In automata theory (a branch of theoretical computer science), DFA minimization is the task of transforming a given deterministic finite automaton (DFA) into an equivalent DFA that has a minimum number of states. Here, two DFAs are called equivalent if they recognize the same regular language. Several different algorithms accomplishing this task are known and described in standard textbooks on automata theory. == Minimal DFA == For each regular language, there also exists a minimal automaton that accepts it, that is, a DFA with a minimum number of states and this DFA is unique (except that states can be given different names). The minimal DFA ensures minimal computational cost for tasks such as pattern matching. There are three classes of states that can be removed or merged from the original DFA without affecting the language it accepts. Unreachable states are the states that are not reachable from the initial state of the DFA, for any input string. These states can be removed. Dead states are the states from which no final state is reachable. These states can be removed unless the automaton is required to be complete. Nondistinguishable states are those that cannot be distinguished from one another for any input string. These states can be merged. DFA minimization is usually done in three steps: remove dead and unreachable states (this will accelerate the following step), merge nondistinguishable states, optionally, re-create a single dead state ("sink" state) if the resulting DFA is required to be complete. == Unreachable states == The state p {\displaystyle p} of a deterministic finite automaton M = ( Q , Σ , δ , q 0 , F ) {\displaystyle M=(Q,\Sigma ,\delta ,q_{0},F)} is unreachable if no string w {\displaystyle w} in Σ ∗ {\displaystyle \Sigma ^{}} exists for which p = δ ∗ ( q 0 , w ) {\displaystyle p=\delta ^{}(q_{0},w)} . In this definition, Q {\displaystyle Q} is the set of states, Σ {\displaystyle \Sigma } is the set of input symbols, δ {\displaystyle \delta } is the transition function (mapping a state and an input symbol to a set of states), δ ∗ {\displaystyle \delta ^{}} is its extension to strings (also known as extended transition function), q 0 {\displaystyle q_{0}} is the initial state, and F {\displaystyle F} is the set of accepting (also known as final) states. Reachable states can be obtained with the following algorithm: Assuming an efficient implementation of the state sets (e.g. new_states) and operations on them (such as adding a state or checking whether it is present), this algorithm can be implemented with time complexity O ( n + m ) {\displaystyle O(n+m)} , where n {\displaystyle n} is the number of states and m {\displaystyle m} is the number of transitions of the input automaton. Unreachable states can be removed from the DFA without affecting the language that it accepts. == Nondistinguishable states == The following algorithms present various approaches to merging nondistinguishable states. === Hopcroft's algorithm === One algorithm for merging the nondistinguishable states of a DFA, due to Hopcroft (1971), is based on partition refinement, partitioning the DFA states into groups by their behavior. These groups represent equivalence classes of the Nerode congruence, whereby every two states are equivalent if they have the same behavior for every input sequence. That is, for every two states p1 and p2 that belong to the same block of the partition P, and every input word w, the transitions determined by w should always take states p1 and p2 to either states that both accept or states that both reject. It should not be possible for w to take p1 to an accepting state and p2 to a rejecting state or vice versa. The following pseudocode describes the form of the algorithm as given by Xu. Alternative forms have also been presented. The algorithm starts with a partition that is too coarse: every pair of states that are equivalent according to the Nerode congruence belong to the same set in the partition, but pairs that are inequivalent might also belong to the same set. It gradually refines the partition into a larger number of smaller sets, at each step splitting sets of states into pairs of subsets that are necessarily inequivalent. The initial partition is a separation of the states into two subsets of states that clearly do not have the same behavior as each other: the accepting states and the rejecting states. The algorithm then repeatedly chooses a set A from the current partition and an input symbol c, and splits each of the sets of the partition into two (possibly empty) subsets: the subset of states that lead to A on input symbol c, and the subset of states that do not lead to A. Since A is already known to have different behavior than the other sets of the partition, the subsets that lead to A also have different behavior than the subsets that do not lead to A. When no more splits of this type can be found, the algorithm terminates. Lemma. Given a fixed character c and an equivalence class Y that splits into equivalence classes B and C, only one of B or C is necessary to refine the whole partition. Example: Suppose we have an equivalence class Y that splits into equivalence classes B and C. Suppose we also have classes D, E, and F; D and E have states with transitions into B on character c, while F has transitions into C on character c. By the Lemma, we can choose either B or C as the distinguisher, let's say B. Then the states of D and E are split by their transitions into B. But F, which doesn't point into B, simply doesn't split during the current iteration of the algorithm; it will be refined by other distinguisher(s). Observation. All of B or C is necessary to split referring classes like D, E, and F correctly—subsets won't do. The purpose of the outermost if statement (if Y is in W) is to patch up W, the set of distinguishers. We see in the previous statement in the algorithm that Y has just been split. If Y is in W, it has just become obsolete as a means to split classes in future iterations. So Y must be replaced by both splits because of the Observation above. If Y is not in W, however, only one of the two splits, not both, needs to be added to W because of the Lemma above. Choosing the smaller of the two splits guarantees that the new addition to W is no more than half the size of Y; this is the core of the Hopcroft algorithm: how it gets its speed, as explained in the next paragraph. The worst case running time of this algorithm is O(ns log n), where n is the number of states and s is the size of the alphabet. This bound follows from the fact that, for each of the ns transitions of the automaton, the sets drawn from Q that contain the target state of the transition have sizes that decrease relative to each other by a factor of two or more, so each transition participates in O(log n) of the splitting steps in the algorithm. The partition refinement data structure allows each splitting step to be performed in time proportional to the number of transitions that participate in it. This remains the most efficient algorithm known for solving the problem, and for certain distributions of inputs its average-case complexity is even better, O(n log log n). Once Hopcroft's algorithm has been used to group the states of the input DFA into equivalence classes, the minimum DFA can be constructed by forming one state for each equivalence class. If S is a set of states in P, s is a state in S, and c is an input character, then the transition in the minimum DFA from the state for S, on input c, goes to the set containing the state that the input automaton would go to from state s on input c. The initial state of the minimum DFA is the one containing the initial state of the input DFA, and the accepting states of the minimum DFA are the ones whose members are accepting states of the input DFA. === Moore's algorithm === Moore's algorithm for DFA minimization is due to Edward F. Moore (1956). Like Hopcroft's algorithm, it maintains a partition that starts off separating the accepting from the rejecting states, and repeatedly refines the partition until no more refinements can be made. At each step, it replaces the current partition with the coarsest common refinement of s + 1 partitions, one of which is the current one and the rest of which are the preimages of the current partition under the transition functions for each of the input symbols. The algorithm terminates when this replacement does not change the current partition. Its worst-case time complexity is O(n2s): each step of the algorithm may be performed in time O(ns) using a variant of radix sort to reorder the states so that states in the same set of the new partition are consecutive in the ordering, and there are at most n steps since each one but the last increases the number of sets in the partition. The instances of the DFA minimization problem that cause the worst-case behavior are the same as for Hopcroft's algorithm. The number of steps th
Azure Stream Analytics
Microsoft Azure Stream Analytics is a serverless scalable complex event processing engine by Microsoft that enables users to develop and run real-time analytics on multiple streams of data from sources such as devices, sensors, web sites, social media, and other applications. Users can set up alerts to detect anomalies, predict trends, trigger necessary workflows when certain conditions are observed, and make data available to other downstream applications and services for presentation, archiving, or further analysis. == Query Language == Users can author real-time analytics using a simple declarative SQL-like language with embedded support for temporal logic. Callouts to custom code with JavaScript user defined functions extend the streaming logic written in SQL. Callouts to Azure Machine Learning helps with predictive scoring on streaming data. == Scalability == Azure Stream Analytics is a serverless job service on Azure that eliminates the need for infrastructure, servers, virtual machines, or managed clusters. Users only pay for the processing used for the running jobs. == IoT applications == Azure Stream Analytics integrates with Azure IoT Hub to enable real-time analytics on data from IoT devices and applications. == Real-time Dashboards == Users can build real-time dashboards with Power BI for a live command and control view. Real-time dashboards help transform live data into actionable and insightful visuals. == Data Input Sources == Stream Analytics supports three different types of input sources - Azure Event Hubs, Azure IoT Hubs, and Azure Blob Storage. Additionally, stream analytics supports Azure Blob storage as the input reference data to help augment fast moving event data streams with static data. Stream analytics supports a wide variety of output targets. Support for Power BI allows for real-time dashboarding. Event Hub, Service bus topics and queues help trigger downstream workflows. Support for Azure Table Storage, Azure SQL Databases, Azure SQL Data Warehouse, Azure SQL, Document DB, Azure Data Lake Store enable a variety of downstream analysis and archiving capabilities.
ZipBooks
ZipBooks is a free online accounting software company based in American Fork, Utah. The cloud-based software is an accounting and bookkeeping tool that helps business owners process credit cards, track finances, and send invoices, among other features. == History == ZipBooks was founded by Tim Chaves in June 2015, backed by venture capital firm Peak Ventures. The company secured an additional $2 million of funding in July 2016, and in 2017 it was awarded a $100,000 economic grant by the Utah Governor's Office of Economic Development Technology Commercialization and Innovation Program. == Products == ZipBooks' core modules are invoicing, transactions, bills, reporting, time tracking, contacts, and payroll. Accrual accounting was added in 2017. The application is available on G Suite, iOS, Slack, and as a web application. == Reception == Computerworld compared ZipBooks favorably with other accounting software. PC Magazine praised its user experience, but stated it lacked "a lot of features that competing sites offer".
Geo-replication
Geo-replication systems are designed to provide improved availability and disaster tolerance by using geographically distributed data centers. This is intended to improve the response time for applications such as web portals. Geo-replication can be achieved using software, hardware or a combination of the two. == Software == Geo-replication software is a network performance-enhancing technology that is designed to provide improved access to portal or intranet content for users at the most remote parts of large organizations. It is based on the principle of storing complete replicas of portal content on local servers, and then keeping the content on those servers up-to-date using heavily compressed data updates. === Portal acceleration === Geo-replication technologies are used to provide replication of the content of portals, intranets, web applications, content and data between servers, across wide area networks WAN to allow users at remote sites to access central content at LAN speeds. Geo-replication software can improve the performance of data networks that suffer limited bandwidth, latency and periodic disconnection. Terabytes of data can be replicated over a wide area network, giving remote sites faster access to web applications. Geo-replication software uses a combination of data compression and content caching technologies. differencing technologies can also be employed to reduce the volume of data that has to be transmitted to keep portal content accurate across all servers. This update compression can reduce the load that portal traffic places on networks, and improve the response time of a portal. === Portal replication === Remote users of web portals and collaboration environments will frequently experience network bandwidth and latency problems which will slow down their experience of opening and closing files, and otherwise interacting with the portal. Geo-replication technology is deployed to accelerate the remote end user portal performance to be equivalent to that experienced by users locally accessing the portal in the central office. === Differencing engine technologies === To deliver this reduction in the size of the required data updates across a portal, geo-replication systems often use differencing engine technologies. These systems are able to difference the content of each portal server right down to the byte level. This knowledge of the content that is already on each server enables the system to rebuild any changes to the content on one server, across each of the other servers in the deployment from content already hosted on those other servers. This type of differencing system ensures that no content, at the byte level, is ever sent to a server twice. === Offline portal replication on laptops === Geo-replication systems are often extended to deliver local replication beyond the server and down to the laptop used by a single user. Server to laptop replication enables mobile users to have access to a local replica of their business portal on a standard laptop. This technology may be employed to provide in the field access to portal content by, for example, sales forces and combat forces. == Geo-replication systems ==
RCUDA
rCUDA, which stands for Remote CUDA, is a type of middleware software framework for remote GPU virtualization. Fully compatible with the CUDA application programming interface (API), it allows the allocation of one or more CUDA-enabled GPUs to a single application. Each GPU can be part of a cluster or running inside of a virtual machine. The approach is aimed at improving performance in GPU clusters that are lacking full utilization. GPU virtualization reduces the number of GPUs needed in a cluster, and in turn, leads to a lower cost configuration – less energy, acquisition, and maintenance. The recommended distributed acceleration architecture is a high performance computing cluster with GPUs attached to only a few of the cluster nodes. When a node without a local GPU executes an application needing GPU resources, remote execution of the kernel is supported by data and code transfers between local system memory and remote GPU memory. rCUDA is designed to accommodate this client-server architecture. On one end, clients employ a library of wrappers to the high-level CUDA Runtime API, and on the other end, there is a network listening service that receives requests on a TCP port. Several nodes running different GPU-accelerated applications can concurrently make use of the whole set of accelerators installed in the cluster. The client forwards the request to one of the servers, which accesses the GPU installed in that computer and executes the request in it. Time-multiplexing the GPU, or in other words sharing it, is accomplished by spawning different server processes for each remote GPU execution request. == rCUDA v20.07 == The rCUDA middleware enables the concurrent usage of CUDA-compatible devices remotely. rCUDA employs either the InfiniBand network or the socket API for the communication between clients and servers. rCUDA can be useful in three different environments: Clusters. To reduce the number of GPUs installed in High Performance Clusters. This leads to energy savings, as well as other related savings like acquisition costs, maintenance, space, cooling, etc. Academia. In commodity networks, to offer access to a few high performance GPUs concurrently to many students. Virtual Machines. To enable the access to the CUDA facilities on the physical machine. The current version of rCUDA (v20.07) supports CUDA version 9.0, excluding graphics interoperability. rCUDA v20.07 targets the Linux OS (for 64-bit architectures) on both client and server sides. CUDA applications do not need any change in their source code in order to be executed with rCUDA.
Box blur
A box blur (also known as a box linear filter) is a spatial domain linear filter in which each pixel in the resulting image has a value equal to the average value of its neighboring pixels in the input image. It is a form of low-pass ("blurring") filter. A 3 by 3 box blur ("radius 1") can be written as matrix 1 9 [ 1 1 1 1 1 1 1 1 1 ] . {\displaystyle {\frac {1}{9}}{\begin{bmatrix}1&1&1\\1&1&1\\1&1&1\end{bmatrix}}.} Due to its property of using equal weights, it can be implemented using a much simpler accumulation algorithm, which is significantly faster than using a sliding-window algorithm. Box blurs are frequently used to approximate a Gaussian blur. By the central limit theorem, repeated application of a box blur will approximate a Gaussian blur. In the frequency domain, a box blur has zeros and negative components. That is, a sine wave with a period equal to the size of the box will be blurred away entirely, and wavelengths shorter than the size of the box may be phase-reversed, as seen when two bokeh circles touch to form a bright spot where there would be a dark spot between two bright spots in the original image. == Extensions == Gwosdek, et al. has extended Box blur to take a fractional radius: the edges of the 1-D filter are expanded with a fraction. It makes slightly better gaussian approximation possible due to the elimination of integer-rounding error. Mario Klingemann has a "stack blur" that tries to better emulate gaussian's look in one pass by stacking weights: 1 9 [ 1 2 3 2 1 ] {\displaystyle {\frac {1}{9}}{\begin{bmatrix}1&2&3&2&1\end{bmatrix}}} The triangular impulse response it forms decomposes to two rounds of box blur. Stacked Integral Image by Bhatia et al. takes the weighted average of a few box blurs to fit the gaussian response curve. == Implementation == The following pseudocode implements a 3x3 box blur. The example does not handle the edges of the image, which would not fit inside the kernel, so that these areas remain unblurred. In practice, the issue is better handled by: Introducing an alpha channel to represent the absence of colors; Extending the boundary by filling in values, ranked by quality: Fill in a mirrored image at the border Fill in a constant color extending from the last pixel Pad in a fixed color A number of optimizations can be applied when implementing the box blur of a radius r and N pixels: The box blur is a separable filter, so that only two 1D passes of averaging 2 r + 1 pixels will be needed, one horizontal and one vertical, for each pixel. This lowers the complexity from O(Nr2) to O(Nr). In digital signal processing terminology, each pass is a moving-average filter. Accumulation. Instead of discarding the sum for each pixel, the algorithm re-uses the previous sum, and updates it by subtracting away the old pixel and adding the new pixel in the blurring range. A summed-area table can be used similarly. This lowers the complexity from O(Nr) to O(N). When being used in multiple passes to approximate a Gaussian blur, the cascaded integrator–comb filter construction allows for doing the equivalent operation in a single pass.
Too Good To Go
Too Good To Go is a service with a mobile application that connects customers to restaurants and stores that have surplus unsold food. The service covers major European cities, and in October 2020 started operations in North America. As part of the initiatives taken on the International Day of Awareness of Food Loss and Waste to reduce food loss and waste, the app is suggested alongside OLIO among many others. In 2023 Too Good To Go was the fastest-growing sustainable food app startup by number of downloads. As of August 2023, it claimed 164,000 businesses, serving 62 million users, have saved 155 million bags of food. As of March 2023, it claimed to have saved over 200 million meals. == History == The company was created in 2015 in Denmark by Thomas Bjørn Momsen, Klaus Bagge Pedersen, Adam Sigbrand and Brian Christensen. In 2017, Mette Lykke (co-founder of Endomondo) joined as CEO. In February 2019, the company raised an additional 6 million euros in a new round of investment. In August 2019, Too Good To Go was re-launched in Austria. In September 2019, Too Good To Go acquired the Spanish startup weSAVEeat and merged it into its own brand. In November 2019, the offer of Too Good To Go extended to plants through a partnership with the French retail plants company Jardiland. In December 2019, Too Good To Go partnered with the French grocery retail stores Intermarché, and donated 60K euros to the French charity Restaurants du Cœur. In October 2021, Bonnie Wright teamed up with Too Good To Go to drive the initiative to reduce food waste. == Corporate affairs == The key trends for the Danish entity Too Good To Go ApS are (as of the financial year ending December 31): == International expansion == As of March 2026 the company serves the European countries Austria, Belgium, Czechia, Denmark, the Faroe Islands, France, Germany, Ireland, Italy, the Netherlands, Norway, Poland, Portugal, Spain, Sweden, Switzerland, the United Kingdom. Outside of Europe the service is available in Australia, Canada, Japan, New Zealand and the United States. == Purpose == The purpose of Too Good To Go is to reduce food waste worldwide. It developed a mobile application that connects restaurants and stores that have unsold, surplus food, with customers who can then buy whatever food the outlet considers surplus to requirements—without being able to choose—at a much lower price than normal. The food on the app is priced at one-third its original price. The company claims this reduces the waste of food that would otherwise be discarded; food waste is a global problem that affects the environment. In three years active, the app reached more than 9.5 million users. As of 2022, more than 57.7 million users and 154,000 establishments have signed up, and 139 million meals have been collected. In 2019, the company had 350 employees in Europe. As of June 2023 the company was estimated to have 1,289 employees. == Use == Food outlets must notify the TGTG company about what they have available on each day, stating what sort of food they have (baked foods, meals, produce, vegan food), and the price for a 'surprise bag', whose contents they determine; the user cannot choose, but the original prices will be three or more times the TGTG price. Notification is made early based upon the quantity predicted to be left over, not at the end of a selling period. Users must register to use the service. A mobile phone with an Internet connection running Android or iOS is needed. The user runs the TGTG app, which lists outlets available within a chosen distance and time range. The customer can then order and pay for a 'surprise bag'. The supplier can cancel an order at any time if the expected surplus is not available—the purchaser is notified by text message—and the purchaser can cancel with two hours' notice. The phone must be taken to the food supplier in a specified pickup time window, often 30 or 60 minutes long, and the transaction is finalised by swiping the app—connected to the Internet—to confirm collection.