ajou University UCRi & CUS. == Examples == One of the earliest ubiquitous systems was artist Natalie Jeremijenko's "Live Wire", also known as "Dangling String", installed at Xerox PARC during Mark Weiser's time there. This was a piece of string attached to a stepper motor and controlled by a LAN connection; network activity caused the string to twitch, yielding a peripherally noticeable indication of traffic. Weiser called this an example of calm technology. A present manifestation of this trend is the widespread diffusion of mobile phones. Many mobile phones support high speed data transmission, video services, and other services with powerful computational ability. Although these mobile devices are not necessarily manifestations of ubiquitous computing, there are examples, such as Japan's Yaoyorozu ("Eight Million Gods") Project in which mobile devices, coupled with radio frequency identification tags demonstrate that ubiquitous computing is already present in some form. Ambient Devices has produced an "orb", a "dashboard", and a "weather beacon": these decorative devices receive data from a wireless network and report current events, such as stock prices and the weather, like the Nabaztag, which was invented by Rafi Haladjian and Olivier Mével, and manufactured by the company Violet. The Australian futurist Mark Pesce has produced a highly configurable 52-LED LAMP enabled lamp which uses Wi-Fi named MooresCloud after Gordon Moore. The Unified Computer Intelligence Corporation launched a device called Ubi – The Ubiquitous Computer designed to allow voice interaction with the home and provide constant access to information. Ubiquitous computing research has focused on building an environment in which computers allow humans to focus attention on select aspects of the environment and operate in supervisory and policy-making roles. Ubiquitous computing emphasizes the creation of a human computer interface that can interpret and support a user's intentions. For example, MIT's Project Oxygen seeks to create a system in which computation is as pervasive as air: In the future, computation will be human centered. It will be freely available everywhere, like batteries and power sockets, or oxygen in the air we breathe...We will not need to carry our own devices around with us. Instead, configurable generic devices, either handheld or embedded in the environment, will bring computation to us, whenever we need it and wherever we might be. As we interact with these "anonymous" devices, they will adopt our information personalities. They will respect our desires for privacy and security. We won't have to type, click, or learn new computer jargon. Instead, we'll communicate naturally, using speech and gestures that describe our intent... This is a fundamental transition that does not seek to escape the physical world and "enter some metallic, gigabyte-infested cyberspace" but rather brings computers and communications to us, making them "synonymous with the useful tasks they perform". Network robots link ubiquitous networks with robots, contributing to the creation of new lifestyles and solutions to address a variety of social problems including the aging of population and nursing care. The "Continuity" set of features, introduced by Apple in OS X Yosemite, can be seen as an example of ubiquitous computing. == Issues == Privacy is easily the most often-cited criticism of ubiquitous computing (ubicomp), and may be the greatest barrier to its long-term success. == Research centres == This is a list of notable institutions who claim to have a focus on Ubiquitous computing sorted by country: Canada Topological Media Lab, Concordia University, Canada Finland Community Imaging Group, University of Oulu, Finland Germany Telecooperation Office (TECO), Karlsruhe Institute of Technology, Ger
NewSQL
NewSQL is a class of relational database management systems that seek to provide the scalability of NoSQL systems for online transaction processing (OLTP) workloads while maintaining the ACID guarantees of a traditional database system. Many enterprise systems that handle high-profile data (e.g., financial and order processing systems) are too large for conventional relational databases, but have transactional and consistency requirements that are not practical for NoSQL systems. The only options previously available for these organizations were to either purchase more powerful computers or to develop custom middleware that distributes requests over conventional DBMS. Both approaches feature high infrastructure costs and/or development costs. NewSQL systems attempt to reconcile the conflicts. == History == The term was first used by 451 Group analyst Matthew Aslett in a 2011 research paper discussing the rise of a new generation of database management systems. One of the first NewSQL systems was the H-Store parallel database system. == Applications == Typical applications are characterized by heavy OLTP transaction volumes. OLTP transactions; are short-lived (i.e., no user stalls) touch small amounts of data per transaction use indexed lookups (no table scans) have a small number of forms (a small number of queries with different arguments). However, some support hybrid transactional/analytical processing (HTAP) applications. Such systems improve performance and scalability by omitting heavyweight recovery or concurrency control. == List of NewSQL-databases == Apache Trafodion Clustrix CockroachDB Couchbase CrateDB Google Spanner MySQL Cluster NuoDB OceanBase Pivotal GemFire XD SequoiaDB SingleStore was formerly known as MemSQL. TIBCO Active Spaces TiDB TokuDB TransLattice Elastic Database VoltDB YDB YugabyteDB == Features == The two common distinguishing features of NewSQL database solutions are that they support online scalability of NoSQL databases and the relational data model (including ACID consistency) using SQL as their primary interface. NewSQL systems can be loosely grouped into three categories: === New architectures === NewSQL systems adopt various internal architectures. Some systems employ a cluster of shared-nothing nodes, in which each node manages a subset of the data. They include components such as distributed concurrency control, flow control, and distributed query processing. === SQL engines === The second category are optimized storage engines for SQL. These systems provide the same programming interface as SQL, but scale better than built-in engines. === Transparent sharding === These systems automatically split databases across multiple nodes using Raft or Paxos consensus algorithm.
Inception score
The Inception Score (IS) is an algorithm used to assess the quality of images created by a generative image model such as a generative adversarial network (GAN). The score is calculated based on the output of a separate, pretrained Inception v3 image classification model applied to a sample of (typically around 30,000) images generated by the generative model. The Inception Score is maximized when the following conditions are true: The entropy of the distribution of labels predicted by the Inceptionv3 model for the generated images is minimized. In other words, the classification model confidently predicts a single label for each image. Intuitively, this corresponds to the desideratum of generated images being "sharp" or "distinct". The predictions of the classification model are evenly distributed across all possible labels. This corresponds to the desideratum that the output of the generative model is "diverse". It has been somewhat superseded by the related Fréchet inception distance. While the Inception Score only evaluates the distribution of generated images, the FID compares the distribution of generated images with the distribution of a set of real images ("ground truth"). == Definition == Let there be two spaces, the space of images Ω X {\displaystyle \Omega _{X}} and the space of labels Ω Y {\displaystyle \Omega _{Y}} . The space of labels is finite. Let p g e n {\displaystyle p_{gen}} be a probability distribution over Ω X {\displaystyle \Omega _{X}} that we wish to judge. Let a discriminator be a function of type p d i s : Ω X → M ( Ω Y ) {\displaystyle p_{dis}:\Omega _{X}\to M(\Omega _{Y})} where M ( Ω Y ) {\displaystyle M(\Omega _{Y})} is the set of all probability distributions on Ω Y {\displaystyle \Omega _{Y}} . For any image x {\displaystyle x} , and any label y {\displaystyle y} , let p d i s ( y | x ) {\displaystyle p_{dis}(y|x)} be the probability that image x {\displaystyle x} has label y {\displaystyle y} , according to the discriminator. It is usually implemented as an Inception-v3 network trained on ImageNet. The Inception Score of p g e n {\displaystyle p_{gen}} relative to p d i s {\displaystyle p_{dis}} is I S ( p g e n , p d i s ) := exp ( E x ∼ p g e n [ D K L ( p d i s ( ⋅ | x ) ‖ ∫ p d i s ( ⋅ | x ) p g e n ( x ) d x ) ] ) {\displaystyle IS(p_{gen},p_{dis}):=\exp \left(\mathbb {E} _{x\sim p_{gen}}\left[D_{KL}\left(p_{dis}(\cdot |x)\|\int p_{dis}(\cdot |x)p_{gen}(x)dx\right)\right]\right)} Equivalent rewrites include ln I S ( p g e n , p d i s ) := E x ∼ p g e n [ D K L ( p d i s ( ⋅ | x ) ‖ E x ∼ p g e n [ p d i s ( ⋅ | x ) ] ) ] {\displaystyle \ln IS(p_{gen},p_{dis}):=\mathbb {E} _{x\sim p_{gen}}\left[D_{KL}\left(p_{dis}(\cdot |x)\|\mathbb {E} _{x\sim p_{gen}}[p_{dis}(\cdot |x)]\right)\right]} ln I S ( p g e n , p d i s ) := H [ E x ∼ p g e n [ p d i s ( ⋅ | x ) ] ] − E x ∼ p g e n [ H [ p d i s ( ⋅ | x ) ] ] {\displaystyle \ln IS(p_{gen},p_{dis}):=H[\mathbb {E} _{x\sim p_{gen}}[p_{dis}(\cdot |x)]]-\mathbb {E} _{x\sim p_{gen}}[H[p_{dis}(\cdot |x)]]} ln I S {\displaystyle \ln IS} is nonnegative by Jensen's inequality. Pseudocode:INPUT discriminator p d i s {\displaystyle p_{dis}} . INPUT generator g {\displaystyle g} . Sample images x i {\displaystyle x_{i}} from generator. Compute p d i s ( ⋅ | x i ) {\displaystyle p_{dis}(\cdot |x_{i})} , the probability distribution over labels conditional on image x i {\displaystyle x_{i}} . Sum up the results to obtain p ^ {\displaystyle {\hat {p}}} , an empirical estimate of ∫ p d i s ( ⋅ | x ) p g e n ( x ) d x {\displaystyle \int p_{dis}(\cdot |x)p_{gen}(x)dx} . Sample more images x i {\displaystyle x_{i}} from generator, and for each, compute D K L ( p d i s ( ⋅ | x i ) ‖ p ^ ) {\displaystyle D_{KL}\left(p_{dis}(\cdot |x_{i})\|{\hat {p}}\right)} . Average the results, and take its exponential. RETURN the result. === Interpretation === A higher inception score is interpreted as "better", as it means that p g e n {\displaystyle p_{gen}} is a "sharp and distinct" collection of pictures. ln I S ( p g e n , p d i s ) ∈ [ 0 , ln N ] {\displaystyle \ln IS(p_{gen},p_{dis})\in [0,\ln N]} , where N {\displaystyle N} is the total number of possible labels. ln I S ( p g e n , p d i s ) = 0 {\displaystyle \ln IS(p_{gen},p_{dis})=0} iff for almost all x ∼ p g e n {\displaystyle x\sim p_{gen}} p d i s ( ⋅ | x ) = ∫ p d i s ( ⋅ | x ) p g e n ( x ) d x {\displaystyle p_{dis}(\cdot |x)=\int p_{dis}(\cdot |x)p_{gen}(x)dx} That means p g e n {\displaystyle p_{gen}} is completely "indistinct". That is, for any image x {\displaystyle x} sampled from p g e n {\displaystyle p_{gen}} , discriminator returns exactly the same label predictions p d i s ( ⋅ | x ) {\displaystyle p_{dis}(\cdot |x)} . The highest inception score N {\displaystyle N} is achieved if and only if the two conditions are both true: For almost all x ∼ p g e n {\displaystyle x\sim p_{gen}} , the distribution p d i s ( y | x ) {\displaystyle p_{dis}(y|x)} is concentrated on one label. That is, H y [ p d i s ( y | x ) ] = 0 {\displaystyle H_{y}[p_{dis}(y|x)]=0} . That is, every image sampled from p g e n {\displaystyle p_{gen}} is exactly classified by the discriminator. For every label y {\displaystyle y} , the proportion of generated images labelled as y {\displaystyle y} is exactly E x ∼ p g e n [ p d i s ( y | x ) ] = 1 N {\displaystyle \mathbb {E} _{x\sim p_{gen}}[p_{dis}(y|x)]={\frac {1}{N}}} . That is, the generated images are equally distributed over all labels.
Reservoir sampling
Reservoir sampling is a family of randomized algorithms for choosing a simple random sample, without replacement, of k items from a population of unknown size n in a single pass over the items. The size of the population n is not known to the algorithm and is typically too large for all n items to fit into main memory. The population is revealed to the algorithm over time, and the algorithm cannot look back at previous items. At any point, the current state of the algorithm must permit extraction of a simple random sample without replacement of size k over the part of the population seen so far. == Motivation == Suppose we see a sequence of items, one at a time. We want to keep 10 items in memory, and we want them to be selected at random from the sequence. If we know the total number of items n and can access the items arbitrarily, then the solution is easy: select 10 distinct indices i between 1 and n with equal probability, and keep the i-th elements. The problem is that we do not always know the exact n in advance. == Simple: Algorithm R == A simple and popular but slow algorithm, Algorithm R, was created by Jeffrey Vitter. Initialize an array R {\displaystyle R} indexed from 1 {\displaystyle 1} to k {\displaystyle k} , containing the first k items of the input x 1 , . . . , x k {\displaystyle x_{1},...,x_{k}} . This is the reservoir. For each new input x i {\displaystyle x_{i}} , generate a random number j uniformly in { 1 , . . . , i } {\displaystyle \{1,...,i\}} . If j ∈ { 1 , . . . , k } {\displaystyle j\in \{1,...,k\}} , then set R [ j ] := x i . {\displaystyle R[j]:=x_{i}.} Otherwise, discard x i {\displaystyle x_{i}} . Return R {\displaystyle R} after all inputs are processed. This algorithm works by induction on i ≥ k {\displaystyle i\geq k} . While conceptually simple and easy to understand, this algorithm needs to generate a random number for each item of the input, including the items that are discarded. The algorithm's asymptotic running time is thus O ( n ) {\displaystyle O(n)} . Generating this amount of randomness and the linear run time causes the algorithm to be unnecessarily slow if the input population is large. This is Algorithm R, implemented as follows: == Optimal: Algorithm L == If we generate n {\displaystyle n} random numbers u 1 , . . . , u n ∼ U [ 0 , 1 ] {\displaystyle u_{1},...,u_{n}\sim U[0,1]} independently, then the indices of the smallest k {\displaystyle k} of them is a uniform sample of the k {\displaystyle k} -subsets of { 1 , . . . , n } {\displaystyle \{1,...,n\}} . The process can be done without knowing n {\displaystyle n} : Keep the smallest k {\displaystyle k} of u 1 , . . . , u i {\displaystyle u_{1},...,u_{i}} that has been seen so far, as well as w i {\displaystyle w_{i}} , the index of the largest among them. For each new u i + 1 {\displaystyle u_{i+1}} , compare it with u w i {\displaystyle u_{w_{i}}} . If u i + 1 < u w i {\displaystyle u_{i+1}