A System Requirements Specification (SysRS) (abbreviated SysRS to be distinct from a software requirements specification (SRS)) is a structured collection of information that embodies the requirements of a system. A business analyst (BA), sometimes titled system analyst, is responsible for analyzing the business needs of their clients and stakeholders to help identify business problems and propose solutions. Within the systems development life cycle domain, the BA typically performs a liaison function between the business side of an enterprise and the information technology department or external service providers.
Systems development life cycle
The systems development life cycle (SDLC) describes the typical phases and progression between phases during the development of a computer-based system. These phases progress from inception to retirement. At base, there is just one life cycle, but the taxonomy used to describe it may vary; the cycle may be classified into different numbers of phases and various names may be used for those phases. The SDLC is analogous to the life cycle of a living organism from its birth to its death. In particular, the SDLC varies by system in much the same way that each living organism has a unique path through its life. The SDLC does not prescribe how engineers should go about their work to move the system through its life cycle. Prescriptive techniques are referred to using various terms such as methodology, model, framework, and formal process. Other terms are used for the same concept as SDLC, including software development life cycle (also SDLC), application development life cycle (ADLC), and system design life cycle (also SDLC). These other terms focus on a different scope of development and are associated with different prescriptive techniques, but are about the same essential life cycle. The term "life cycle" is often written without a space, as "lifecycle", with the former more popular in the past and in non-engineering contexts. The acronym SDLC was coined when the longer form was more popular and has remained associated with the expansion, even though the shorter form is popular in engineering. Also, SDLC is relatively unique as opposed to the TLA SDL, which is highly overloaded. == Phases == Depending on the source, the SDLC is described as having different phases and using different terms. Even so, there are common aspects. The following attempts to describe notable phases using notable terminology. The phases are somewhat ordered by the natural sequence of development, although they can be overlapping and iterative. === Conceptualization === During conceptualization (a.k.a. conceptual design, system investigation, feasibility), options and priorities are considered. A feasibility study can determine whether the development effort is worthwhile via activities such as understanding user needs, cost estimation, benefit analysis, and resource analysis. A study should address operational, financial, technical, human factors, and legal/political concerns. === Requirements analysis === Requirements analysis (a.k.a. preliminary design) involves understanding the problem and determining what is needed. Often this involves engaging users to define the requirements and recording them in a document known as a requirements specification. === Design === During the design phase (a.k.a. detail design), a solution is planned. The plan can include relatively high-level information such as describing the major components of the system. The plan can include relatively low-level information such as describing functions, screen layout, business rules, and process flow. The design phase is informed by the requirements of the system. The design must satisfy each requirement. The design may be recorded in textual documents as well as functional hierarchy diagrams, example screen images, business rules, process diagrams, pseudo-code, and data models. === Construction === During construction (a.k.a. implementation, production), the system is realized. Based on the design, hardware and software components are created and integrated. This phase includes testing sub-components, components and the integration of some components, but typically does not include testing at the complete system level. This phase may include the development of training materials, including user manuals and help files. === Acceptance === The acceptance phase (a.k.a. system testing) is about testing the complete system to ensure that it meets customer expectations (requirements). === Deployment === The deployment phase (a.k.a. implementation) involves the logistics of delivery to the customer. Some systems are deployed as a single instance (i.e. in the cloud), and deployment may be ad hoc and manual. Some systems are built in quantity and are associated with manufacturing process and commissioning. This phase may include training users to use the system. It may include transitioning future development to support staff. === Maintenance === During the maintenance phase (a.k.a. operation, utilization, support) development is largely inactive, although this phase does include customer support for resolving user issues and recording suggestions for improvement. Fixes and enhancements are handled by returning to the first phase, conceptualization. For minor changes, the cycle may be significantly abbreviated compared to initial development. === Decommission === Decommission (a.k.a. disposition, retirement, phase-out) is when the system is removed from use, i.e., when it reaches end-of-life. == Practices == === Management and control === SDLC phase objectives are described in this section with key deliverables, a description of recommended tasks, and a summary of related control objectives for effective management. It is critical for the project manager to establish and monitor control objectives while executing projects. Control objectives are clear statements of the desired result or purpose and should be defined and monitored throughout a project. Control objectives can be grouped into major categories (domains), and relate to the SDLC phases as shown in the figure. To manage and control a substantial SDLC initiative, a work breakdown structure (WBS) captures and schedules the work. The WBS and all programmatic material should be kept in the "project description" section of the project notebook. The project manager chooses a WBS format that best describes the project. The diagram shows that coverage spans numerous phases of the SDLC, but the associated MCD (Management Control Domains) shows mappings to SDLC phases. For example, Analysis and Design is primarily performed as part of the Acquisition and Implementation Domain, and System Build and Prototype is primarily performed as part of delivery and support. === Work breakdown structured organization === The upper section of the WBS provides an overview of the project scope and timeline. It should also summarize the major phases and milestones. The middle section is based on the SDLC phases. WBS elements consist of milestones and tasks to be completed rather than activities to be undertaken, and have a deadline. Each task has a measurable output (e.g., an analysis document). A WBS task may rely on one or more activities (e.g., coding). Parts of the project needing support from contractors should have a statement of work (SOW). The development of an SOW does not occur during a specific phase of SDLC but is developed to include the work from the SDLC process that may be conducted by contractors. === Baselines === Baselines are established after four of the five phases of the SDLC, and are critical to the iterative nature of the model. Baselines become milestones. functional baseline: established after the conceptual design phase. allocated baseline: established after the preliminary design phase. product baseline: established after the detailed design and development phase. updated product baseline: established after the production construction phase. In the following diagram, these stages are divided into ten steps, from definition to creation and modification of IT work products:
Magic state distillation
Magic state distillation is a method for creating more accurate quantum states from multiple noisy ones, which is important for building fault tolerant quantum computers. It has also been linked to quantum contextuality, a concept thought to contribute to quantum computers' power. The technique was first proposed by Emanuel Knill in 2004, and further analyzed by Sergey Bravyi and Alexei Kitaev the same year. Thanks to the Gottesman–Knill theorem, it is known that some quantum operations (operations in the Clifford group) can be perfectly simulated in polynomial time on a classical computer. In order to achieve universal quantum computation, a quantum computer must be able to perform operations outside this set. Magic state distillation achieves this, in principle, by concentrating the usefulness of imperfect resources, represented by mixed states, into states that are conducive for performing operations that are difficult to simulate classically. A variety of qubit magic state distillation routines and distillation routines for qubits with various advantages have been proposed. == Stabilizer formalism == The Clifford group consists of a set of n {\displaystyle n} -qubit operations generated by the gates {H, S, CNOT} (where H is Hadamard and S is [ 1 0 0 i ] {\displaystyle {\begin{bmatrix}1&0\\0&i\end{bmatrix}}} ) called Clifford gates. The Clifford group generates stabilizer states which can be efficiently simulated classically, as shown by the Gottesman–Knill theorem. This set of gates with a non-Clifford operation is universal for quantum computation. == Magic states == Magic states are purified from n {\displaystyle n} copies of a mixed state ρ {\displaystyle \rho } . These states are typically provided via an ancilla to the circuit. A magic state for the π / 6 {\displaystyle \pi /6} rotation operator is | M ⟩ = cos ( β / 2 ) | 0 ⟩ + e i π 4 sin ( β / 2 ) | 1 ⟩ {\displaystyle |M\rangle =\cos(\beta /2)|0\rangle +e^{i{\frac {\pi }{4}}}\sin(\beta /2)|1\rangle } where β = arccos ( 1 3 ) {\displaystyle \beta =\arccos \left({\frac {1}{\sqrt {3}}}\right)} . A non-Clifford gate can be generated by combining (copies of) magic states with Clifford gates. Since a set of Clifford gates combined with a non-Clifford gate is universal for quantum computation, magic states combined with Clifford gates are also universal. == Purification algorithm for distilling |M〉 == The first magic state distillation algorithm, invented by Sergey Bravyi and Alexei Kitaev, is as follows. Input: Prepare 5 imperfect states. Output: An almost pure state having a small error probability. repeat Apply the decoding operation of the five-qubit error correcting code and measure the syndrome. If the measured syndrome is | 0000 ⟩ {\displaystyle |0000\rangle } , the distillation attempt is successful. else Get rid of the resulting state and restart the algorithm. until The states have been distilled to the desired purity.
Tagsistant
Tagsistant is a semantic file system for the Linux kernel, written in C and based on FUSE. Unlike traditional file systems that use hierarchies of directories to locate objects, Tagsistant introduces the concept of tags. == Design and differences with hierarchical file systems == In computing, a file system is a type of data store which could be used to store, retrieve and update files. Each file can be uniquely located by its path. The user must know the path in advance to access a file and the path does not necessarily include any information about the content of the file. Tagsistant uses a complementary approach based on tags. The user can create a set of tags and apply those tags to files, directories and other objects (devices, pipes, ...). The user can then search all the objects that match a subset of tags, called a query. This kind of approach is well suited for managing user contents like pictures, audio recordings, movies and text documents but is incompatible with system files (like libraries, commands and configurations) where the univocity of the path is a security requirement to prevent the access to a wrong content. == The tags/ directory == A Tagsistant file system features four main directories: archive/ relations/ stats/ tags/ Tags are created as sub directories of the tags/ directory and can be used in queries complying to this syntax: tags/subquery/[+/subquery/[+/subquery/]]/@/ where a subquery is an unlimited list of tags, concatenated as directories: tag1/tag2/tag3/.../tagN/ The portion of a path delimited by tags/ and @/ is the actual query. The +/ operator joins the results of different sub-queries in one single list. The @/ operator ends the query. To be returned as a result of the following query: tags/t1/t2/+/t1/t4/@/ an object must be tagged as both t1/ and t2/ or as both t1/ and t4/. Any object tagged as t2/ or t4/, but not as t1/ will not be retrieved. The query syntax deliberately violates the POSIX file system semantics by allowing a path token to be a descendant of itself, like in tags/t1/t2/+/t1/t4/@ where t1/ appears twice. As a consequence a recursive scan of a Tagsistant file system will exit with an error or endlessly loop, as done by Unix find: This drawback is balanced by the possibility to list the tags inside a query in any order. The query tags/t1/t2/@/ is completely equivalent to tags/t2/t1/@/ and tags/t1/+/t2/t3/@/ is equivalent to tags/t2/t3/+/t1/@/. The @/ element has the precise purpose of restoring the POSIX semantics: the path tags/t1/@/directory/ refers to a traditional directory and a recursive scan of this path will properly perform. == The reasoner and the relations/ directory == Tagsistant features a simple reasoner which expands the results of a query by including objects tagged with related tags. A relation between two tags can be established inside the relations/ directory following a three level pattern: relations/tag1/rel/tag2/ The rel element can be includes or is_equivalent. To include the rock tag in the music tag, the Unix command mkdir can be used: mkdir -p relations/music/includes/rock The reasoner can recursively resolve relations, allowing the creation of complex structures: mkdir -p relations/music/includes/rock mkdir -p relations/rock/includes/hard_rock mkdir -p relations/rock/includes/grunge mkdir -p relations/rock/includes/heavy_metal mkdir -p relations/heavy_metal/includes/speed_metal The web of relations created inside the relations/ directory constitutes a basic form of ontology. == Autotagging plugins == Tagsistant features an autotagging plugin stack which gets called when a file or a symlink is written. Each plugin is called if its declared MIME type matches The list of working plugins released with Tagsistant 0.6 is limited to: text/html: tags the file with each word in
Manufacture Modules Technologies
Manufacture Modules Technologies Sarl (MMT) is a Swiss company established in Geneva in 2015 which originally specialised in the development and commercialization of "Horological Smartwatch modules", firmware, apps and cloud. Located at Geneva's Skylab high-tech hub, it expanded into the development and manufacturing of "E-Straps" operated with a mobile application. Philippe Fraboulet is the CEO. == History == In June 2015, Fullpower Technologies and Union Horlogère Suisse (Swiss Watchmakers Corporation) formed MMT as a joint venture, which then launched the MotionX Horological Smartwatch Open Platform for the Swiss watch industry. The initial licensees were Frederique Constant, Alpina and Mondaine, brands owned by Union Horlogère Suisse. Fullpower created and managed the circuit design, firmware, smartphone applications (including sleep activity), as well as the cloud Infrastructure. MMT managed the Swiss watch movement development and production as well as licensing and support. In July 2016, Union Horlogere Holding and MMT were spun-out of the Frédérique Constant Group. Fullpower Technologies' 19.99% share was acquired by Union Horlogere Holding BV, giving it 100% of MMT's shares. == Business == The company offers firmware, a cloud, manufacturing, service and over-the-air facilities for upgrades. The company also offers its own apps, which bear the label “Swiss Made software”.
Alipay
Alipay (simplified Chinese: 支付宝; traditional Chinese: 支付寶; pinyin: zhīfùbǎo) is a third-party mobile and online payment platform, established in Hangzhou, China, in February 2004 by Alibaba Group and its founder Jack Ma. In 2015, Alipay moved its headquarters to Pudong, Shanghai, although its parent company Ant Financial remains Hangzhou-based. Alipay overtook PayPal as the world's largest mobile (digital) payment platform in 2013. As of June 2020, Alipay serves over 1.3 billion users and 80 million merchants. According to the statistics of the fourth quarter of 2018, Alipay has a 55.32% share of the third-party payment market in mainland China, and it continues to grow. Along with WeChat, Alipay has been described to be China's super-app with a wide range of functionalities including ridesharing, travel booking and medical appointments. == History == The service was first launched in 2003, by Taobao. The People's Bank of China, China's central bank, issued licensing regulations in June 2010 for third-party payment providers. It also issued separate guidelines for foreign-funded payment institutions. Because of this, Alipay, which accounted for half of China's non-bank online payment market, was restructured as a domestic company controlled by Alibaba CEO Jack Ma in order to facilitate the regulatory approval for the license. The 2010 transfer of Alipay's ownership was controversial, with media reports in 2011 that Yahoo! and Softbank (Alibaba Group's controlling shareholders) were not informed of the sale for nominal value. Chinese business publication Century Weekly criticised Ma, who stated that Alibaba Group's board of directors was aware of the transaction. The incident was criticised in foreign and Chinese media as harming foreign trust in making Chinese investments. The ownership dispute was resolved by Alibaba Group, Yahoo!, and Softbank in July 2011. In 2013, Alipay launched a financial product platform called Yu'e Bao. Alipay partnered with Tianhong Asset Management to launch the it. Yu'e Bao offers an online money market account in which Alipay customers can deposit money and receive a higher interest rate than that available from banks. It soon became China's largest online money market fund and prompted competitors like Baidu and Tencent to introduce alternatives. Alibaba (the parent company of Alipay) reported having 152 million Yu'e Bao users in mid-2016, with 810 billion RMB (US$117 billion) in funds under management. In 2015, Alipay's parent company was re-branded as Ant Financial Services Group. In 2017, Alipay unveiled their facial recognition payment service. In 2020, Alipay upgraded from a payment financial instrument to an open platform for digital life. In 2021, the mandate by the Ministry of Industry and Information Technology (MIIT) to open up the "walled garden" ecosystems of the major tech companies has led to the introduction of interoperability of payment QR codes of Alipay and competing WeChat Pay and UnionPay's Cloud QuickPass platforms. In response to the increase in Alipay's payment volume due to use on Alibaba's e-commerce sites and others, Chinese regulators introduced new rules in 2020. The new rules focused on Alipay because the payment volume exploded due to its use on Alibaba's e-commerce sites and other platforms. By the second quarter in 2020, Alipay held 55.6% of China's third party mobile payment market. The People's Bank of China made rules that required payment firms to place money with regulators and anti-monopoly reviews would be triggered if the amount exceeded 50% market share. The rules included that the People's Bank of China mandate an online-payment clearing route through the NetsUnion Clearing Corporation, a centralized, state-overseen clearing body, and that unused consumer funds be held by a third-party payment provider in a non-interest-bearing account. These measures increased transparency and reduced systemic risk. When Alipay operates outside of China, it must comply with local financial regulations, which may treat specific functions such as money-market funds or investment-linked products. In Singapore, such services may require prior authorization from securities or financial-services regulators before they can be offered to residents. == Services == Alipay states that it operates with more than 65 financial institutions including Visa and MasterCard to provide payment services for Taobao and Tmall as well as more than 460,000 online and local Chinese businesses. Alipay is used in smartphones with their Alipay Wallet app. QR code payment codes are used for local in-store payments. The Alipay app also provides features such as credit card bill payments, bank account managements, P2P transfer, prepay mobile phone top-up, bus and train ticket purchases, food orders, vehicles for hire, insurance selections and a digital identification document storage. Alipay also allows online check-out on most Chinese-based websites such as Taobao and Tmall. The Alipay app allows users to add their own services provided from different companies to create a more personalised experience. Since late 2008, Alipay has promoted public service payment services and has covered more than 300 cities nationwide, supporting more than 1,200 partner organizations. In addition to utility bills such as water and electricity, Alipay also extends their services to areas such as paying transportation fines, property fees, and cable television fees. Common online payment services also include hydropower coal payment, tuition payment and traffic fine. On 15 January 2009, Alipay launched a credit card repayment service, supporting 39 domestic bank-issued credit cards. It is currently the most popular third-party repayment platform. The main advantages are free credit card bills checking, repayments with no administrative fee, as well as automatic repayment, repayment reminders and other value-added services. In the first quarter of 2014, 76% of credit cards were also paid by Alipay Wallet. From December 2013, several chain convenience store companies, including Meiyijia, Hongqi Chain, and Qishiduo C-STORE and 7-Eleven, have successively supported Alipay payment; in December, Beijing taxi drivers began to accept Alipay to pay the fare. Subsequently, Wanda Cinema, Joy City, Wangfujing and other large-scale retail companies as well as movie theaters, KTV, and catering companies have access to Alipay. From 26 March 2019, the service fee will be charged for the payment of credit card through Alipay. Customers only pay the portion of the payment that exceeds 2,000 yuan at 0.1%. In addition to this, in 2019, Walgreens accepted Alipay as payment in 3,000 US stores. Walgreen's products are available to Chinese customers through Alibaba's Tmall online marketplace. The payment application can also be used on Alibaba.com's site and Taobao as a means of payment. A Nielsen report suggests that over 90% of Chinese tourists would be willing to use mobile payment overseas if given the option. Many Chinese tourists do not have international credit cards, and so Alipay is a payment option. Digital payments have become the norm in China as the government pushes a cashless system even in rural and village areas. In November 2019, Alipay introduced Tourpass, a service component that allows non-Chinese users to use its mobile payment feature by pre-loading Chinese Yuan equivalent foreign currency into the app. In 2020, Alipay used a QR code system to help in containing the COVID-19 outbreak. The health code system tags users one of three colors according to their location, basic health information and travel history. "Beauty filters" were included to Alipay's face-scan payment system in a new upgrade that was released in July 2019. The market has responded well to the "beauty filters," which make users seem better when they use the program to make payments. Alipay Tap is a payment function launched by Alipay in July 2024. Alipay+ NFC enables wallets to offer tap-to-pay acceptance across Mastercard's global contactless network, all within your existing wallet infrastructure. == Foreign expansion == Outside of China, more than 300 worldwide merchants use Alipay to sell directly to consumers in China. It currently supports transactions in 18 foreign currencies. Since the launch of Alipay in the Mainland China, Ant Financial introduced a series of expansion of the services to other countries. Other than expanding into individual countries, the system would also be integrated with online payment platform providers. Ant Group had acquired a majority stake into 2C2P, a Singapore-based provider used by merchants worldwide in April 2022, and would eventually integrate Alipay with 2C2P. === Asia === ==== Bangladesh ==== In 2018, Alipay bought 20% shares in Bangladeshi mobile financial service provider bKash Limited. ==== Hong Kong ==== In 2017, Ant Financial expanded to Hong Kong. In a joint venture with CK Hutchison, as Alipay Payment Ser
Randomized rounding
In computer science and operations research, randomized rounding is a widely used approach for designing and analyzing approximation algorithms. Many combinatorial optimization problems are computationally intractable to solve exactly (to optimality). For such problems, randomized rounding can be used to design fast (polynomial time) approximation algorithms—that is, algorithms that are guaranteed to return an approximately optimal solution given any input. The basic idea of randomized rounding is to convert an optimal solution of a relaxation of the problem into an approximately-optimal solution to the original problem. The resulting algorithm is usually analyzed using the probabilistic method. == Overview == The basic approach has three steps: Formulate the problem to be solved as an integer linear program (ILP). Compute an optimal fractional solution x {\displaystyle x} to the linear programming relaxation (LP) of the ILP. Round the fractional solution x {\displaystyle x} of the LP to an integer solution x ′ {\displaystyle x'} of the ILP. (Although the approach is most commonly applied with linear programs, other kinds of relaxations are sometimes used. For example, see Goemans' and Williamson's semidefinite programming-based Max-Cut approximation algorithm.) In the first step, the challenge is to choose a suitable integer linear program. Familiarity with linear programming, in particular modelling using linear programs and integer linear programs, is required. For many problems, there is a natural integer linear program that works well, such as in the Set Cover example below. (The integer linear program should have a small integrality gap; indeed randomized rounding is often used to prove bounds on integrality gaps.) In the second step, the optimal fractional solution can typically be computed in polynomial time using any standard linear programming algorithm. In the third step, the fractional solution must be converted into an integer solution (and thus a solution to the original problem). This is called rounding the fractional solution. The resulting integer solution should (provably) have cost not much larger than the cost of the fractional solution. This will ensure that the cost of the integer solution is not much larger than the cost of the optimal integer solution. The main technique used to do the third step (rounding) is to use randomization, and then to use probabilistic arguments to bound the increase in cost due to the rounding (following the probabilistic method from combinatorics). Therein, probabilistic arguments are used to show the existence of discrete structures with desired properties. In this context, one uses such arguments to show the following: Given any fractional solution x {\displaystyle x} of the LP, with positive probability the randomized rounding process produces an integer solution x ′ {\displaystyle x'} that approximates x {\displaystyle x} according to some desired criterion. Finally, to make the third step computationally efficient, one either shows that x ′ {\displaystyle x'} approximates x {\displaystyle x} with high probability (so that the step can remain randomized) or one derandomizes the rounding step, typically using the method of conditional probabilities. The latter method converts the randomized rounding process into an efficient deterministic process that is guaranteed to reach a good outcome. == Example: the set cover problem == The following example illustrates how randomized rounding can be used to design an approximation algorithm for the set cover problem. Fix any instance ⟨ c , S ⟩ {\displaystyle \langle c,{\mathcal {S}}\rangle } of set cover over a universe U {\displaystyle {\mathcal {U}}} . === Computing the fractional solution === For step 1, let IP be the standard integer linear program for set cover for this instance. For step 2, let LP be the linear programming relaxation of IP, and compute an optimal solution x ∗ {\displaystyle x^{}} to LP using any standard linear programming algorithm. This takes time polynomial in the input size. The feasible solutions to LP are the vectors x {\displaystyle x} that assign each set s ∈ S {\displaystyle s\in {\mathcal {S}}} a non-negative weight x s {\displaystyle x_{s}} , such that, for each element e ∈ U {\displaystyle e\in {\mathcal {U}}} , x ′ {\displaystyle x'} covers e {\displaystyle e} —the total weight assigned to the sets containing e {\displaystyle e} is at least 1, that is, ∑ s ∋ e x s ≥ 1. {\displaystyle \sum _{s\ni e}x_{s}\geq 1.} The optimal solution x ∗ {\displaystyle x^{}} is a feasible solution whose cost ∑ s ∈ S c ( S ) x s ∗ {\displaystyle \sum _{s\in {\mathcal {S}}}c(S)x_{s}^{}} is as small as possible. Note that any set cover C {\displaystyle {\mathcal {C}}} for S {\displaystyle {\mathcal {S}}} gives a feasible solution x {\displaystyle x} (where x s = 1 {\displaystyle x_{s}=1} for s ∈ C {\displaystyle s\in {\mathcal {C}}} , x s = 0 {\displaystyle x_{s}=0} otherwise). The cost of this C {\displaystyle {\mathcal {C}}} equals the cost of x {\displaystyle x} , that is, ∑ s ∈ C c ( s ) = ∑ s ∈ S c ( s ) x s . {\displaystyle \sum _{s\in {\mathcal {C}}}c(s)=\sum _{s\in {\mathcal {S}}}c(s)x_{s}.} In other words, the linear program LP is a relaxation of the given set-cover problem. Since x ∗ {\displaystyle x^{}} has minimum cost among feasible solutions to the LP, the cost of x ∗ {\displaystyle x^{}} is a lower bound on the cost of the optimal set cover. === Randomized rounding step === In step 3, we must convert the minimum-cost fractional set cover x ∗ {\displaystyle x^{}} into a feasible integer solution x ′ {\displaystyle x'} (corresponding to a true set cover). The rounding step should produce an x ′ {\displaystyle x'} that, with positive probability, has cost within a small factor of the cost of x ∗ {\displaystyle x^{}} .Then (since the cost of x ∗ {\displaystyle x^{}} is a lower bound on the cost of the optimal set cover), the cost of x ′ {\displaystyle x'} will be within a small factor of the optimal cost. As a starting point, consider the most natural rounding scheme: For each set s ∈ S {\displaystyle s\in {\mathcal {S}}} in turn, take x s ′ = 1 {\displaystyle x'_{s}=1} with probability min ( 1 , x s ∗ ) {\displaystyle \min(1,x_{s}^{})} , otherwise take x s ′ = 0 {\displaystyle x'_{s}=0} . With this rounding scheme, the expected cost of the chosen sets is at most ∑ s c ( s ) x s ∗ {\displaystyle \sum _{s}c(s)x_{s}^{}} , the cost of the fractional cover. This is good. Unfortunately the coverage is not good. When the variables x s ∗ {\displaystyle x_{s}^{}} are small, the probability that an element e {\displaystyle e} is not covered is about ∏ s ∋ e 1 − x s ∗ ≈ ∏ s ∋ e exp ( − x s ∗ ) = exp ( − ∑ s ∋ e x s ∗ ) ≈ exp ( − 1 ) . {\displaystyle \prod _{s\ni e}1-x_{s}^{}\approx \prod _{s\ni e}\exp(-x_{s}^{})=\exp {\Big (}-\sum _{s\ni e}x_{s}^{}{\Big )}\approx \exp(-1).} So only a constant fraction of the elements will be covered in expectation. To make x ′ {\displaystyle x'} cover every element with high probability, the standard rounding scheme first scales up the rounding probabilities by an appropriate factor λ > 1 {\displaystyle \lambda >1} . Here is the standard rounding scheme: Fix a parameter λ ≥ 1 {\displaystyle \lambda \geq 1} . For each set s ∈ S {\displaystyle s\in {\mathcal {S}}} in turn, take x s ′ = 1 {\displaystyle x'_{s}=1} with probability min ( λ x s ∗ , 1 ) {\displaystyle \min(\lambda x_{s}^{},1)} , otherwise take x s ′ = 0 {\displaystyle x'_{s}=0} . Scaling the probabilities up by λ {\displaystyle \lambda } increases the expected cost by λ {\displaystyle \lambda } , but makes coverage of all elements likely. The idea is to choose λ {\displaystyle \lambda } as small as possible so that all elements are provably covered with non-zero probability. Here is a detailed analysis. ==== Lemma (approximation guarantee for rounding scheme) ==== Fix λ = ln ( 2 | U | ) {\displaystyle \lambda =\ln(2|{\mathcal {U}}|)} . With positive probability, the rounding scheme returns a set cover x ′ {\displaystyle x'} of cost at most 2 ln ( 2 | U | ) c ⋅ x ∗ {\displaystyle 2\ln(2|{\mathcal {U}}|)c\cdot x^{}} (and thus of cost O ( log | U | ) {\displaystyle O(\log |{\mathcal {U}}|)} times the cost of the optimal set cover). (Note: with care the O ( log | U | ) {\displaystyle O(\log |{\mathcal {U}}|)} can be reduced to ln ( | U | ) + O ( log log | U | ) {\displaystyle \ln(|{\mathcal {U}}|)+O(\log \log |{\mathcal {U}}|)} .) ==== Proof ==== The output x ′ {\displaystyle x'} of the random rounding scheme has the desired properties as long as none of the following "bad" events occur: the cost c ⋅ x ′ {\displaystyle c\cdot x'} of x ′ {\displaystyle x'} exceeds 2 λ c ⋅ x ∗ {\displaystyle 2\lambda c\cdot x^{}} , or for some element e {\displaystyle e} , x ′ {\displaystyle x'} fails to cover e {\displaystyle e} . The expectation of each x s ′ {\displaystyle x'_{s}} is at most λ x s ∗ {\displaystyle \lambda x_{s