In computer science, an enumeration algorithm is an algorithm that enumerates the answers to a computational problem. Formally, such an algorithm applies to problems that take an input and produce a list of solutions, similarly to function problems. For each input, the enumeration algorithm must produce the list of all solutions, without duplicates, and then halt. The performance of an enumeration algorithm is measured in terms of the time required to produce the solutions, either in terms of the total time required to produce all solutions, or in terms of the maximal delay between two consecutive solutions and in terms of a preprocessing time, counted as the time before outputting the first solution. This complexity can be expressed in terms of the size of the input, the size of each individual output, or the total size of the set of all outputs, similarly to what is done with output-sensitive algorithms. == Formal definitions == An enumeration problem P {\displaystyle P} is defined as a relation R {\displaystyle R} over strings of an arbitrary alphabet Σ {\displaystyle \Sigma } : R ⊆ Σ ∗ × Σ ∗ {\displaystyle R\subseteq \Sigma ^{}\times \Sigma ^{}} An algorithm solves P {\displaystyle P} if for every input x {\displaystyle x} the algorithm produces the (possibly infinite) sequence y {\displaystyle y} such that y {\displaystyle y} has no duplicate and z ∈ y {\displaystyle z\in y} if and only if ( x , z ) ∈ R {\displaystyle (x,z)\in R} . The algorithm should halt if the sequence y {\displaystyle y} is finite. == Common complexity classes == Enumeration problems have been studied in the context of computational complexity theory, and several complexity classes have been introduced for such problems. A very general such class is EnumP, the class of problems for which the correctness of a possible output can be checked in polynomial time in the input and output. Formally, for such a problem, there must exist an algorithm A which takes as input the problem input x, the candidate output y, and solves the decision problem of whether y is a correct output for the input x, in polynomial time in x and y. For instance, this class contains all problems that amount to enumerating the witnesses of a problem in the class NP. Other classes that have been defined include the following. In the case of problems that are also in EnumP, these problems are ordered from least to most specific: Output polynomial, the class of problems whose complete output can be computed in polynomial time. Incremental polynomial time, the class of problems where, for all i, the i-th output can be produced in polynomial time in the input size and in the number i. Polynomial delay, the class of problems where the delay between two consecutive outputs is polynomial in the input (and independent from the output). Strongly polynomial delay, the class of problems where the delay before each output is polynomial in the size of this specific output (and independent from the input or from the other outputs). The preprocessing is generally assumed to be polynomial. Constant delay, the class of problems where the delay before each output is constant, i.e., independent from the input and output. The preprocessing phase is generally assumed to be polynomial in the input. == Common techniques == Backtracking: The simplest way to enumerate all solutions is by systematically exploring the space of possible results (partitioning it at each successive step). However, performing this may not give good guarantees on the delay, i.e., a backtracking algorithm may spend a long time exploring parts of the space of possible results that do not give rise to a full solution. Flashlight search: This technique improves on backtracking by exploring the space of all possible solutions but solving at each step the problem of whether the current partial solution can be extended to a partial solution. If the answer is no, then the algorithm can immediately backtrack and avoid wasting time, which makes it easier to show guarantees on the delay between any two complete solutions. In particular, this technique applies well to self-reducible problems. Closure under set operations: If we wish to enumerate the disjoint union of two sets, then we can solve the problem by enumerating the first set and then the second set. If the union is non disjoint but the sets can be enumerated in sorted order, then the enumeration can be performed in parallel on both sets while eliminating duplicates on the fly. If the union is not disjoint and both sets are not sorted then duplicates can be eliminated at the expense of a higher memory usage, e.g., using a hash table. Likewise, the cartesian product of two sets can be enumerated efficiently by enumerating one set and joining each result with all results obtained when enumerating the second step. == Examples of enumeration problems == The vertex enumeration problem, where we are given a polytope described as a system of linear inequalities and we must enumerate the vertices of the polytope. Enumerating the minimal transversals of a hypergraph. This problem is related to monotone dualization and is connected to many applications in database theory and graph theory. Enumerating the answers to a database query, for instance a conjunctive query or a query expressed in monadic second-order. There have been characterizations in database theory of which conjunctive queries could be enumerated with linear preprocessing and constant delay. The problem of enumerating maximal cliques in an input graph, e.g., with the Bron–Kerbosch algorithm Listing all elements of structures such as matroids and greedoids Several problems on graphs, e.g., enumerating independent sets, paths, cuts, etc. Enumerating the satisfying assignments of representations of Boolean functions, e.g., a Boolean formula written in conjunctive normal form or disjunctive normal form, a binary decision diagram such as an OBDD, or a Boolean circuit in restricted classes studied in knowledge compilation, e.g., NNF. == Connection to computability theory == The notion of enumeration algorithms is also used in the field of computability theory to define some high complexity classes such as RE, the class of all recursively enumerable problems. This is the class of sets for which there exist an enumeration algorithm that will produce all elements of the set: the algorithm may run forever if the set is infinite, but each solution must be produced by the algorithm after a finite time.
TSheets
TSheets was a web-based and mobile time tracking and employee scheduling app. The service was accessed via a web browser or a mobile app. TSheets was an alternative to a paper timesheet or punch cards. == History == Based in Eagle, Idaho, TSheets was co-founded in 2006 by CEO Matt Rissell and CTO Brandon Zehm. In 2008, TSheets released a native employee time tracking app for the iPhone. In 2012, TSheets released an integration with accounting and payroll software QuickBooks. In 2015, TSheets accepted $15 million in growth equity funding from Summit Partners, bought a building in Eagle, Idaho, and opened a second location in Sydney, Australia. On 5 December 2017, Intuit announced an agreement to acquire TSheets. The transaction was valued at approximately $340 million of cash and other consideration and closed on 11 January 2018. After the transaction closed, Time Capture became a new business unit within Intuit's Small Business and Self-Employed Group with Matt Rissell assuming the leader role reporting to Alex Chriss. TSheets's Eagle, Idaho site became an Intuit location.
Vx-underground
vx-underground, also known as VXUG, is an educational website about malware and cybersecurity. It claims to have the largest online repository of malware. The site was launched in May, 2019 and has grown to host over 35 million pieces of malware samples. On their account on Twitter, VXUG reports on and verifies cybersecurity breaches. == Reception == Kim Crawley compared the site to VirusTotal and states that vx-underground is more susceptible to suspicion for law enforcement. == Data breach reports == In May 2024, the International Baccalaureate organizations faced allegations over supposed breaches in their IT infrastructure after an incident of examination leaks. Upon inspecting leaked data, VXUG were the first to report that the breach seemed legitimate on the morning of May 6.
Centurion Guard
Centurion Guard is a PC hardware and software-based security product, developed by Centurion Technologies. It was first released in 1996. There were several different releases and versions of this product, and many were distributed in computers donated to libraries by the Bill & Melinda Gates Foundation. == Operating system compatibility == Microsoft Windows 7 Microsoft Windows Vista Microsoft Windows XP
Imieliński–Lipski algebra
In database theory, Imieliński–Lipski algebra is an extension of relational algebra onto tables with different types of null values. It is used to operate on relations with incomplete information. Imieliński–Lipski algebras are defined to satisfy precise conditions for semantically meaningful extension of the usual relational operators, such as projection, selection, union, and join, from operators on relations to operators on relations with various kinds of "null values". These conditions require that the system be safe in the sense that no incorrect conclusion is derivable by using a specified subset F of the relational operators; and that it be complete in the sense that all valid conclusions expressible by relational expressions using operators in F are in fact derivable in this system. For example, it is well known that the three-valued logic approach to deal with null values, supported treatment of nulls values by SQL is not complete, see Ullman book. To show this, let T be: Take SQL query Q SQL query Q will return empty set (no results) under 3-valued semantics currently adopted by all variants of SQL. This is the case because in SQL, NULL is never equal to any constant – in this case, neither to “Spring” nor “Fall” nor “Winter” (if there is Winter semester in this school). NULL='Spring' will evaluate to MAYBE and so will NULL='Fall'. The disjunction MAYBE OR MAYBE evaluates to MAYBE (not TRUE). Thus Igor will not be part of the answer (and of course neither will Rohit). But Igor should be returned as the answer. Indeed, regardless what semester Igor took the Networks class (no matter what was the unknown value of NULL), the selection condition will be true. This “Igor” will be missed by SQL and the SQL answer would be incomplete according to completeness requirements specified in Tomasz Imieliński, Witold Lipski, 'Incomplete Information in Relational Databases'. It is also argued there that 3-valued logic (TRUE, FALSE, MAYBE) can never provide guarantee of complete answer for tables with incomplete information. Three algebras which satisfy conditions of safety and completeness are defined as Imielinski–Lipski algebras: the Codd-Tables algebra, the V-tables algebra and the Conditional tables (C-tables) algebra. == Codd-tables algebra == Codd-tables algebra is based on the usual Codd's single NULL values. The table T above is an example of Codd-table. Codd-table algebra supports projection and positive selections only. It is also demonstrated in [IL84 that it is not possible to correctly extend more relational operators over Codd-Tables. For example, such basic operation as join is not extendable over Codd-tables. It is not possible to define selections with Boolean conditions involving negation and preserve completeness. For example, queries like the above query Q cannot be supported. In order to be able to extend more relational operators, more expressive form of null value representation is needed in tables which are called V-table. == V-tables algebra == V-tables algebra is based on many different ("marked") null values or variables allowed to appear in a table. V-tables allow to show that a value may be unknown but the same for different tuples. For example, in the table below Gaurav and Igor order the same (but unknown) beer in two unknown bars (which may, or may not be different – but remain unknown). Gaurav and Jane frequent the same unknown bar (Y1). Thus, instead one NULL value, we use indexed variables, or Skolem constants . V-tables algebra is shown to correctly support projection, positive selection (with no negation occurring in the selection condition), union, and renaming of attributes, which allows for processing arbitrary conjunctive queries. A very desirable property enjoyed by the V-table algebra is that all relational operators on tables are performed in exactly the same way as in the case of the usual relations. === Conditional tables (c-tables) algebra === Example of conditional table (c-table) is shown below. It has additional column “con” which is a Boolean condition involving variables, null values – same as in V-tables. over the following table c-table Conditional tables algebra, mainly of theoretical interest, supports projection, selection, union, join, and renaming. Under closed-world assumption, it can also handle the operator of difference, thus it can support all relational operators. == History == Imieliński–Lipski algebras were introduced by Tomasz Imieliński and Witold Lipski Jr. in Incomplete Information in Relational Databases.
Lucy–Hook coaddition method
The Lucy–Hook coaddition method is an image processing technique for combining sub-stepped astronomical image data onto a finer grid. The method allows the option of resolution and contrast enhancement or the choice of a conservative, re-convolved, output. Tests with very deep Hubble Space Telescope Wide Field and Planetary Camera 2 (WFPC2) imaging data of excellent quality show that these methods can be very effective and allow fine-scale features to be studied better than on the unprocessed images. The Lucy–Hook coaddition method is an extension of the standard Richardson–Lucy deconvolution iterative restoration method. For many purposes it may be more convenient to combine dithered datasets using the Drizzle method.
VK Video
VK Video is an internet video hosting service launched by VK (formerly known as Mail.ru Group) in 2021. It is positioned as a Russian alternative to the international platform YouTube. == History == The "VK Video" service began operations on October 15, 2021, following the merger of video platforms belonging to the social networks "VKontakte" and "Odnoklassniki". The launch of "VK Video" was managed by a team of executives led by VKontakte CEO Marina Krasnova, who worked at the company until 2023. Its launch was intended as an alternative to the international platform YouTube, which Russian authorities sought to replace with "domestic analogs. Key differences of the Russian service became the presence of pirated materials. Videos from the American video hosting site were uploaded en masse to "VK Video," which even caused the service to be temporarily blocked by YouTube. From 2022, to attract users, VKontakte's management bet on working with famous bloggers, specifically purchasing the shows "What Happened Next?" (ChBD) and "Vnutri Lapenko". Among the bloggers recruited to promote the service was the popular video blogger Vlad A4. An additional advantage for creators was the availability of monetization, which had been unavailable on YouTube for users from the Russian Federation since 2022. In September 2023, a separate "VK Video" mobile app appeared. In total, by the end of 2023, the monthly audience of "VK Video" reached 67.9 million users (which is almost 30 million less than YouTube). In the summer of 2024, following the blocking of YouTube in Russia, the service's traffic grew sharply: in August, its audience increased by more than two times compared to July. In the same month, "VK Video" took second place in downloads among free apps in the App Store and third in Google Play. In December 2024, the service received its own domain: vkvideo.ru. For the first time, "VK Video" managed to surpass YouTube in monthly audience in Russia in July 2025: the Russian service attracted 76.4 million viewers, whereas YouTube's reach amounted to 74.9 million people. == Platform features == On "VK Video," a view is recorded from the first second, whereas on YouTube it is only from the thirtieth. At the same time, a significant portion of comments are left by bots. For videos from the platform's most popular bloggers, the engagement level (likes to views) does not reach 4%. The "Trends" section most often features videos from large channels where the ratio of likes to views does not exceed 2%. == Management == In April 2025, the post of General Director of "VK Video" was taken by Marianna Maksimovskaya. From June 2022 to July 2024, the development of the platform was led by Fyodor Yezhov, who was primarily responsible for its technical direction. == Awards == In 2023, VK Video was awarded the Runet Prize in the "Science, Technology and Innovation" category.