Growth Rates in Algorithm Complexity: the Missing Link

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1 Ieriol Jourl O Eieeri Ad Copuer Siee ISSN:9-74 Volue 6 Issue Noveber 07, Pe No Idex Coperius vlue (05): 58.0 DOI: 0.855/ijes/v6i.5 Grow es i Alori Coplexiy: e Missi Li D Vrjioru Idi Uiversiy Sou Bed, Depre o Copuer d Iorio Siees 700 Misw Ave, Sou Bed, IN 4664, USA Absr: I is pper, we propose explii esure or e row re o lori oplexiy uio. Tis esure oplees e usul ie or spe oplexiy lysis o loris d ill p i e udersdi o e sypoi oio d us, provide eduiol beeis. Firs, we disuss soe properies o e row esure, su s is bevior wi respe o lier operors. Seod, we lyze is oeio o e sypoi oplexiy oios d disuss is ipliios. Keywords: lori oplexiy, row re, sypoi oios, eduio.. Iroduio Te sypoi lori bevior oios o O, Ω, Θ, d orrespodi o d ω, re ooly used o desribe e ie d spe oplexiy o loris. Tey re ow s e Ldu ily o oios []. Bi O, lled e B- Ldu oio, ws irs irodued by P. B i 89 []. Bi Ω d Θ, s ey re used ow, were deied by D. Ku i 976 [], d were bsed o e Hrdy-Lilewood oios [4]. I is pper, we will use e oo lori oplexiy deiiios or ese oios, s oud i [5] d [6]. Alysis o loris is o e os diiul opis i e opuer siee urriulu. A ypil llee or e sudes osiss o udersdi e eed or e os i e deiiio o e sypoi oios d ow o oose i. A soure o ousio is e eve ou we use e equliy oio, O(()) represes se or ily o uios. I is lso diiul o rsp i lori is O(), e i is lso O( ), bu we oly se e slles su ily s e lori's order o oplexiy. Te usul lue used wi ese oios posis ey desribe e row o uio, or e row o lori eor. We oe expli o sudes e exeuio ie is o ipor i isel, beuse i deped o e plor. Tus, i our lysis we w o ow ow e exeuio ie will e we e proble rows, espeilly by ive or su s or 0. I is w we ll e row o e uio. I e exper lierure s well s i iorl oplexiy disussios, i is oe sid i () = O(()), e e uio rows os s s s e uio or lre eou. I is eve irly oo o sy e row re o e uio is less or equl o e row re o e uio, su s i [7]. Currely, ere is o orl deiiio o e er o row re. Te sypoi oios eselves represe oly idire esure o e row o uio. I is sudy, we propose dire esure o i. Te er row re is used i oeio o e ie or spe oplexiy o loris, s well s i oer siuios reled o eoois or bioloy. I soe ses, su s [8], i is used o represe (+)/(). Tus, or lori o oplexiy O( ), e os is lled e row re i is oex. I s lso bee used i oer sudies [9] o illusre e evoluio o proessor speed d Liux erel size ro yer o yer. Soe uors siply use i ierebly wi e er oplexiy o lori, or o d sruure [0]. Tus, i is pper, we propose o irodue dire esure o e row o uio, or e row o e oplexiy o lori wi e size o e proble. We lso lyze e oeio bewee e dire row re d e sypoi bevior oios. Our ol is o ive e ide o row re orl deiiio be used bo i oplexiy lysis d s pedoil ool. We e row re or e oplexiy lysis re eioed, e oio o opri () wi (0) or () is oe prese [7], []. Oer uors, su s i [], iquire bou e size o proble be solved by vrious loris i ive ou o ie, s or exple, i oe seod. We explore is ide urer i our deiiio o e row re uio. Te ide is lso ipliily prese i lori slbiliy disussios []. Oeies, we rue or lori bei slble is de, e perore esures re preseed o lorii sle. For exple, i [4], Fiure 9 sows e perore o ie rerievl lori or size o e dbse oi ro 0 o 00 d e o M. I [5], e ru ie perore o e lori is preseed o lorii sle over e uber o proessors bei used. I sees or slbiliy purposes, i is ipor o ow w will ppe we preer i e lori is uliplied by os. Te pper is orized e ollowi wy. Seio deies e row re uio d provides soe properies or i. Seio lyzes e oeio bewee e opriso o row res or wo uios d eir sypoi reliosip. Seio 4 disusses e ipliios o e eores i is pper bo i ers o loris oplexiy D Vrjioru, IJECS Volue 6 Issue Noveber 07 Pe No Pe 089

2 d i ers o pedoil uses. Te pper eds wi soe olusios.. Grow e Mesure I is seio, we deie our row esure or oplexiy uio d exie is properies. Deiiio. Le () be posiive uio deied o posiive ieers d le > be rel uber. Te we deie e row re uio, Tus, we esure e row o lori by ei ow u is oplexiy rows we e size o e proble ireses by or. We ll e preer i is deiiio e row or. I e bove deiiio, we ssue e uio s exesio deied o ll rel ubers. Prilly, i is o e se, d i is o ieer, we pply e loor o i e deiiio. We lso r wi udersore ised o usi i s orl preer beuse we usully osider i os. Tble sows e row res or vriey o oo uios. Tis ble illusres ow e esure be used s pedoil ool. Here, e row re be see o irese lerly ro oe ily o uios o e ex. Tble. Grow re uio exples For eduiol purposes, sudes beei ro bei irodued o is ble or soe vlue o $$ su s, 5, or 0. As sed i e iroduio, soeies sudes srule o rsp e eed d ipore o e os i e $O$ deiiio. Tis ble sows direly ow uio rows ser oer wiou vi o sele su os. I elp e quire iuiive sese o ow e os oo uios re ordered by oplexiy. Aoer ipor elee esily eeres ro is ble is e disiio bewee polyoil uios d e oers. Tus, e row re o polyoil is boud by oss i os, wile or lrer uios, i depeds o. Teore. I () d () re posiive uios deied o posiive ieers, e or y rel > ) I () = () / () is oooous sedi uio, e (, ) (, ). b) I () = () / () is oooous desedi uio, e Proo. (, ) (, ). ) I () = () / () is oooous sedi uio, e or y posiive ieers p, q su p < q, (p) (q) or (p) / (p) (q) / (q). Te we wrie,, ( ) ( ) Te ls iequliy is rue beuse is oooous sedi uio d > d >0 e <. b) Te proo is siilr o ). Teore. I () d () re posiive uios deied o posiive ieers, e or y rel > ) (, ) = (, ), or y rel posiive os, b) (, ) = (, ) (, ), ) (+, ) < (, ) + (, ), d) i (, ) = (, ) i (, ), or y rel os i >. Proo. ), Tis es e esure ivri o e uio bei uliplied by os. b),,,. Tis es e esure disribuive wi respe o uio ulipliio. ), ( ) ( ). ( ) ( ) ( ) Sie bo () d () re posiive uios, we wrie Tis es ( ) d. ( ), ( ),,. Tus, e esure sisies e riulr propery. d) i i i, i,, Teore ) be used i lssroo o expli e presee o e os i e deiiio o e $O$ oio. D Vrjioru, IJECS Volue 6 Issue Noveber 07 Pe No Pe 090

3 Tis eore ells us i we uliply uio by os, is row re reis e se. Tus, i ers o opri row res, i does o e dieree i we opre uio wi uio or wi, were is os.. Coeio wi Asypoi Noios I is seio, we will exie e oeio bewee e uio d e oplexiy oios o O, Ω, Θ, d o. Teore. I () d () re wo posiive uios o posiive ieers su (, ) (, ) or y >, > 0, e eier () = o(()), or () = Θ(())$. I bo ses, () = O(()). Proo. Le us use e oio () = () / (). Te irs pr o e proo osiss o sowi e posiive uio () is oooous sedi. Le < be wo posiive ieers. Te i we e o be equl o /, e > d =. Te i us be rue (, ) (, ) (, ). Tis es ( ) ( ) Tis proves e uio is oooous sedi. I is is e se, e () ()$ or y $. I we deoe by = (), e we ve ( ) ( ) ( ). I priulr, is es () = O(()). Nex, () bei posiive uio is oooous sedi, eier () overes o iiiy, or ere exiss os d uber 0 su () or y > 0. I () =, e wi es $() = o(())$. 0 I e seod se, () or y > 0 es ( ) ( ) wi ells us () = O(()) d () =Ω(()). Sie we lredy ow () = O(()), is es () = Θ(()). Teore 4. I () = Θ(()), e (, ) = Θ( (, )) or y >. Proo. I () = Θ(()), e ere exiss, > 0 su Te we wrie () () (). () () () d ( ) ( ) ( ) By uliplyi e wo iequliies we e ) ( ) ( ) (,,, (, ) = Θ( (, )) Teore 5. Le () = A()+ (), were A() d () re posiive uios su () = o(a()). Te Proo. A, (, ) ~ (A, ). A A A A A, A A A A A A,. A A We ow () = o(a()), wi es () / A() = 0. Fro is, we dedue A wi leds o (, ) ~ (A, ) + (, ) () / A(). Now le's opue e i o (A, ) divided by e expressio o e ri-d side we oes o iiiy. A, A,, A A, A, A beuse () = o(a()) iplies () / A() = 0. Filly, rsiiviy o e sypoi oio ives us e olusio o e eore. Here is exple. Le () = + + 5, were A()= d () = + 5. We we opue e re uios or d or A we e d,, 5( ) 5 Sie e seod er i (, ) overes o 0 s oes o iiiy, (A, ) ~ (, ) ideed. Here is seod exple were (, ) is o siple D Vrjioru, IJECS Volue 6 Issue Noveber 07 Pe No Pe 09

4 uio o. Le () = +, were A() = d () =. We we opue e row re uio or A d we pply e se resoi s i e proo or Teore 5, we e (A, ) = (-) d (, ) ~ (-) + (-) (/) = (-) + ( /) We see or > lo (), e seod er overes o iiiy we oes o iiiy. However, i we opue e i o (, ) divided by is ls expressio, we e ( ) ( ) ( ) so eve i is se, (A, ) ~ (, ). So r, e eores we ve proved see o sow our esure beves prey well d i be oeder o e sypoi oios desribe e oplexiy o lori i sipler wy. Te reii o is seio will sow soe rues o e orry. Preiry Observio. Le d be posiive uios o posiive ieers su () < () or lre eou. I is possible (, ) > (, ) or soe priulr vlue. Le us develop is iequliy:,, Fro () < () we ow ( ) / ( )} >. Le us ssue e wo uios be exeded o rel ubers d (x) < (x) or y rel uber x lre eou. Fiure illusres visully e vlue o ( ) ( ) / ( ) i oeio o e oer ivolved vlues. Fiure. Grow res or e uios d I we drw sri lie rou ( ) d ( ), s lo s ese vlues re o equl o e oer, i will ierse e Ox xis i poi. Te i we drw oer lie oei d ( ), i ierses e veril lie oi up ro e veril vlue ( ) / ( ). Tis oes ro e rile siilriy: sie e wo veril lies re prllel, i we deoe by L e veril oordie o e op orer, e ollowi equliies us be rue: wi we solve or L o id e vlue eioed bove. Iidelly, is lso llows us o opue e vlue $\displysyle L Fro is ie we see ere is wole iervl o possible vlues or ( )$ eep i lrer ( ) wiou oi over L. For ll o ese vlues, (, ) < (, ). Tis observio suess e uio ould be osisely less e uio, bu is row re ould be lrer o i ood uber o pois. Teore 6. I () d () re posiive uios o posiive ieers, e () = O(()) does o iply (, ) (, ) or lre eou d or y posiive rel uber. Tus, ) ere exis pirs o su uios d su or soe vlues o, (, ) > (, ) or iiie uber o vlues o ; b) ere exis pirs o su uios d su bo d re oooous sedi, d su or soe vlues o, (, ) > (, ) or iiie uber o vlues o ; ) ere exis pirs o su uios d su () = o(()) d or soe vlues o, (, ) > (, ) or iiie uber o vlues o. Proo. ) Le d () =. i i Te uio is lier or odd ubers d qudri or eve ubers. I is esily ler () = O(()). Te row re or e uio is (, ) = or y > 0, s sow i Tble. Le =. Te e row re or e uio or odd ubers is 4 4, 4( ) Tus, (, +) > (, +) = 4 or y. Furerore, (, +) is o boud by y os ies (, +), or by y os i eerl. As oe, e uio () is eier Θ(()), or o(()). b) Le us deoe by 4 e sequee o powers o 4 were e expoe is power o. Noe = +. D Vrjioru, IJECS Volue 6 Issue Noveber 07 Pe No Pe 09

5 i i d () =. Te uio is qudri i e ubers o e sequee. I bewee e, e uio rows lierly. Ai, i's quie esy o see ()= O(()). Tus, e uio is qudri or ubers irese i qudri sio, d i rows lierly i bewee ese vlues. Te uio is oooi sedi. Le =. We re oi o exie e row re or = ½ +., Jus lie or e uio poi ), (, ½ + ) > (, ½ + ) = 4 or y. Te row re o is uio is lso o boud by y os. ) Le () be eier e uio ro e proo o ) or o b). I we e () =, e () = o(()) i bo ses. (, ) = 8, wile or bo o ese uios, e vlue o or e se o ubers i e previous proos rows wi, so eveully i will be lrer (, ) d y os ies (, ). Teores d 6 sow us e bi O oio is ore eerl e row re uio. Tus, i e row re o is less e row re o, e = O(), bu e oer wy roud is o lwys rue. Eve e odiio = o() is o suiie or e row res o be i order. 4. Disussio d Fuure Wor A observio we drw ro e eores preseed ere is syi i () = O(()), e e uio rows ser e uio is ie, bu o i rows ser re. I vs joriy o e uios observed ro lyzi e oplexiy o loris, we () = O(()), e row re o is lower e row re o. Teore 6 sows is is o eessrily e se beuse e uios re oooous. I y ses, e rio ()/() be sow o be oooous, wi oes o e rows res rou Teore. A direio o uure reser would be o id ore suiie odiios or e row re o uio o be lrer o oer oe. Furerore, e uio used i e proo or Teore 6 ) be used pedoilly s exple were e row re siply ils o desribe e bevior o uio. Tis uio is lerly boud by wo polyoils, bu is row re i soe iruses is u lrer o eier polyoil. Eve ore, i is o os, wi es i lrer y polyoil. Tis be used i lssroo s rue explii wy e row re is o suiie. Aoer eduiol rue i vor o e sypoi oios is e soe loris do o lwys ve sile uio desribi eir uber o operios. For exple, e lier ser o vlue i rry require oly os uber o operios i e re is e beii o e rry d lier uio o e rry size i e re is o i e rry. I is rd o express e row re o is lori s wole. Te O oio is ore pproprie i is se. As sed i severl ples, e row re esure be used s eduiol ool, o elp sudes udersd vrious spes o loris oplexiy ore esily. A direio o uure develope will be o sudy e ip e ide o row re ve o ei oplexiy lysis. I is ipor, ou, o esblis e eil oudios o is esure irs, wi is e purpose o is pper. 5. Colusios I is pper, we irodued explii esure or e row re or lori oplexiy uios, (, ), deied s e re bewee () d (), were is e size o e proble d e row or. I Seio, we orlly deied e esure d sowed i is ivrible wi respe o os ulipliio, siilrly o e sypoi oios. Te row esure is disribuive wi respe o uio ulipliio, d sisies e riulr propery wi respe o uio ddiio. I Seio, we sowed i e row re o uio is less or equl o e row re o oer, e e irs uio is O o e seod. Te opposie oeio is o s sriorwrd, ou. I uio is Θ o oer oe, e eir row res re lso Θ o e oer. More eerlly, e row re o uio is sypoi o e row re o is lres er. Tis eoes e siilr propery o e Θ oio. However, uio be O o oer oe, bu is row re be lrer e row re o e seod uio or soe vlue o e row or d or iiie uber o vlues o. I soe o ese ses, e irs uio is eier o o e seod, or Θ o e seod. However, eve e o oio does o uree sller row re. For os o e uios observed i lysis o loris, e O oio is ied o sller row re. Teore 6 uios us is is o rue or y pir o uios. I is beer o use e o or Θ oios ised o O we possible. Filly, e row re sows poeil o use s pedoil ool i ei oplexiy o loris d e pper s ilied severl ides iprove e ei eeiveess o is diiul opi. eerees. E. Ldu, Hdbu der Lere vo der Vereilu der Prizle, Leipzi: B. G. Teuber, P. B, Alyise Zleeorie, Leipzi: B. G. Teuber, 89.. D. Ku, Bi Oiro d Bi Oe d Bi Te, i SIGACT News, pp. 8 4, G. H. Hrdy d J. E. Lilewood, Soe Probles o Diopie pproxiio, i A Mei, p. 5, D. Vrjioru d W. Ki, Pril Alysis o Aloris, Teoreil Copuer Siee Series, Sprier, T. Core, C. Leiserso,. ives, d C. Sei, Iroduio o Aloris, MIT Press, ed., M. A. Weiss, D Sruures d Proble Solvi Usi C++, Perso, d ed., 999. D Vrjioru, IJECS Volue 6 Issue Noveber 07 Pe No Pe 09

6 8. M. Cy, H. Dell, D. Losov, D. Mrx, J. Nederlo, Y. Ooo,. Puri, S. Surb, d M. Wlsro, O Probles s Hrd s CNF-SAT, i ACM Trsios o Aloris,, L. Yu, Coevoluio o Iorio Eosyses: Sudy o e Sisil elios Ao e Grow es o Hrdwre, Syse Sowre, d Appliio Sowre, i ACM SIGSOFT Sowre Eieeri Noes, 6, L. Bbi, Te Grow e o Verex-rsiive Plr Grps, i Proeedis o e 8 Aul ACM-SIAM Syposiu o Disree Aloris,, 5-7 Jury 997, pp A. Drozde, D Sruures d Aloris i C++, Cee, 4 ed., 0.. M. Goodri,. Tssi, d D. Mou, D Sruures d Aloris i C++, Wiley, d ed., 0.. W. P d Y. Coi, Hi Perore d Hi Slble Pe Clssiiio Alori or Newor Seuriy Syses, i IEEE Trs. Depedble Seuriy Copui, 4, pp. 7 49, D. Nisér d H. Sewéius, Slble eoiio wi Vobulry Tree, i Proeedis o CVP, 006, pp J. Srssbur d V. Alexdrov, O Slbiliy Beviour o Moe Crlo Sprse Approxie Iverse or Mrix Copuios, i Proeedis o e Worsop o Les Adves i Slble Aloris or Lre-Sle Syses, SlA, New Yor, NY, USA, 0, ACM, pp. 6: 6:8. Auor Proile Dr. Vrjioru s obied er ile o Door o Siees ro e Uiversiy o Neuel, Swizerld, i 997. Se is urrely ssoie proessor o opuer siee Idi Uiversiy Sou Bed. Her reser ieres ilude iellie syses, evoluiory opuio, opuer rpis, d sieii visulizio. D Vrjioru, IJECS Volue 6 Issue Noveber 07 Pe No Pe 094