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t known quantum algorithm for this problem shor s algorithm does run in polynomial time unfortunately this fact doesn t say much about where the problem lies with respect to non quantum complexity classes separations between other complexity classes edit many known complexity classes are suspected to be unequal but this has not been proved for instance p np pp pspace displaystyle textsf p subseteq textsf np subseteq textsf pp subseteq textsf pspace but it is possible that p pspace displaystyle textsf p textsf pspace if p displaystyle textsf p is not equal to np displaystyle textsf np then p displaystyle textsf p is not equal to pspace displaystyle textsf pspace either since there are many known complexity classes between p displaystyle textsf p and pspace displaystyle textsf pspace such as rp displaystyle textsf rp bpp displaystyle textsf bpp pp displaystyle textsf pp bqp displaystyle textsf bqp ma displaystyle textsf ma ph displaystyle textsf ph etc it is possible that all these complexity classes collapse to one class proving that any of these classes are unequal would be a major breakthrough in complexity theory along the same lines co np displaystyle textsf co np is the class containing the complement problems i e problems with the yes no answers reversed of np displaystyle textsf np problems it is believed 13 that np displaystyle textsf np is not equal to co np displaystyle textsf co np however it has not yet been proven it is clear that if these two complexity classes are not equal then p displaystyle textsf p is not equal to np displaystyle textsf np since p co p displaystyle textsf p textsf co p thus if p n p displaystyle p np we would have co p co np displaystyle textsf co p textsf co np whence np p co p co np displaystyle textsf np textsf p textsf co p textsf co np similarly it is not known if l displaystyle textsf l the set of all problems that can be solved in logarithmic space is strictly contained in p displaystyle textsf p or equal to p displaystyle textsf p again there are many complexity classes between the two such as nl displaystyle textsf nl and nc displaystyle textsf nc and it is not known if they are distinct or equal classes it is suspected that p displaystyle textsf p and bpp displaystyle textsf bpp are equal however it is currently open if bpp nexp displaystyle textsf bpp textsf nexp intractability edit see also combinatorial explosion a problem that can theoretically be solved but requires impractically large near infinite amount of resources e g time to do so is known as an intractable problem 14 conversely a problem that can be solved in practice is called a tractable problem literally a problem that can be handled the term infeasible literally cannot be done is sometimes used interchangeably with intractable 15 though this risks confusion with a feasible solution in mathematical optimization 16 tractable problems are frequently identified with problems that have polynomial time solutions p displaystyle textsf p ptime displaystyle textsf ptime this is known as the cobham edmonds thesis problems that are known to be intractable in this sense include those that are exptime hard if np displaystyle textsf np is not the same as p displaystyle textsf p then np hard problems are also intractable in this sense however this identification is inexact a polynomial time solution with large degree or large leading coefficient grows quickly and may be impractical for practical size problems conversely an exponential time solution that grows slowly may be practical on realistic input or a solution that takes a long time in the worst case may take a short time in most cases or the average case and thus still be practical saying that a problem is not in p displaystyle textsf p does not imply that all large cases of the problem are hard or even that most of them are for example the decision problem in presburger arithmetic has been shown not to be in p displaystyle textsf p yet algorithms have been written that solve the problem in reasonable times in most cases similarly algorithms can solve the np complete knapsack problem over a wide range of sizes in less than quadratic time and sat solvers routinely handle large instances of the np complete boolean satisfiability problem to see why exponential time algorithms are generally unusable in practice consider a program that makes 2 n displaystyle 2 n operations before halting for small n displaystyle n say 100 and assuming for the sake of example that the computer does 10 12 displaystyle 10 12 operations each second the program would run for about 4 10 10 displaystyle 4 times 10 10 years which is the same order of magnitude as the age of the universe even with a much faster computer the program would only be useful for very small instances and in that sense the intractability of a problem is somewhat independent of technological progress however an exponential time algorithm that takes 1 0001 n displaystyle 1 0001 n operations is practical until n displaystyle n gets relatively large similarly a polynomial time algorithm is not always practical if its running time is say n 15 displaystyle n 15 it is unreasonable to consider it efficient and it is still useless except on small instances indeed in practice even n 3 displaystyle n 3 or n 2 displaystyle n 2 algorithms are often impractical on realistic sizes of problems continuous complexity theory edit continuous complexity theory can refer to complexity theory of problems that involve continuous functions that are approximated by discretizations as studied in numerical analysis one approach to complexity theory of numerical analysis 17 is information based complexity continuous complexity theory can also refer to complexity theory of the use of analog computation which uses continuous dynamical systems and differential equations 18 control theory can be considered a form of computation and differential equations are used in the modelling of continuous time and hybrid discrete continuous time systems 19 history edit an early example of algorithm complexity analysis is the running time analysis of the euclidean algorithm done by gabriel lamé in 1844 before the actual research explicitly devoted to the complexity of algorithmic problems started off numerous foundations were laid out by various researchers most influential among these was the definition of turing machines by alan turing in 1936 which turned out to be a very robust and flexible simplification of a computer the beginning of systematic studies in computational complexity is attributed to the seminal 1965 paper on the computational complexity of algorithms by juris hartmanis and richard e stearns which laid out the definitions of time complexity and space complexity and proved the hierarchy theorems 20 in addition in 1965 edmonds suggested to consider a good algorithm to be one with running time bounded by a polynomial of the input size 21 earlier papers studying problems solvable by turing machines with specific bounded resources include 20 john myhill s definition of linear bounded automata myhill 1960 raymond smullyan s study of rudimentary sets 1961 as well as hisao yamada s paper 22 on real time computations 1962 somewhat earlier boris trakhtenbrot 1956 a pioneer in the field from the ussr studied another specific complexity measure 23 as he remembers however my initial interest in automata theory was increasingly set aside in favor of computational complexity an exciting fusion of combinatorial methods inherited from switching theory with the conceptual arsenal of the theory of algorithms these ideas had occurred to me earlier in 1955 when i coined the term signalizing function which is nowadays commonly known as complexity measure 24 in 1967 manuel blum formulated a set of axioms now known as blum axioms specifying desirable properties of complexity measures on the set of computable functions and proved an important result the so called speed up theorem the field began to flourish in 1971 when stephen cook and leonid levin proved the existence of practically relevant problems that are np complete in 1972 richard karp took this idea a leap forward with his landmark paper reducibility among combinatorial problems in which he showed that 21 diverse combinatorial and graph theoretical problems each infamous for its computational intractability are np complete 25 see also edit computational complexity descriptive complexity theory game complexity leaf language limits of computation list of complexity classes list of computability and complexity topics list of unsolved problems in computer science parameterized complexity proof complexity quantum complexity theory structural complexity theory transcomputational problem computational complexity of mathematical operations works on complexity edit wuppuluri shyam doria francisco a eds 2020 unravelling complexity the life and work of gregory chaitin world scientific doi 10 1142 11270 isbn 978 981 12 0006 9 s2cid 198790362 references edit citations edit p vs np problem clay mathematics institute www claymath org archived from the original on july 6 2018 retrieved july 6 2018 see arora barak 2009 chapter 1 the computational model and why it doesn t matter 1 2 see sipser 2006 chapter 7 time complexity 1 2 ladner richard e 1975 on the structure of polynomial time reducibility journal of the acm 22 1 151 171 doi 10 1145 321864 321877 s2cid 14352974 berger bonnie a leighton t 1998 protein folding in the hydrophobic hydrophilic hp model is np complete journal of computational biology 5 1 27 40 doi 10 1089 cmb 1998 5 27 pmid 9541869 cook stephen april 2000 the p versus np problem pdf clay mathematics institute archived from the original pdf on december 12 2010 retrieved october 18 2006 jaffe arthur m 2006 the millennium grand challenge in mathematics pdf notices of the ams 53 6 archived pdf from the original on june 12 2006 retrieved october 18 2006 arvind vikraman kurur piyush p 2006 graph isomorphism is in spp information and computation 204 5 835 852 doi 10 1016 j ic 2006 02 002 schöning uwe 1988 graph isomorphism is in the low hierarchy journal of computer and system sciences 37 3 312 323 doi 10 1016 0022 0000 88 90010 4 babai lászló 2016 graph isomorphism in quasipolynomial time arxiv 1512 03547 cs ds fortnow lance september 13 2002 computational complexity blog factoring weblog fortnow com wolfram mathworld number field sieve boaz barak s course on computational complexity lecture 2 hopcroft j e motwani r and ullman j d 2007 introduction to automata theory languages and computation addison wesley boston san francisco new york page 368 meurant gerard 2014 algorithms and complexity elsevier p p 4 isbn 978 0 08093391 7 zobel justin 2015 writing for computer science springer p 132 isbn 978 1 44716639 9 smale steve 1997 complexity theory and numerical analysis acta numerica 6 cambridge univ press 523 551 bibcode 1997acnum 6 523s doi 10 1017 s0962492900002774 s2cid 5949193 babai lászló campagnolo manuel 2009 a survey on continuous time computations arxiv 0907 3117 cs cc tomlin claire j mitchell ian bayen alexandre m oishi meeko july 2003 computational techniques for the verification of hybrid systems proceedings of the ieee 91 7 986 1001 bibcode 2003ieeep 91 986t doi 10 1109 jproc 2003 814621 1 2 fortnow homer 2003 richard m karp combinatorics complexity and randomness 1985 turing award lecture yamada h 1962 real time computation and recursive functions not real time computable ieee transactions on electronic computers ec 11 6 753 760 bibcode 1962irtec 11 753y doi 10 1109 tec 1962 5219459 trakhtenbrot b a signalizing functions and tabular operators uchionnye zapiski penzenskogo pedinstituta transactions of the penza pedagogoical institute 4 75 87 1956 in russian boris trakhtenbrot from logic to theoretical computer science an update in pillars of computer science lncs 4800 springer 2008 karp richard m 1972 reducibility among combinatorial problems pdf in miller r e thatcher j w eds complexity of computer computations new york plenum pp 85 103 archived from the original pdf on june 29 2011 retrieved september 28 2009 textbooks edit arora sanjeev barak boaz 2009 computational complexity a modern approach cambridge university press isbn 978 0 521 42426 4 zbl 1193 68112 downey rod fellows michael 1999 parameterized complexity monographs in computer science berlin new york springer verlag isbn 978 0 387 94883 6 du ding zhu ko ker i 2000 theory of computational complexity john wiley sons isbn 978 0 471 34506 0 garey michael r johnson david s 1979 computers and intractability a guide to the theory of np completeness series of books in the mathematical sciences 1st ed new york w h freeman and company isbn 978 0 7167 1045 5 mr 0519066 oclc 247570676 goldreich oded 2008 computational complexity a conceptual perspective cambridge university press van leeuwen jan ed 1990 handbook of theoretical computer science vol a algorithms and complexity mit press isbn 978 0 444 88071 0 papadimitriou christos 1994 computational complexity 1st ed addison wesley isbn 978 0 201 53082 7 sipser michael 2006 introduction to the theory of computation 2nd ed usa thomson course technology isbn 978 0 534 95097 2 surveys edit khalil hatem ulery dana 1976 a review of current studies on complexity of algorithms for partial differential equations proceedings of the annual conference on acm 76 pp 197 201 doi 10 1145 800191 805573 isbn 978 1 4503 7489 7 s2cid 15497394 cook stephen 1983 an overview of computational complexity communications of the acm 26 6 400 408 doi 10 1145 358141 358144 issn 0001 0782 s2cid 14323396 fortnow lance homer steven 2003 a short history of computational complexity pdf bulletin of the eatcs 80 95 133 mertens stephan 2002 computational complexity for physicists computing in science engineering 4 3 31 47 arxiv cond mat 0012185 bibcode 2002cse 4c 31m doi 10 1109 5992 998639 issn 1521 9615 s2cid 633346 external links edit look up tractable feasible intractability or infeasible in wiktionary the free dictionary wikimedia commons has media related to computational complexity theory the complexity zoo computational complexity classes encyclopedia of mathematics ems press 2001 1994 scott aaronson why philosophers should care about computational complexity v t e complexity classes considered feasible dlogtime ac 0 acc 0 reg tc tc 0 l sl rl fl nl nl complete nc sc cc p p complete zpp rp bpp bqp apx fp suspected infeasible up np np complete np hard co np co np complete tfnp fnp am qma ph p pp p p complete ip pspace pspace complete considered infeasible exptime nexptime expspace 2 exptime elementary nonelementary pr r re all other complexity classes polyl qp class hierarchies polynomial hierarchy exponential hierarchy grzegorczyk hierarchy arithmetical hierarchy boolean hierarchy families of classes dtime ntime dspace nspace probabilistically checkable proof interactive proof system list of complexity classes v t e computer scienc...
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