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73-450: OLAP tools enable users to analyse multidimensional data interactively from multiple perspectives. OLAP consists of three basic analytical operations: consolidation (roll-up), drill-down, and slicing and dicing. Consolidation involves the aggregation of data that can be accumulated and computed in one or more dimensions. For example, all sales offices are rolled up to the sales department or sales division to anticipate sales trends. By contrast,

146-415: A dimension . Each Sale has a Date/Time label that describes more about that sale. For example: Multidimensional structure is defined as "a variation of the relational model that uses multidimensional structures to organize data and express the relationships between data". The structure is broken into cubes and the cubes are able to store and access data within the confines of each cube. "Each cell within

219-489: A graph isomorphism exists between two graphs. Two graphs are isomorphic if one can be transformed into the other simply by renaming vertices . Consider these two problems: The Subgraph Isomorphism problem is NP-complete. The graph isomorphism problem is suspected to be neither in P nor NP-complete, though it is in NP. This is an example of a problem that is thought to be hard , but is not thought to be NP-complete. This class

292-426: A relational database . Measures are derived from the records in the fact table and dimensions are derived from the dimension tables . Each measure can be thought of as having a set of labels , or meta-data associated with it. A dimension is what describes these labels ; it provides information about the measure . A simple example would be a cube that contains a store's sales as a measure , and Date/Time as

365-616: A broader perspective of a problem unlike other models. It has been claimed that for complex queries OLAP cubes can produce an answer in around 0.1% of the time required for the same query on OLTP relational data. The most important mechanism in OLAP which allows it to achieve such performance is the use of aggregations . Aggregations are built from the fact table by changing the granularity on specific dimensions and aggregating up data along these dimensions, using an aggregate function (or aggregation function ). The number of possible aggregations

438-445: A cache. Advantages of MOLAP Disadvantages of MOLAP Examples of commercial products that use MOLAP are Cognos Powerplay, Oracle Database OLAP Option , MicroStrategy , Microsoft Analysis Services , Essbase , TM1 , Jedox , and icCube. ROLAP works directly with relational databases and does not require pre-computation. The base data and the dimension tables are stored as relational tables and new tables are created to hold

511-428: A company might wish to summarize financial data by product, by time-period, and by city to compare actual and budget expenses. Product, time, city and scenario (actual and budget) are the data's dimensions. Cube is a shorthand for multidimensional dataset , given that data can have an arbitrary number of dimensions . The term hypercube is sometimes used, especially for data with more than three dimensions. A cube

584-484: A cube with hierarchical dimensions leads to conceptually straightforward operations to facilitate analysis. Aligning the data content with a familiar visualization enhances analyst learning and productivity. The user-initiated process of navigating by calling for page displays interactively, through the specification of slices via rotations and drill down/up is sometimes called "slice and dice". Common operations include slice and dice, drill down, roll up, and pivot. Slice

657-807: A database will divide data between relational and specialized storage. For example, for some vendors, a HOLAP database will use relational tables to hold the larger quantities of detailed data and use specialized storage for at least some aspects of the smaller quantities of more-aggregate or less-detailed data. HOLAP addresses the shortcomings of MOLAP and ROLAP by combining the capabilities of both approaches. HOLAP tools can utilize both pre-calculated cubes and relational data sources. In this mode HOLAP stores aggregations in MOLAP for fast query performance, and detailed data in ROLAP to optimize time of cube processing . In this mode HOLAP stores some slice of data, usually

730-421: A dicing operation: The new cube shows the sales figures of a limited number of product categories, the time and region dimensions cover the same range as before. Drill Down/Up allows the user to navigate among levels of data ranging from the most summarized (up) to the most detailed (down). The picture shows a drill-down operation: The analyst moves from the summary category "Outdoor protective equipment" to see

803-473: A dimension can be organized as a hierarchy , a set of parent-child relationships, typically where a parent member summarizes its children. Parent elements can further be aggregated as the children of another parent. For example, May 2005's parent is Second Quarter 2005 which is in turn the child of Year 2005. Similarly cities are the children of regions; products roll into product groups and individual expense items into types of expenditure. Conceiving data as

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876-560: A multidimensional structure contains aggregated data related to elements along each of its dimensions". Even when data is manipulated it remains easy to access and continues to constitute a compact database format. The data still remains interrelated. Multidimensional structure is quite popular for analytical databases that use online analytical processing (OLAP) applications. Analytical databases use these databases because of their ability to deliver answers to complex business queries swiftly. Data can be viewed from different angles, which gives

949-428: A particular quarter. Pivoting could replace products with time periods to see data across time for a single product. The picture shows a pivoting operation: The whole cube is rotated, giving another perspective on the data. In database theory , an OLAP cube is an abstract representation of a projection of an RDBMS relation. Given a relation of order N , consider a projection that subtends X , Y , and Z as

1022-428: A polynomial time algorithm, all problems in NP do. The set of NP-complete problems is often denoted by NP-C or NPC . Although a solution to an NP-complete problem can be verified "quickly", there is no known way to find a solution quickly. That is, the time required to solve the problem using any currently known algorithm increases rapidly as the size of the problem grows. As a consequence, determining whether it

1095-504: A result, Codd's "twelve laws of online analytical processing" were explicit in their reference to Essbase. There was some ensuing controversy and when Computerworld learned that Codd was paid by Arbor, it retracted the article. The OLAP market experienced strong growth in the late 1990s with dozens of commercial products going into market. In 1998, Microsoft released its first OLAP Server – Microsoft Analysis Services , which drove wide adoption of OLAP technology and moved it into

1168-400: A subroutine that solves Y {\displaystyle \scriptstyle Y} in polynomial time, one could write a program that calls this subroutine and solves X {\displaystyle \scriptstyle X} in polynomial time. This contrasts with many-one reducibility, which has the restriction that the program can only call the subroutine once, and the return value of

1241-401: Is NP-complete if: C {\displaystyle \scriptstyle C} can be shown to be in NP by demonstrating that a candidate solution to C {\displaystyle \scriptstyle C} can be verified in polynomial time. Note that a problem satisfying condition 2 is said to be NP-hard , whether or not it satisfies condition 1. A consequence of this definition

1314-482: Is NP-complete is first to prove that it is in NP, and then to reduce some known NP-complete problem to it. Therefore, it is useful to know a variety of NP-complete problems. The list below contains some well-known problems that are NP-complete when expressed as decision problems. To the right is a diagram of some of the problems and the reductions typically used to prove their NP-completeness. In this diagram, problems are reduced from bottom to top. Note that this diagram

1387-414: Is a cycle or is bipartite is very easy (in L ), but finding a maximum bipartite or a maximum cycle subgraph is NP-complete. A solution of the knapsack problem within any fixed percentage of the optimal solution can be computed in polynomial time, but finding the optimal solution is NP-complete. An interesting example is the graph isomorphism problem , the graph theory problem of determining whether

1460-426: Is a multi-dimensional array of data. Online analytical processing (OLAP) is a computer-based technique of analyzing data to look for insights. The term cube here refers to a multi-dimensional dataset, which is also sometimes called a hypercube if the number of dimensions is greater than three. A cube can be considered a multi-dimensional generalization of a two- or three-dimensional spreadsheet . For example,

1533-511: Is a many-one reduction that can be computed with only a logarithmic amount of space. Since every computation that can be done in logarithmic space can also be done in polynomial time it follows that if there is a logarithmic-space many-one reduction then there is also a polynomial-time many-one reduction. This type of reduction is more refined than the more usual polynomial-time many-one reductions and it allows us to distinguish more classes such as P-complete . Whether under these types of reductions

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1606-419: Is a spreadsheet-style display, where values of X populate row $ 1; values of Y populate column $ A; and values of g  : ( X , Y ) → W populate the individual cells at intersections of X -labeled columns and Y -labeled rows, "southeast", so to speak, of $ B$ 2, with $ B$ 2 itself included. NP-Complete In computational complexity theory , a problem is NP-complete when: The name "NP-complete"

1679-407: Is called NP-Intermediate problems and exists if and only if P≠NP. At present, all known algorithms for NP-complete problems require time that is superpolynomial in the input size. The vertex cover problem has O ( 1.2738 k + n k ) {\displaystyle O(1.2738^{k}+nk)} for some k > 0 {\displaystyle k>0} and it

1752-508: Is called the P versus NP problem . But if any NP-complete problem can be solved quickly, then every problem in NP can, because the definition of an NP-complete problem states that every problem in NP must be quickly reducible to every NP-complete problem (that is, it can be reduced in polynomial time). Because of this, it is often said that NP-complete problems are harder or more difficult than NP problems in general. A decision problem C {\displaystyle \scriptstyle C}

1825-435: Is determined by every possible combination of dimension granularities. The combination of all possible aggregations and the base data contains the answers to every query which can be answered from the data. Because usually there are many aggregations that can be calculated, often only a predetermined number are fully calculated; the remainder are solved on demand. The problem of deciding which aggregations (views) to calculate

1898-413: Is empty. The complexity class of problems of this form is called NP , an abbreviation for "nondeterministic polynomial time". A problem is said to be NP-hard if everything in NP can be transformed in polynomial time into it even though it may not be in NP. A problem is NP-complete if it is both in NP and NP-hard. The NP-complete problems represent the hardest problems in NP. If some NP-complete problem has

1971-518: Is generally characterized by much less complex queries, in a larger volume, to process transactions rather than for the purpose of business intelligence or reporting. Whereas OLAP systems are mostly optimized for read, OLTP has to process all kinds of queries (read, insert, update and delete). At the core of any OLAP system is an OLAP cube (also called a 'multidimensional cube' or a hypercube ). It consists of numeric facts called measures that are categorized by dimensions . The measures are placed at

2044-575: Is known as the view selection problem. View selection can be constrained by the total size of the selected set of aggregations, the time to update them from changes in the base data, or both. The objective of view selection is typically to minimize the average time to answer OLAP queries, although some studies also minimize the update time. View selection is NP-Complete . Many approaches to the problem have been explored, including greedy algorithms , randomized search, genetic algorithms and A* search algorithm . Some aggregation functions can be computed for

2117-440: Is known, however, that AC reductions define a strictly smaller class than polynomial-time reductions. According to Donald Knuth , the name "NP-complete" was popularized by Alfred Aho , John Hopcroft and Jeffrey Ullman in their celebrated textbook "The Design and Analysis of Computer Algorithms". He reports that they introduced the change in the galley proofs for the book (from "polynomially-complete"), in accordance with

2190-411: Is misleading as a description of the mathematical relationship between these problems, as there exists a polynomial-time reduction between any two NP-complete problems; but it indicates where demonstrating this polynomial-time reduction has been easiest. There is often only a small difference between a problem in P and an NP-complete problem. For example, the 3-satisfiability problem, a restriction of

2263-473: Is not a "cube" in the strict mathematical sense, as the sides are not all necessarily equal. But this term is used widely. A Slice is a term for a subset of the data, generated by picking a value for one dimension and only showing the data for that value (for instance only the data at one point in time). Spreadsheets are only 2-dimensional, so by (continued) slicing or other techniques, it becomes possible to visualise multidimensional data in them. Each cell of

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2336-805: Is not obvious. The Cook–Levin theorem states that the Boolean satisfiability problem is NP-complete, thus establishing that such problems do exist. In 1972, Richard Karp proved that several other problems were also NP-complete (see Karp's 21 NP-complete problems ); thus, there is a class of NP-complete problems (besides the Boolean satisfiability problem). Since the original results, thousands of other problems have been shown to be NP-complete by reductions from other problems previously shown to be NP-complete; many of these problems are collected in Garey & Johnson (1979) . The easiest way to prove that some new problem

2409-524: Is possible to solve these problems quickly, called the P versus NP problem , is one of the fundamental unsolved problems in computer science today. While a method for computing the solutions to NP-complete problems quickly remains undiscovered, computer scientists and programmers still frequently encounter NP-complete problems. NP-complete problems are often addressed by using heuristic methods and approximation algorithms . NP-complete problems are in NP ,

2482-401: Is short for "nondeterministic polynomial-time complete". In this name, "nondeterministic" refers to nondeterministic Turing machines , a way of mathematically formalizing the idea of a brute-force search algorithm. Polynomial time refers to an amount of time that is considered "quick" for a deterministic algorithm to check a single solution, or for a nondeterministic Turing machine to perform

2555-406: Is that if we had a polynomial time algorithm (on a UTM , or any other Turing-equivalent abstract machine ) for C {\displaystyle \scriptstyle C} , we could solve all problems in NP in polynomial time. The concept of NP-completeness was introduced in 1971 (see Cook–Levin theorem ), though the term NP-complete was introduced later. At the 1971 STOC conference, there

2628-473: Is the act of picking a rectangular subset of a cube by choosing a single value for one of its dimensions, creating a new cube with one fewer dimension. The picture shows a slicing operation: The sales figures of all sales regions and all product categories of the company in the year 2005 and 2006 are "sliced" out of the data cube. Dice : The dice operation produces a subcube by allowing the analyst to pick specific values of multiple dimensions. The picture shows

2701-436: Is the classic form of OLAP and is sometimes referred to as just OLAP. MOLAP stores this data in an optimized multi-dimensional array storage, rather than in a relational database. Some MOLAP tools require the pre-computation and storage of derived data, such as consolidations – the operation known as processing. Such MOLAP tools generally utilize a pre-calculated data set referred to as a data cube . The data cube contains all

2774-409: Is unknown whether there are any faster algorithms. The following techniques can be applied to solve computational problems in general, and they often give rise to substantially faster algorithms: One example of a heuristic algorithm is a suboptimal O ( n log ⁡ n ) {\displaystyle O(n\log n)} greedy coloring algorithm used for graph coloring during

2847-463: The XML for Analysis specification, which was endorsed by most of the OLAP vendors. Since this also used MDX as a query language, MDX became the de facto standard. Since September-2011 LINQ can be used to query SSAS OLAP cubes from Microsoft .NET. The first product that performed OLAP queries was Express, which was released in 1970 (and acquired by Oracle in 1995 from Information Resources). However,

2920-426: The register allocation phase of some compilers, a technique called graph-coloring global register allocation . Each vertex is a variable, edges are drawn between variables which are being used at the same time, and colors indicate the register assigned to each variable. Because most RISC machines have a fairly large number of general-purpose registers, even a heuristic approach is effective for this application. In

2993-461: The Boolean satisfiability problem, remains NP-complete, whereas the slightly more restricted 2-satisfiability problem is in P (specifically, it is NL-complete ), but the slightly more general max. 2-sat. problem is again NP-complete. Determining whether a graph can be colored with 2 colors is in P, but with 3 colors is NP-complete, even when restricted to planar graphs . Determining if a graph

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3066-432: The OLAP cube and then rolled up, since on overall sum (or count etc.) is the sum of sub-sums, but it is difficult to support MEDIAN , as that must be computed for every view separately: the median of a set is not the median of medians of subsets. Pivot allows an analyst to rotate the cube in space to see its various faces. For example, cities could be arranged vertically and products horizontally while viewing data for

3139-532: The aggregate function can be computed by computing auxiliary numbers for cells, aggregating these auxiliary numbers, and finally computing the overall number at the end; examples include AVERAGE (tracking sum and count, dividing at the end) and RANGE (tracking max and min, subtracting at the end). In other cases, the aggregate function cannot be computed without analyzing the entire set at once, though in some cases approximations can be computed; examples include DISTINCT COUNT, MEDIAN, and MODE ; for example,

3212-456: The aggregated information. It depends on a specialized schema design. This methodology relies on manipulating the data stored in the relational database to give the appearance of traditional OLAP's slicing and dicing functionality. In essence, each action of slicing and dicing is equivalent to adding a "WHERE" clause in the SQL statement. ROLAP tools do not use pre-calculated data cubes but instead pose

3285-511: The classic vector analytic sense of dimensional reduction, not in the SQL sense, although the two are conceptually similar), which may suppress a primary key, but still have some semantic significance, perhaps a slice of the triadic functional representation for a given Z value of interest. The motivation behind OLAP displays harks back to the cross-tabbed report paradigm of 1980s DBMS , and to earlier contingency tables from 1904. The result

3358-414: The cube holds a number that represents some measure of the business, such as sales, profits, expenses, budget and forecast. OLAP data is typically stored in a star schema or snowflake schema in a relational data warehouse or in a special-purpose data management system. Measures are derived from the records in the fact table and dimensions are derived from the dimension tables . The elements of

3431-412: The cube, they must be computed from the base data, either computing them online (slow) or precomputing them for possible rollouts (large space). Aggregation functions that can be determined from the cells are known as decomposable aggregation functions , and allow efficient computation. For example, it is easy to support COUNT, MAX, MIN, and SUM in OLAP, since these can be computed for each cell of

3504-498: The data to be re-loaded into an optimal OLAP design. The undesirable trade-off between additional ETL cost and slow query performance has ensured that most commercial OLAP tools now use a "Hybrid OLAP" (HOLAP) approach, which allows the model designer to decide which portion of the data will be stored in MOLAP and which portion in ROLAP. There is no clear agreement across the industry as to what constitutes "Hybrid OLAP", except that

3577-527: The definition of NP-complete changes is still an open problem. All currently known NP-complete problems are NP-complete under log space reductions. All currently known NP-complete problems remain NP-complete even under much weaker reductions such as A C 0 {\displaystyle AC_{0}} reductions and N C 0 {\displaystyle NC_{0}} reductions. Some NP-Complete problems such as SAT are known to be complete even under polylogarithmic time projections. It

3650-403: The definition of NP-complete given above, the term reduction was used in the technical meaning of a polynomial-time many-one reduction . Another type of reduction is polynomial-time Turing reduction . A problem X {\displaystyle \scriptstyle X} is polynomial-time Turing-reducible to a problem Y {\displaystyle \scriptstyle Y} if, given

3723-411: The displayed view. Clients can also offer a variety of graphical widgets such as sliders, geographic maps, heat maps and more which can be grouped and coordinated as dashboards. An extensive list of clients appears in the visualization column of the comparison of OLAP servers table. Below is a list of top OLAP vendors in 2006, with figures in millions of US Dollars . OLAP cube An OLAP cube

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3796-459: The drill-down is a technique that allows users to navigate through the details. For instance, users can view the sales by individual products that make up a region's sales. Slicing and dicing is a feature whereby users can take out (slicing) a specific set of data of the OLAP cube and view (dicing) the slices from different viewpoints. These viewpoints are sometimes called dimensions (such as looking at

3869-626: The entire OLAP cube by precomputing values for each cell, and then computing the aggregation for a roll-up of cells by aggregating these aggregates, applying a divide and conquer algorithm to the multidimensional problem to compute them efficiently. For example, the overall sum of a roll-up is just the sum of the sub-sums in each cell. Functions that can be decomposed in this way are called decomposable aggregation functions , and include COUNT, MAX, MIN, and SUM , which can be computed for each cell and then directly aggregated; these are known as self-decomposable aggregation functions. In other cases,

3942-501: The greater scalability of ROLAP and the faster computation of MOLAP. For example, a HOLAP server may store large volumes of detailed data in a relational database, while aggregations are kept in a separate MOLAP store. The Microsoft SQL Server 7.0 OLAP Services supports a hybrid OLAP server Each type has certain benefits, although there is disagreement about the specifics of the benefits between providers. The following acronyms are also sometimes used, although they are not as widespread as

4015-421: The intersections of the hypercube, which is spanned by the dimensions as a vector space . The usual interface to manipulate an OLAP cube is a matrix interface, like Pivot tables in a spreadsheet program, which performs projection operations along the dimensions, such as aggregation or averaging. The cube metadata is typically created from a star schema or snowflake schema or fact constellation of tables in

4088-413: The key and W as the residual attribute . Characterizing this as a function , the attributes X , Y , and Z correspond to the axes of the cube, while the W value corresponds to the data element that populates each cell of the cube. Insofar as two-dimensional output devices cannot readily characterize three dimensions, it is more practical to project "slices" of the data cube (we say project in

4161-702: The largest independent survey across all major OLAP products, being conducted for 6 years (2001 to 2006) have consistently found that companies using ROLAP report slower performance than those using MOLAP even when data volumes were taken into consideration. However, as with any survey there are a number of subtle issues that must be taken into account when interpreting the results. Some companies select ROLAP because they intend to re-use existing relational database tables—these tables will frequently not be optimally designed for OLAP use. The superior flexibility of ROLAP tools allows this less-than-optimal design to work, but performance suffers. MOLAP tools in contrast would force

4234-439: The mainstream. OLAP clients include many spreadsheet programs like Excel, web application, SQL, dashboard tools, etc. Many clients support interactive data exploration where users select dimensions and measures of interest. Some dimensions are used as filters (for slicing and dicing the data) while others are selected as the axes of a pivot table or pivot chart. Users can also vary aggregation level (for drilling-down or rolling-up)

4307-420: The median of a set is not the median of medians of subsets. These latter are difficult to implement efficiently in OLAP, as they require computing the aggregate function on the base data, either computing them online (slow) or precomputing them for possible rollouts (large space). OLAP systems have been traditionally categorized using the following taxonomy. MOLAP (multi-dimensional online analytical processing)

4380-593: The more recent one (i.e. sliced by Time dimension) in MOLAP for fast query performance, and older data in ROLAP . Moreover, we can store some dices in MOLAP and others in ROLAP , leveraging the fact that in a large cuboid, there will be dense and sparse subregions. The first product to provide HOLAP storage was Holos , but the technology also became available in other commercial products such as Microsoft Analysis Services , Oracle Database OLAP Option , MicroStrategy and SAP AG BI Accelerator. The hybrid OLAP approach combines ROLAP and MOLAP technology, benefiting from

4453-513: The ones above: Unlike relational databases , which had SQL as the standard query language, and widespread APIs such as ODBC , JDBC and OLEDB , there was no such unification in the OLAP world for a long time. The first real standard API was OLE DB for OLAP specification from Microsoft which appeared in 1997 and introduced the MDX query language. Several OLAP vendors – both server and client – adopted it. In 2001 Microsoft and Hyperion announced

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4526-436: The other. This is known as "the question of whether P=NP". Nobody has yet been able to determine conclusively whether NP-complete problems are in fact solvable in polynomial time, making this one of the great unsolved problems of mathematics . The Clay Mathematics Institute is offering a US$ 1 million reward ( Millennium Prize ) to anyone who has a formal proof that P=NP or that P≠NP. The existence of NP-complete problems

4599-483: The possible answers to a given range of questions. As a result, they have a very fast response to queries. On the other hand, updating can take a long time depending on the degree of pre-computation. Pre-computation can also lead to what is known as data explosion. Other MOLAP tools, particularly those that implement the functional database model do not pre-compute derived data but make all calculations on demand other than those that were previously requested and stored in

4672-462: The query to the standard relational database and its tables in order to bring back the data required to answer the question. ROLAP tools feature the ability to ask any question because the methodology is not limited to the contents of a cube. ROLAP also has the ability to drill down to the lowest level of detail in the database. While ROLAP uses a relational database source, generally the database must be carefully designed for ROLAP use. A database which

4745-479: The results of a poll he had conducted of the theoretical computer science community. Other suggestions made in the poll included " Herculean ", "formidable", Steiglitz 's "hard-boiled" in honor of Cook, and Shen Lin's acronym "PET", which stood for "probably exponential time", but depending on which way the P versus NP problem went, could stand for "provably exponential time" or "previously exponential time". The following misconceptions are frequent. Viewing

4818-405: The sales figures for the individual products. Roll-up : A roll-up involves summarizing the data along a dimension. The summarization rule might be an aggregate function , such as computing totals along a hierarchy or applying a set of formulas such as "profit = sales - expenses". General aggregation functions may be costly to compute when rolling up: if they cannot be determined from the cells of

4891-425: The same sales by salesperson, or by date, or by customer, or by product, or by region, etc.). Databases configured for OLAP use a multidimensional data model, allowing for complex analytical and ad hoc queries with a rapid execution time. They borrow aspects of navigational databases , hierarchical databases and relational databases. OLAP is typically contrasted to OLTP (online transaction processing), which

4964-429: The set of all decision problems whose solutions can be verified in polynomial time; NP may be equivalently defined as the set of decision problems that can be solved in polynomial time on a non-deterministic Turing machine . A problem p in NP is NP-complete if every other problem in NP can be transformed (or reduced) into p in polynomial time. It is not known whether every problem in NP can be quickly solved—this

5037-405: The subroutine must be the return value of the program. If one defines the analogue to NP-complete with Turing reductions instead of many-one reductions, the resulting set of problems won't be smaller than NP-complete; it is an open question whether it will be any larger. Another type of reduction that is also often used to define NP-completeness is the logarithmic-space many-one reduction which

5110-417: The term did not appear until 1993 when it was coined by Edgar F. Codd , who has been described as "the father of the relational database". Codd's paper resulted from a short consulting assignment which Codd undertook for former Arbor Software (later Hyperion Solutions , and in 2007 acquired by Oracle), as a sort of marketing coup. The company had released its own OLAP product, Essbase , a year earlier. As

5183-402: The whole search. " Complete " refers to the property of being able to simulate everything in the same complexity class . More precisely, each input to the problem should be associated with a set of solutions of polynomial length, the validity of each of which can be tested quickly (in polynomial time ), such that the output for any input is "yes" if the solution set is non-empty and "no" if it

5256-422: Was a fierce debate between the computer scientists about whether NP-complete problems could be solved in polynomial time on a deterministic Turing machine . John Hopcroft brought everyone at the conference to a consensus that the question of whether NP-complete problems are solvable in polynomial time should be put off to be solved at some later date, since nobody had any formal proofs for their claims one way or

5329-419: Was designed for OLTP will not function well as a ROLAP database. Therefore, ROLAP still involves creating an additional copy of the data. However, since it is a database, a variety of technologies can be used to populate the database. In the OLAP industry ROLAP is usually perceived as being able to scale for large data volumes but suffering from slower query performance as opposed to MOLAP . The OLAP Survey ,

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