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Weak supervision
Weak supervision (also known as semi-supervised learning) is a paradigm in machine learning, the relevance and notability of which increased with the advent of large language models due to the large amount of data required to train them. It is characterized by using a combination of a small amount of human-labeled data (exclusively used in more expensive and time-consuming supervised learning paradigm), followed by a large amount of unlabeled data (used exclusively in unsupervised learning paradigm). In other words, the desired output values are provided only for a subset of the training data. The remaining data is unlabeled or imprecisely labeled. Intuitively, it can be seen as an exam and labeled data as sample problems that the teacher solves for the class as an aid in solving another set of problems. In the transductive setting, these unsolved problems act as exam questions. In the inductive setting, they become practice problems of the sort that will make up the exam. == Problem == The acquisition of labeled data for a learning problem often requires a skilled human agent (e.g. to transcribe an audio segment) or a physical experiment (e.g. determining the 3D structure of a protein or determining whether there is oil at a particular location). The cost associated with the labeling process thus may render large, fully labeled training sets infeasible, whereas acquisition of unlabeled data is relatively inexpensive. In such situations, semi-supervised learning can be of great practical value. Semi-supervised learning is also of theoretical interest in machine learning and as a model for human learning. == Technique == More formally, semi-supervised learning assumes a set of l {\displaystyle l} independently identically distributed examples x 1 , … , x l ∈ X {\displaystyle x_{1},\dots ,x_{l}\in X} with corresponding labels y 1 , … , y l ∈ Y {\displaystyle y_{1},\dots ,y_{l}\in Y} and u {\displaystyle u} unlabeled examples x l + 1 , … , x l + u ∈ X {\displaystyle x_{l+1},\dots ,x_{l+u}\in X} are processed. Semi-supervised learning combines this information to surpass the classification performance that can be obtained either by discarding the unlabeled data and doing supervised learning or by discarding the labels and doing unsupervised learning. Semi-supervised learning may refer to either transductive learning or inductive learning. The goal of transductive learning is to infer the correct labels for the given unlabeled data x l + 1 , … , x l + u {\displaystyle x_{l+1},\dots ,x_{l+u}} only. The goal of inductive learning is to infer the correct mapping from X {\displaystyle X} to Y {\displaystyle Y} . It is unnecessary (and, according to Vapnik's principle, imprudent) to perform transductive learning by way of inferring a classification rule over the entire input space; however, in practice, algorithms formally designed for transduction or induction are often used interchangeably. == Assumptions == In order to make any use of unlabeled data, some relationship to the underlying distribution of data must exist. Semi-supervised learning algorithms make use of at least one of the following assumptions: === Continuity / smoothness assumption === Points that are close to each other are more likely to share a label. This is also generally assumed in supervised learning and yields a preference for geometrically simple decision boundaries. In the case of semi-supervised learning, the smoothness assumption additionally yields a preference for decision boundaries in low-density regions, so few points are close to each other but in different classes. === Cluster assumption === The data tend to form discrete clusters, and points in the same cluster are more likely to share a label (although data that shares a label may spread across multiple clusters). This is a special case of the smoothness assumption and gives rise to feature learning with clustering algorithms. === Manifold assumption === The data lie approximately on a manifold of much lower dimension than the input space. In this case learning the manifold using both the labeled and unlabeled data can avoid the curse of dimensionality. Then learning can proceed using distances and densities defined on the manifold. The manifold assumption is practical when high-dimensional data are generated by some process that may be hard to model directly, but which has only a few degrees of freedom. For instance, human voice is controlled by a few vocal folds, and images of various facial expressions are controlled by a few muscles. In these cases, it is better to consider distances and smoothness in the natural space of the generating problem, rather than in the space of all possible acoustic waves or images, respectively. == History == The heuristic approach of self-training (also known as self-learning or self-labeling) is historically the oldest approach to semi-supervised learning, with examples of applications starting in the 1960s. The transductive learning framework was formally introduced by Vladimir Vapnik in the 1970s. Interest in inductive learning using generative models also began in the 1970s. A probably approximately correct learning bound for semi-supervised learning of a Gaussian mixture was demonstrated by Ratsaby and Venkatesh in 1995. == Methods == === Generative models === Generative approaches to statistical learning first seek to estimate p ( x | y ) {\displaystyle p(x|y)} , the distribution of data points belonging to each class. The probability p ( y | x ) {\displaystyle p(y|x)} that a given point x {\displaystyle x} has label y {\displaystyle y} is then proportional to p ( x | y ) p ( y ) {\displaystyle p(x|y)p(y)} by Bayes' rule. Semi-supervised learning with generative models can be viewed either as an extension of supervised learning (classification plus information about p ( x ) {\displaystyle p(x)} ) or as an extension of unsupervised learning (clustering plus some labels). Generative models assume that the distributions take some particular form p ( x | y , θ ) {\displaystyle p(x|y,\theta )} parameterized by the vector θ {\displaystyle \theta } . If these assumptions are incorrect, the unlabeled data may actually decrease the accuracy of the solution relative to what would have been obtained from labeled data alone. However, if the assumptions are correct, then the unlabeled data necessarily improves performance. The unlabeled data are distributed according to a mixture of individual-class distributions. In order to learn the mixture distribution from the unlabeled data, it must be identifiable, that is, different parameters must yield different summed distributions. Gaussian mixture distributions are identifiable and commonly used for generative models. The parameterized joint distribution can be written as p ( x , y | θ ) = p ( y | θ ) p ( x | y , θ ) {\displaystyle p(x,y|\theta )=p(y|\theta )p(x|y,\theta )} by using the chain rule. Each parameter vector θ {\displaystyle \theta } is associated with a decision function f θ ( x ) = argmax y p ( y | x , θ ) {\displaystyle f_{\theta }(x)={\underset {y}{\operatorname {argmax} }}\ p(y|x,\theta )} . The parameter is then chosen based on fit to both the labeled and unlabeled data, weighted by λ {\displaystyle \lambda } : argmax Θ ( log p ( { x i , y i } i = 1 l | θ ) + λ log p ( { x i } i = l + 1 l + u | θ ) ) {\displaystyle {\underset {\Theta }{\operatorname {argmax} }}\left(\log p(\{x_{i},y_{i}\}_{i=1}^{l}|\theta )+\lambda \log p(\{x_{i}\}_{i=l+1}^{l+u}|\theta )\right)} === Low-density separation === Another major class of methods attempts to place boundaries in regions with few data points (labeled or unlabeled). One of the most commonly used algorithms is the transductive support vector machine, or TSVM (which, despite its name, may be used for inductive learning as well). Whereas support vector machines for supervised learning seek a decision boundary with maximal margin over the labeled data, the goal of TSVM is a labeling of the unlabeled data such that the decision boundary has maximal margin over all of the data. In addition to the standard hinge loss ( 1 − y f ( x ) ) + {\displaystyle (1-yf(x))_{+}} for labeled data, a loss function ( 1 − | f ( x ) | ) + {\displaystyle (1-|f(x)|)_{+}} is introduced over the unlabeled data by letting y = sign f ( x ) {\displaystyle y=\operatorname {sign} {f(x)}} . TSVM then selects f ∗ ( x ) = h ∗ ( x ) + b {\displaystyle f^{}(x)=h^{}(x)+b} from a reproducing kernel Hilbert space H {\displaystyle {\mathcal {H}}} by minimizing the regularized empirical risk: f ∗ = argmin f ( ∑ i = 1 l ( 1 − y i f ( x i ) ) + + λ 1 ‖ h ‖ H 2 + λ 2 ∑ i = l + 1 l + u ( 1 − | f ( x i ) | ) + ) {\displaystyle f^{}={\underset {f}{\operatorname {argmin} }}\left(\displaystyle \sum _{i=1}^{l}(1-y_{i}f(x_{i}))_{+}+\lambda _{1}\|h\|_{\mathcal {H}}^{2}+\lambda _{2}\sum _{i=l+1}^{l+u}(1-|f(x_{i})|)_{+}\right)} An exact solution is intractable due to the non-convex term ( 1 − | f ( x ) | ) + {\displayst
Algorithmic game theory
Algorithmic game theory (AGT) is an interdisciplinary field at the intersection of game theory and computer science, focused on understanding and designing algorithms for environments where multiple strategic agents interact. This research area combines computational thinking with economic principles to address challenges that emerge when algorithmic inputs come from self-interested participants. In traditional algorithm design, inputs are assumed to be fixed and reliable. However, in many real-world applications—such as online auctions, internet routing, digital advertising, and resource allocation systems—inputs are provided by multiple independent agents who may strategically misreport information to manipulate outcomes in their favor. AGT provides frameworks to analyze and design systems that remain effective despite such strategic behavior. The field can be approached from two complementary perspectives: Analysis: Evaluating existing algorithms and systems through game-theoretic tools to understand their strategic properties. This includes calculating and proving properties of Nash equilibria (stable states where no participant can benefit by changing only their own strategy), measuring price of anarchy (efficiency loss due to selfish behavior), and analyzing best-response dynamics (how systems evolve when players sequentially optimize their strategies). Design: Creating mechanisms and algorithms with both desirable computational properties and game-theoretic robustness. This sub-field, known as algorithmic mechanism design, develops systems that incentivize truthful behavior while maintaining computational efficiency. Algorithm designers in this domain must satisfy traditional algorithmic requirements (such as polynomial-time running time and good approximation ratio) while simultaneously addressing incentive constraints that ensure participants act according to the system's intended design. == History == === Nisan-Ronen: a new framework for studying algorithms === In 1999, the seminal paper of Noam Nisan and Amir Ronen drew the attention of the Theoretical Computer Science community to designing algorithms for selfish (strategic) users. As they claim in the abstract: We consider algorithmic problems in a distributed setting where the participants cannot be assumed to follow the algorithm but rather their own self-interest. As such participants, termed agents, are capable of manipulating the algorithm, the algorithm designer should ensure in advance that the agents’ interests are best served by behaving correctly. Following notions from the field of mechanism design, we suggest a framework for studying such algorithms. In this model the algorithmic solution is adorned with payments to the participants and is termed a mechanism. The payments should be carefully chosen as to motivate all participants to act as the algorithm designer wishes. We apply the standard tools of mechanism design to algorithmic problems and in particular to the shortest path problem. This paper coined the term algorithmic mechanism design and was recognized by the 2012 Gödel Prize committee as one of "three papers laying foundation of growth in Algorithmic Game Theory". === Price of Anarchy === The other two papers cited in the 2012 Gödel Prize for fundamental contributions to Algorithmic Game Theory introduced and developed the concept of "Price of Anarchy". In their 1999 paper "Worst-case Equilibria", Koutsoupias and Papadimitriou proposed a new measure of the degradation of system efficiency due to the selfish behavior of its agents: the ratio of between system efficiency at an optimal configuration, and its efficiency at the worst Nash equilibrium. (The term "Price of Anarchy" only appeared a couple of years later.) === The Internet as a catalyst === The Internet created a new economy—both as a foundation for exchange and commerce, and in its own right. The computational nature of the Internet allowed for the use of computational tools in this new emerging economy. On the other hand, the Internet itself is the outcome of actions of many. This was new to the classic, ‘top-down’ approach to computation that held till then. Thus, game theory is a natural way to view the Internet and interactions within it, both human and mechanical. Game theory studies equilibria (such as the Nash equilibrium). An equilibrium is generally defined as a state in which no player has an incentive to change their strategy. Equilibria are found in several fields related to the Internet, for instance financial interactions and communication load-balancing. Game theory provides tools to analyze equilibria, and a common approach is then to ‘find the game’—that is, to formalize specific Internet interactions as a game, and to derive the associated equilibria. Rephrasing problems in terms of games allows the analysis of Internet-based interactions and the construction of mechanisms to meet specified demands. If equilibria can be shown to exist, a further question must be answered: can an equilibrium be found, and in reasonable time? This leads to the analysis of algorithms for finding equilibria. Of special importance is the complexity class PPAD, which includes many problems in algorithmic game theory. == Areas of research == === Algorithmic mechanism design === Mechanism design is the subarea of economics that deals with optimization under incentive constraints. Algorithmic mechanism design considers the optimization of economic systems under computational efficiency requirements. Typical objectives studied include revenue maximization and social welfare maximization. === Inefficiency of equilibria === The concepts of price of anarchy and price of stability were introduced to capture the loss in performance of a system due to the selfish behavior of its participants. The price of anarchy captures the worst-case performance of the system at equilibrium relative to the optimal performance possible. The price of stability, on the other hand, captures the relative performance of the best equilibrium of the system. These concepts are counterparts to the notion of approximation ratio in algorithm design. === Complexity of finding equilibria === The existence of an equilibrium in a game is typically established using non-constructive fixed point theorems. There are no efficient algorithms known for computing Nash equilibria. The problem is complete for the complexity class PPAD even in 2-player games. In contrast, correlated equilibria can be computed efficiently using linear programming, as well as learned via no-regret strategies. === Computational social choice === Computational social choice studies computational aspects of social choice, the aggregation of individual agents' preferences. Examples include algorithms and computational complexity of voting rules and coalition formation. Other topics include: Algorithms for computing Market equilibria Fair division Multi-agent systems And the area counts with diverse practical applications: Sponsored search auctions Spectrum auctions Cryptocurrencies Prediction markets Reputation systems Sharing economy Matching markets such as kidney exchange and school choice Crowdsourcing and peer grading Economics of the cloud == Journals and newsletters == ACM Transactions on Economics and Computation (TEAC) SIGEcom Exchanges Algorithmic Game Theory papers are often also published in Game Theory journals such as GEB, Economics journals such as Econometrica, and Computer Science journals such as SICOMP.
Data Science Africa
Data Science Africa (DSA) is a non-profit knowledge sharing professional group that aims at bringing together leading researchers and practitioners working on data science methods or applications relevant to Africa, and providing training on state of the art data science methods to students and others interested in developing practical skills. Since 2013, DSA has been organizing conference, workshops and summer schools on machine learning and data science across East Africa. Facilitators of Summer School and workshops are researchers and practitioners from the academia, private and public institutions across the world. == Summer schools and workshops == The first summer school which started as Gaussian Process Summer School was held at Makerere University in Kampala, Uganda from 6th to 9 August 2013. The First Data Science Summer School and Workshop was held at Dedan Kimathi University of Technology in Nyeri, Kenya from 15th to 19 June 2015. The Second Data Science Summer School was held at Makerere University, Kampala, Uganda from 27th to 29 July 2016, and the workshop was held at Pulse Lab, Kampala, Uganda from 30 July to 1 August 2016. The Third Data Science Summer School and Workshop was held at Nelson Mandela African Institute of Science and Technology, Tanzania from 19th to 21 July 2017. Among the sponsors of the event was ARM
Information flow
In discourse-based grammatical theory, information flow is any tracking of referential information by speakers. Information may be new, i.e., just introduced into the conversation; given, i.e., already active in the speakers' consciousness; or old, i.e., no longer active. The various types of activation, and how these are defined, are model-dependent. Information flow affects grammatical structures such as: Word order (topic, focus, and afterthought constructions). Active, passive, or middle voice. Choice of deixis, such as articles; "medial" deictics such as Spanish ese and Japanese sore are generally determined by the familiarity of a referent rather than by physical distance. Overtness of information, such as whether an argument of a verb is indicated by a lexical noun phrase, a pronoun, or not mentioned at all. Clefting: Splitting a single clause into two clauses, each with its own verb, e.g. ‘The chicken turtles tasted like chicken.’ becomes ‘It was the chicken turtle | that tasted like chicken.’ In this case, clefting is used to shift the focus of the sentence to the subject, the chicken turtle. Front focus: Placing at the start (front) of a sentence information that would normally occur later in the sentence, to give it extra prominence. For example, in pop culture, Yoda's speech often utilizes such syntactic construction, such as when he says 'much to learn you still have' to Luke Skywalker. End focus (or end weight): Given or familiar information followed by new information. This gives prominence to the final part of the sentences and can enable suspense to build, e.g. ‘Through the door came a gigantic wolf’.(Umer Prince)
C-RAN
C-RAN (Cloud-RAN), also referred to as Centralized-RAN, is an architecture for cellular networks. C-RAN is a centralized, cloud computing-based architecture for radio access networks that supports 2G, 3G, 4G, 5G and future wireless communication standards. Its name comes from the four 'C's in the main characteristics of C-RAN system, "Clean, Centralized processing, Collaborative radio, and a real-time Cloud Radio Access Network". == Background == Traditional cellular, or Radio Access Networks (RAN), consist of many stand-alone base stations (BTS). Each BTS covers a small area, whereas a group BTS provides coverage over a continuous area. Each BTS processes and transmits its own signal to and from the mobile terminal, and forwards the data payload to and from the mobile terminal and out to the core network via the backhaul. Each BTS has its own cooling, back haul transportation, backup battery, monitoring system, and so on. Because of limited spectral resources, network operators 'reuse' the frequency among different base stations, which can cause interference between neighboring cells. There are several limitations in the traditional cellular architecture. First, each BTS is costly to build and operate. Moore's law helps reduce the size and power of an electrical system, but the supporting facilities of the BTS are not improved quite as well. Second, when more BTS are added to a system to improve its capacity, interference among BTS is more severe as BTS are closer to each other and more of them are using the same frequency. Third, because users are mobile, the traffic of each BTS fluctuates (called 'tide effect'), and as a result, the average utilization rate of individual BTS is pretty low. However, these processing resources cannot be shared with other BTS. Therefore, all BTS are designed to handle the maximum traffic, not average traffic, resulting in a waste of processing resources and power at idle times. == Evolution of base station architecture == === All-in-one macro base station === In the 1G and 2G cellular networks, base stations had an all-in-one architecture. Analog, digital, and power functions were housed in a single cabinet as large as a refrigerator. Usually the base station cabinet was placed in a dedicated room along with all necessary supporting facilitates such as power, backup battery, air conditioning, environment surveillance, and backhaul transmission equipment. The RF signal is generated by the base station RF unit and propagates through pairs of RF cables up to the antennas on the top of a base station tower or other mounting points. This all-in-one architecture was mostly found in macro cell deployments. === Distributed base station === For 3G, a distributed base station architecture was introduced by Ericsson, Nokia, Huawei, and other leading telecom equipment vendors. In this architecture the radio function unit, also known as the remote radio head (RRH), is separated from the digital function unit, or baseband unit (BBU) by fiber. Digital baseband signals are carried over fiber, using the Open Base Station Architecture Initiative (OBSAI) or Common Public Radio Interface (CPRI) standard. The RRH can be installed on the top of tower close to the antenna, reducing the loss compared to the traditional base station where the RF signal has to travel through a long cable from the base station cabinet to the antenna at the top of the tower. The fiber link between RRH and BBU also allows more flexibility in network planning and deployment as they can be placed a few hundred meters or a few kilometers away. Most modern base stations now use this decoupled architecture. === C-RAN/Cloud-RAN === C-RAN may be viewed as an architectural evolution of the above distributed base station system. It takes advantage of many technological advances in wireless, optical and IT communications systems. For example, it uses the latest CPRI standard, low cost Coarse or Dense Wavelength Division Multiplexing (CWDM/ DWDM) technology, and mmWave to allow transmission of baseband signal over long distance thus achieving large scale centralised base station deployment. It applies recent Data Centre Network technology to allow a low cost, high reliability, low latency and high bandwidth interconnect network in the BBU pool. It utilizes open platforms and real-time virtualization technology rooted in cloud computing to achieve dynamic shared resource allocation and support multi-vendor, multi-technology environments. == Architecture overview == C-RAN architecture has the following characteristics that are distinct from other cellular architectures: Large scale centralized deployment: Allows many RRHs to connect to a centralized BBU pool. The maximum distance can be 20km in fiber link for 4G (LTE/LTE-A) systems, and even longer distances (40~80km) for 3G (WCDMA/TD-SCDMA) and 2G (GSM/CDMA) systems. Native support to Collaborative Radio technologies: Any BBU can talk with any other BBU within the BBU pool with very high bandwidth (10 Gbit/s and above) and low latency (10 μs level). This is enabled by the interconnection of BBUs in the pool. This is one major difference from BBU Hotelling, or base station Hotelling; in the latter case, the BBUs of different base stations are simply stacked together and have no direct link between them to allow physical layer co-ordination. Real-time virtualization capability based on open platform: This is different from traditional base stations built on proprietary hardware, where the software and hardware are close-sourced and provided by single vendors. In contrast, a C-RAN BBU pool is built on open hardware, like x86/ARM CPU based servers, and interface cards that handle fiber links to RRHs and inter-connections in the pool. Real-time virtualization ensures that resources in the pool can be allocated dynamically to base station software stacks, say 4G/3G/2G function modules from different vendors, according to network load. However, to satisfy the strict timing requirements of wireless communication systems, the real-time performance for C-RAN is at the level of tens of microseconds, which is two orders of magnitude better than the millisecond level 'real-time' performance usually seen in Cloud Computing environments. == Similar architecture and systems == KT, a telecom operator in the Republic of Korea, introduced a Cloud Computing Center (CCC) system in their 3G (WCDMA/HSPA) and 4G (LTE/LTE-A) network in 2011 and 2012. The concept of CCC is basically the same as C-RAN. SK Telecom has also deployed Smart Cloud Access Network (SCAN) and Advanced-SCAN in their 4G (LTE/LTE-A) network in Korea no later than 2012. In 2014, Airvana (now CommScope) introduced OneCell, a C-RAN-based small cell system designed for enterprises and public spaces. == Competing architectures in cellular network evolution == === All-in-one BTS === One major alternative solution that is addressing similar challenges of RAN, is the small size, all-in-one outdoor BTS. Thanks to the achievements in the semiconductor industry, all the functionality of a BTS, including RF, baseband processing, MAC processing and package level processing, can now be implemented in a volume of <50 liters. This makes the system small and weatherproof, reduces the difficulty of BTS site choice and construction, eliminates the air conditioning requirement, and thus reduces operational costs. However, because each BTS is still working on its own, it cannot readily make use of the collaboration algorithms to reduce the interference between neighboring BTSs. It is also relatively hard to upgrade or repair because the all-in-one BTS units are usually mounted near the antenna. More processing units in less-protected environments also implies a higher failure rate compared to C-RAN, which only has the RRU deployed outdoors. The advantage of Cloud RAN lies in its ability to implement LTE-Advanced features such as Coordinated MultiPoint (CoMP) with very low latency between multiple radio heads. However, the economic benefit of improvements such as CoMP can be negated by the higher backhaul costs for some operators. === Small cell === The main competition between small cell and C-RAN occurs in two deployment scenarios: outdoor hotspot coverage and indoor coverage. == Academic research and publications == As one of the promising evolution paths for future cellular network architecture, C-RAN has attracted high academic research interest. Meanwhile, because the native support of cooperative radio capability built into the C-RAN architecture, it also enables many advanced algorithms that were hard to implement in cellular networks, including Cooperative Multi-Point Transmission/Receiving, Network Coding, etc. In October 2011, Wireless World Research Forum 27 was hosted in Germany, when China Mobile was invited to give a C-RAN presentation. In August 2012, IEEE C-RAN 2012 workshop was hosted in Kunming, China. CRC Press published a book, "Green Communications: Theore
Sieve of Pritchard
In mathematics, the sieve of Pritchard is an algorithm for finding all prime numbers up to a specified bound. Like the ancient sieve of Eratosthenes, it has a simple conceptual basis in number theory. It is especially suited to quick hand computation for small bounds. Whereas the sieve of Eratosthenes marks off each non-prime for each of its prime factors, the sieve of Pritchard avoids considering almost all non-prime numbers by building progressively larger wheels, which represent the pattern of numbers not divisible by any of the primes processed thus far. It thereby achieves a better asymptotic complexity, and was the first sieve with a running time sublinear in the specified bound. Its asymptotic running-time has not been improved on, and it deletes fewer composites than any other known sieve. It was created in 1979 by Paul Pritchard. Since Pritchard has created a number of other sieve algorithms for finding prime numbers, the sieve of Pritchard is sometimes singled out by being called the wheel sieve (by Pritchard himself) or the dynamic wheel sieve. == Overview == A prime number is a natural number that has no natural number divisors other than the number 1 and itself. To find all the prime numbers less than or equal to a given integer N, a sieve algorithm examines a set of candidates in the range 2, 3, …, N, and eliminates those that are not prime, leaving the primes at the end. The sieve of Eratosthenes examines all of the range, first removing all multiples of the first prime 2, then of the next prime 3, and so on. The sieve of Pritchard instead examines a subset of the range consisting of numbers that occur on successive wheels, which represent the pattern of numbers left after each successive prime is processed by the sieve of Eratosthenes. For i > 0, the ith wheel Wi represents this pattern. It is the set of numbers between 1 and the product Pi = p1 · p2 ⋯ pi of the first i prime numbers that are not divisible by any of these prime numbers (and is said to have an associated length Pi). This is because adding Pi to a number does not change whether it is divisible by one of the first i prime numbers, since the remainder on division by any one of these primes is unchanged. So W1 = {1} with length P1 = 2 represents the pattern of odd numbers; W2 = {1,5} with length P2 = 6 represents the pattern of numbers not divisible by 2 or 3; etc. Wheels are so-called because Wi can be usefully visualized as a circle of circumference Pi with its members marked at their corresponding distances from an origin. Then rolling the wheel along the number line marks points corresponding to successive numbers not divisible by one of the first i prime numbers. The animation shows W2 being rolled up to 30. It is useful to define Wi → n for n > 0 to be the result of rolling Wi up to n. Then the animation generates W2 → 30 = {1,5,7,11,13,17,19,23,25,29}. Note that up to 52 − 1 = 24, this consists only of 1 and the primes between 5 and 25. The sieve of Pritchard is derived from the observation that this holds generally: for all i > 0, the values in Wi → (p2i+1 − 1) are 1 and the primes between pi+1 and p2i+1. It even holds for i = 0, where the wheel has length 1 and contains just 1 (representing all the natural numbers). So the sieve of Pritchard starts with the trivial wheel W0 and builds successive wheels until the square of the wheel's first member after 1 is at least N. Wheels grow very quickly, but only their values up to N are needed and generated. It remains to find a method for generating the next wheel. Note in the animation that W3 = {1,5,7,11,13,17,19,23,25,29} − {5 · 1 , 5 · 5} can be obtained by rolling W2 up to 30 and then removing 5 times each member of W2.This also holds generally: for all i ≥ 0, Wi+1 = (Wi → Pi+1) − {pi+1 · w | w ∈ Wi}. Rolling Wi past Pi just adds values to Wi, so the current wheel is first extended by getting each successive member starting with w = 1, adding Pi to it, and inserting the result in the set. Then the multiples of pi+1 are deleted. Care must be taken to avoid a number being deleted that itself needs to be multiplied by pi+1. The sieve of Pritchard as originally presented does so by first skipping past successive members until finding the maximum one needed, and then doing the deletions in reverse order by working back through the set. This is the method used in the first animation above. A simpler approach is just to gather the multiples of pi+1 in a list, and then delete them. Another approach is given by Gries and Misra. If the main loop terminates with a wheel whose length is less than N, it is extended up to N to generate the remaining primes. The algorithm, for finding all primes up to N, is therefore as follows: Start with a set W = {1} and length = 1 representing wheel 0, and prime p = 2. As long as p2 ≤ N, do the following: if length < N, then extend W by repeatedly getting successive members w of W starting with 1 and inserting length + w into W as long as it does not exceed p · length or N; increase length to the minimum of p · length and N. repeatedly delete p times each member of W by first finding the largest ≤ length and then working backwards. note the prime p, then set p to the next member of W after 1 (or 3 if p was 2). if length < N, then extend W to N by repeatedly getting successive members w of W starting with 1 and inserting length + w into W as long as it does not exceed N; On termination, the rest of the primes up to N are the members of W after 1. === Example === To find all the prime numbers less than or equal to 150, proceed as follows. Start with wheel 0 with length 1, representing all natural numbers 1, 2, 3...: 1 The first number after 1 for wheel 0 (when rolled) is 2; note it as a prime. Now form wheel 1 with length 2 × 1 = 2 by first extending wheel 0 up to 2 and then deleting 2 times each number in wheel 0, to get: 1 2 The first number after 1 for wheel 1 (when rolled) is 3; note it as a prime. Now form wheel 2 with length 3 × 2 = 6 by first extending wheel 1 up to 6 and then deleting 3 times each number in wheel 1, to get 1 2 3 5 The first number after 1 for wheel 2 is 5; note it as a prime. Now form wheel 3 with length 5 × 6 = 30 by first extending wheel 2 up to 30 and then deleting 5 times each number in wheel 2 (in reverse order), to get 1 2 3 5 7 11 13 17 19 23 25 29 The first number after 1 for wheel 3 is 7; note it as a prime. Now wheel 4 has length 7 × 30 = 210, so we only extend wheel 3 up to our limit 150. (No further extending will be done now that the limit has been reached.) We then delete 7 times each number in wheel 3 until we exceed our limit 150, to get the elements in wheel 4 up to 150: 1 2 3 5 7 11 13 17 19 23 25 29 31 37 41 43 47 49 53 59 61 67 71 73 77 79 83 89 91 97 101 103 107 109 113 119 121 127 131 133 137 139 143 149 The first number after 1 for this partial wheel 4 is 11; note it as a prime. Since we have finished with rolling, we delete 11 times each number in the partial wheel 4 until we exceed our limit 150, to get the elements in wheel 5 up to 150: 1 2 3 5 7 11 13 17 19 23 25 29 31 37 41 43 47 49 53 59 61 67 71 73 77 79 83 89 91 97 101 103 107 109 113 119 121 127 131 133 137 139 143 149 The first number after 1 for this partial wheel 5 is 13. Since 13 squared is at least our limit 150, we stop. The remaining numbers (other than 1) are the rest of the primes up to our limit 150. Just 8 composite numbers are removed, once each. The rest of the numbers considered (other than 1) are prime. In comparison, the natural version of Eratosthenes sieve (stopping at the same point) removes composite numbers 184 times. == Pseudocode == The sieve of Pritchard can be expressed in pseudocode, as follows: algorithm Sieve of Pritchard is input: an integer N >= 2. output: the set of prime numbers in {1,2,...,N}. let W and Pr be sets of integer values, and all other variables integer values. k, W, length, p, Pr := 1, {1}, 2, 3, {2}; {invariant: p = pk+1 and W = Wk ∩ {\displaystyle \cap } {1,2,...,N} and length = minimum of Pk,N and Pr = the primes up to pk} while p2 <= N do if (length < N) then Extend W,length to minimum of plength,N; Delete multiples of p from W; Insert p into Pr; k, p := k+1, next(W, 1) if (length < N) then Extend W,length to N; return Pr ∪ {\displaystyle \cup } W - {1}; where next(W, w) is the next value in the ordered set W after w. procedure Extend W,length to n is {in: W = Wk and length = Pk and n > length} {out: W = Wk → {\displaystyle \rightarrow } n and length = n} integer w, x; w, x := 1, length+1; while x <= n do Insert x into W; w := next(W,w); x := length + w; length := n; procedure Delete multiples of p from W,length is integer w; w := p; while pw <= length do w := next(W,w); while w > 1 do w := prev(W,w); Remove pw from W; where prev(W, w) is the previous value in the ordered set W before w. The algorithm can be initialized with W0 instead of W1 at the minor complication of making next(W, 1) a special case when k = 0. This a