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BPP (complexity)

In computational complexity theory, bounded-error probabilistic polynomial time is the class of decision problems solvable by a probabilistic Turing machine in polynomial time with an error probability bounded away from 1/3 for all instances. BPP is one of the largest practical classes of problems, meaning most problems of interest in BPP have efficient probabilistic algorithms that can be run on real modern machines. BPP contains P, the class of problems solvable in polynomial time with a deterministic machine, since a deterministic machine is a special case of a probabilistic machine. Informally, a problem is in BPP if there is an algorithm for it that has the following properties: It is allowed to flip coins and make random decisions It is guaranteed to run in polynomial time On any given run of the algorithm, it has a probability of at most 1/3 of giving the wrong answer, whether the answer is YES or NO. A language L is in BPP if and only if there exists a probabilistic Turing machine M, such that M runs for polynomial time on all inputs For all x in L, M outputs 1 with probability greater than or equal to ​2⁄3 For all x not in L, M outputs 1 with probability less than or equal to ​1⁄3Unlike the complexity class ZPP, the machine M is required to run for polynomial time on all inputs, regardless of the outcome of the random coin flips.

Alternatively, BPP can be defined using only deterministic Turing machines. A language L is in BPP if and only if there exists a polynomial p and deterministic Turing machine M, such that M runs for polynomial time on all inputs For all x in L, the fraction of strings y of length p which satisfy M = 1 is greater than or equal to ​2⁄3 For all x not in L, the fraction of strings y of length p which satisfy M = 1 is less than or equal to ​1⁄3In this definition, the string y corresponds to the output of the random coin flips that the probabilistic Turing machine would have made. For some applications this definition is preferable since it does not mention probabilistic Turing machines. In practice, an error probability of ​1⁄3 might not be acceptable, the choice of ​1⁄3 in the definition is arbitrary, it can be any constant between 0 and ​1⁄2 and the set BPP will be unchanged. It does not have to be constant: the same class of problems is defined by allowing error as high as ​1⁄2 − n−c on the one hand, or requiring error as small as 2−nc on the other hand, where c is any positive constant, n is the length of input.

The idea is that there is a probability of error, but if the algorithm is run many times, the chance that the majority of the runs are wrong drops off exponentially as a consequence of the Chernoff bound. This makes it possible to create a accurate algorithm by running the algorithm several times and taking a "majority vote" of the answers. For example, if one defined the class with the restriction that the algorithm can be wrong with probability at most ​1⁄2100, this would result in the same class of problems. All problems in P are also in BPP. However, many problems have been known to be in BPP but not known to be in P; the number of such problems is decreasing, it is conjectured that P = BPP. For a long time, one of the most famous problems known to be in BPP but not known to be in P was the problem of determining whether a given number is prime. However, in the 2002 paper PRIMES is in P, Manindra Agrawal and his students Neeraj Kayal and Nitin Saxena found a deterministic polynomial-time algorithm for this problem, thus showing that it is in P.

An important example of a problem in BPP still not known to be in P is polynomial identity testing, the problem of determining whether a polynomial is identically equal to the zero polynomial, when you have access to the value of the polynomial for any given input, but not to the coefficients. In other words, is there an assignment of values to the variables such that when a nonzero polynomial is evaluated on these values, the result is nonzero? It suffices to choose each variable's value uniformly at random from a finite subset of at least d values to achieve bounded error probability, where d is the total degree of the polynomial. If the access to randomness is removed from the definition of BPP, we get the complexity class P. In the definition of the class, if we replace the ordinary Turing machine with a quantum computer, we get the class BQP. Adding postselection to BPP, or allowing computation paths to have different lengths, gives the class BPPpath. BPPpath is known to contain NP, it is contained in its quantum counterpart PostBQP.

A Monte Carlo algorithm is a randomized algorithm, to be correct. Problems in the class BPP have Monte Carlo algorithms with polynomial bounded running time; this is compared to a Las Vegas algorithm, a randomized algorithm which either outputs the correct answer, or outputs "fail" with low probability. Las Vegas algorithms with polynomial bound running times are used to define the class ZPP. Alternatively, ZPP contains probabilistic algorithms that are always correct and have expected polynomial running time; this is weaker than saying it is a polynomial time algorithm, since it may run for super-polynomial time, but with low probability. It is known. BPP is low for itself, meaning that a BPP machine with the power to solve BPP problems is not any more powerful than the machine without this extra power. In symbols, BPPBPP = BPP; the relationship between BPP and NP is unknown: it is not known whether BPP is a subset of NP, NP is a

Kung för en dag

Kung för en dag or Kung i baren is the second track of Magnus Uggla's 1997 album Karaoke, written by Magnus Uggla together with Anders Henriksson. Lyrical dealing with partying when the wage arrives for the month, the song was the album's major hit; the music video shows Olle Sarri acting as partying. The single peaked at 4th position at the Swedish singles chart and charted at Svensktoppen for eight weeks between 11 October-29 November 1997, peaking at 5th position; the song charted at Trackslistan for four weeks between 13 September-4 October 1997, peaking at eight position. On 1 July 2008, when Magnus Uggla appeared as guest, the song was performed as a sing-along song at Allsång på Skansen on 1 July 2008. On 7 July 2009, when Magnus Uggla appeared together with Rolandz, the song was once again performed as a sing-along song; the song became recurring at TV 3's Baren. At Dansbandskampen 2008 the song was performed by Larz-Kristerz

Comparison of online backup services

This is a comparison of online backup services. Online backup is a special kind of online storage service. Online Backup requires a backup client program. A browser-only online storage service is not considered a valid online backup service. Online folder sync services can be used for backup purposes. However, some Online Folder Sync services may not provide a safe Online Backup. If a file is accidentally locally corrupted or deleted, it depends on the versioning features of a Folder Sync service, whether this file will still be retrievable. ^1 Windows/Linux/Mac/iOS/Android/BlackBerry: Supported operating systems for thick client, which provide background data transmission and setting services. ^2 Zero knowledge: The service provider has no knowledge of the user's encryption key, ensuring privacy of the backup data. ^3 Secure Key Management: If yes, the user holds and controls the encryption key. If no, the service provider controls the encryption key. ^4 Payment options/plans: Limited MB plan: Pay per computer.

Additional fee for storage over a threshold. Unlimited MB plan: Pay per computer. Storage per computer is unlimited. $ / MB plan: Pay per unit of storage. ^5 Cloud hosted Net Drive: Cloud can serve storage over WebDAV, SMB/CIFS, NFS, AFP or other NAS protocol, allowing files to be streamed from the cloud. A change made to the cloud is accessible to applications on all clients without needing to pre-download the file in full. ^7 Sync: Synchronization between computers, and/or mobile devices ^8 Public Internet file hosting ^9 Restore via physical media ^10 Server location: Countries where physical servers are located. Where the data will be located. ^11 Still in Beta version ^12 Whether the desktop client can detect and upload changes without scanning all files. ^13 Many backup services offer a limited free plan for personal use. It is possible to increase the free backup limit through coupons, referrals, or other means that are not included in this column; this column does not include free trials that are only available for a limited period of time.

^14 External hard drive support: Can refer to an alternate backup destination or whether the service can back up external drives. ^15 Hybrid Online Backup works by storing data to local disk so that the backup can be captured at high speed, either the backup software or a D2D2C appliance encrypts and transmits data to a service provider. Recent backups are retained locally. ^16 Unlimited BW: If bandwidth capping or limits are used on accounts. Acronis Up to five PCs, always incremental backups, remote access from the web Backblaze Data de-duplication. Barracuda Backup Service Data de-duplication. Blaucloud No User Limit each User get his ownCloud Instance in an encrypted Storage. BullGuard Backup 5 PC/license, fast upload speeds, mobile access, encrypted transfer and storage, password-protected settings, free 24/7 support. Carbonite Block-level incremental, Home or Pro editions. IPhone/ Blackberry/ Android App available to remotely access data from the online backup. Can manually select files to upload that are larger than 4 GB.

Cloudberry Backup Image & File Based backups, data de-duplication, block-level and multiple cloud providers supported. CloudJuncxion Decentralized multi-cloud backup with integrated sharing, sync and Cloud NAS. Fault-tolerance against failure of a constituent cloud. Crashplan Unlimited destinations. Data de-duplication. Can run server-free, exchanging backup space with friends and family. Datashield High-level encryption, personalized encryption key, shared cloud drive, sync folder functionality. Dolly Drive Cloud storage, designed for the Mac. Allows users to store files in the cloud for seamless access on any computer or mobile device. Diino iPhone/Android app available. Dropbox Data de-duplication, delta sync, iPhone/Android/Blackberry app available. Dropmysite website backup, database backup, SFTP support, free up to 2 GB. Egnyte Delta sync, Google Docs sync and group management ElephantDrive Auto-transfer from defunct Xdrive. F-Secure Humyo Humyo will become part of Trend Micro SafeSync. Humyo no longer accepts new clients.

IASO Backup Advanced data reduction technology. Data de-duplication mechanism. High level of scalability and cost effectiveness. ICFiles Secure File Share Storage. Proprietary license download client. High level security, SOC 2 TYPE II, ISO 27001,27017, 27018, CSA, PCI, HIPAA, CJIS, EU Model Clauses, on request private servers for FISMA and FedRAMP. IDrive Proprietary license download client. Automatic Selection. Continuous Data Protection. "Virtual drive" explorer. Jungle Disk Proprietary license download client sample code. KeepVault Real-time hybrid offsite data backup. Memopal Cross-user de-duplication, delta sync. MiMedia Initial seed via a MiMedia-owned external hard drive available. Mozy Data de-duplication. "Mozy Data Shuttle" physical seeding service available for extra fee. Replicalia Professional Backup for Professional Data. SpiderOak Data de-duplication."Zero Knowledge" encryption. StoreGrid Cloud Byte-level incremental backup, local backup, Disk Image backup—BMR and physical seeding. Syncplicity Google Docs s