Some Terminologies. Some Terminologies. Trees. Example: UNIX Directory. Trees. Binary Trees, Binary Search Trees 1/9/2014

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1 // Som Trminoogis Binry Trs, Binry Srch Trs /st.ppt Chid nd prnt Evry nod xcpt th root hs on prnt A nod cn hv n ritrry numr of chidrn Lvs Nods with no chidrn Siing nods with sm prnt Trs Linr ccss tim of inkd ists is prohiitiv Dos thr xist ny simp dt structur for which th running tim of most oprtions (srch, insrt, dt) is O(og N)? Som Trminoogis Pth Lngth numr of dgs on th pth Dpth of nod ngth of th uniqu pth from th root to tht nod Th dpth of tr is qu to th dpth of th dpst f Hight of nod ngth of th ongst pth from tht nod to f vs r t hight Th hight of tr is qu to th hight of th root Ancstor nd dscndnt Propr ncstor nd propr dscndnt Trs Exmp: UNIX Dirctory A tr is coction of nods Th coction cn mpty (rcursiv dfinition) If not mpty, tr consists of distinguishd nod r (th root), nd zro or mor nonmpty sutrs T, T,..., T k, ch of whos roots r connctd y dirctd dg from r

xprssion c*+d*f+g*+ Lvs r oprnds (constnts or vris) Th othr nods (intrn nods) contin oprtors Wi not inry tr if som oprtors r not inry Inordr trvrs ft, nod, right.

2 // Binry Trs A tr in which no nod cn hv mor thn two chidrn Th dpth of n vrg inry tr is considry smr thn N, vnthough in th worst cs, th dpth cn s rg s N. Prordr, Postordr nd Inordr Prordr trvrs nod, ft, right prfix xprssion ++*c*+*dfg Exmp: Exprssion Trs Prordr, Postordr nd Inordr Postordr trvrs ft, right, nod postfix xprssion c*+d*f+g*+ Lvs r oprnds (constnts or vris) Th othr nods (intrn nods) contin oprtors Wi not inry tr if som oprtors r not inry Inordr trvrs ft, nod, right. infix xprssion +*c+d*+f*g Tr trvrs Usd to print out th dt in tr in crtin ordr Pr-ordr trvrs Print th dt t th root Rcursivy print out dt in th ft sutr Rcursivy print out dt in th right sutr Prordr

Binry srch tr proprty For vry nod X, th kys in its ft sutr r smr thn th ky vu in X, nd th kys in its right sutr r rgr thn th

3 // Postordr compr: Impmnttion of gnr tr Prordr, Postordr nd Inordr Binry Srch Trs Stors kys in th nods in wy so tht srching, insrtion nd dtion cn don fficinty. Binry srch tr proprty For vry nod X, th kys in its ft sutr r smr thn th ky vu in X, nd th kys in its right sutr r rgr thn th ky vu in X Binry Trs Binry Srch Trs Possi oprtions on th Binry Tr ADT prnt ft_chid, right_chid siing root, tc Impmnttion Bcus inry tr hs t most two chidrn, w cn kp dirct pointrs to thm A inry srch tr Not inry srch tr

// Binry srch trs Two inry srch trs rprsnting th sm st: Avrg dpth of nod is

pointr to th nod tht hs ky X, or NULL if thr is no such nod Tim compxity

If w r srching for ky <, thn w shoud srch in th ft sutr.

4 // Binry srch trs Two inry srch trs rprsnting th sm st: Avrg dpth of nod is O(og N); mximum dpth of nod is O(N) Impmnttion Srching (Find) Find X: rturn pointr to th nod tht hs ky X, or NULL if thr is no such nod Tim compxity O(hight of th tr) Srching BST If w r srching for, thn w r don. If w r srching for ky <, thn w shoud srch in th ft sutr. If w r srching for ky >, thn w shoud srch in th right sutr. Inordr trvrs of BST Print out th kys in sortd ordr Inordr:,,,,,,,,,,

5 // findmin/ findmx Rturn th nod contining th smst mnt in th tr Strt t th root nd go ft s ong s thr is ft chid. Th stopping point is th smst mnt dt Thr css: () th nod is f Dt it immdity () th nod hs on chid Adjust pointr from th prnt to ypss tht nod Simiry for findmx Tim compxity = O(hight of th tr) insrt Procd down th tr s you woud with find If X is found, do nothing (or updt somthing) Othrwis, insrt X t th st spot on th pth trvrsd dt () th nod hs chidrn rpc th ky of tht nod with th minimum mnt t th right sutr dt th minimum mnt Hs ithr no chid or ony right chid cus if it hs ft chid, tht ft chid woud smr nd woud hv n chosn. So invok cs or. Tim compxity = O(hight of th tr) Tim compxity = O(hight of th tr) dt Whn w dt nod, w nd to considr how w tk cr of th chidrn of th dtd nod. This hs to don such tht th proprty of th srch tr is mintind. Priority Quus Binry Hps homs.cs.wshington.du/~ndrson/ iuc/sids_.../hps.ppt

6 // Rc Quus FIFO: First-In, First-Out Som contxts whr this sms right? Som contxts whr som things shoud owd to skip hd in th in? Appictions of th Priority Quu Sct print jos in ordr of dcrsing ngth Forwrd pckts on routrs in ordr of urgncy Sct most frqunt symos for comprssion Sort numrs, picking minimum first Anything grdy Quus tht Aow Lin Jumping Nd nw ADT Oprtions: Insrt n Itm, Rmov th Bst Itm Potnti Impmnttions insrt dtmin Unsortd ist (Arry) O() O(n) Unsortd ist (Linkd-List) O() O(n) insrt dtmin Sortd ist (Arry) O(n) O()* Sortd ist (Linkd-List) O(n) O() Priority Quu ADT. PQuu dt : coction of dt with priority. PQuu oprtions insrt dtmin. PQuu proprty: for two mnts in th quu, x nd y, if x hs owr priority vu thn y, x wi dtd for y Rc From Lists, Quus, Stcks Us n ADT tht corrsponds to your nds Th right ADT is fficint, whi n ovry gnr ADT provids functionity you rn t using, ut r pying for nywys Hps provid O(og n) worst cs for oth insrt nd dtmin, O() vrg insrt

7 // Binry Hp Proprtis. Structur Proprty. Ordring Proprty Brif intrud: Som Dfinitions: A Prfct inry tr A inry tr with f nods t th sm dpth. A intrn nods hv chidrn. hight h h+ nods h non-vs h vs root(t): vs(t): chidrn(b): prnt(h): siings(e): ncstors(f): dscndnts(g): sutr(c): Tr Rviw A B D E F H J K L Tr T C G I M N Hp Structur Proprty A inry hp is compt inry tr. Compt inry tr inry tr tht is compty fid, with th possi xcption of th ottom v, which is fid ft to right. Exmps: Mor Tr Trminoogy dpth(b): hight(g): dgr(b): rnching fctor(t): Tr T A B C D E F G H I J K L M N Rprsnting Compt Binry Trs in n Arry B C D E F H I J K L impicit (rry) impmnttion: A G From nod i: ft chid: right chid: prnt: A B C D E F G H I J K L

8 // Why this pproch to storg? Hp Insrt(v) Bsic Id:. Put v t nxt f position. Prcot up y rptdy xchnging nod unti no ongr ndd Hp Ordr Proprty Hp ordr proprty: For vry non-root nod X, th vu in th prnt of X is ss thn (or qu to) th vu in X. Insrt: prcot up not hp Hp Oprtions findmin: insrt(v): prcot up. dtmin: prcot down. Insrt Cod (optimizd) void insrt(ojct o) { ssrt(!isfu()); siz++; nwpos = prcotup(siz,o); Hp[nwPos] = o; } int prcotup(int ho, Ojct v) { whi (ho > && v < Hp[ho/]) Hp[ho] = Hp[ho/]; ho /= ; } rturn ho; } runtim: (Cod in ook)

9 // Hp Dtmin Bsic Id:. Rmov root (tht is wys th min!). Put st f nod t root. Find smst chid of nod. Swp nod with its smst chid if ndd.. Rpt stps & unti no swps ndd. Insrt:,,,,,, DtMin: prcot down Dt Structurs Binry Hps DtMin Cod (Optimizd) Ojct dtmin() { ssrt(!isempty()); rturnv = Hp[]; siz--; nwpos = prcotdown(, Hp[siz+]); Hp[nwPos] = Hp[siz + ]; rturn rturnv; } runtim: (cod in ook) int prcotdown(int ho, Ojct v) { whi (*ho <= siz) { ft = *ho; right = ft + ; if (right siz && Hp[right] < Hp[ft]) trgt = right; s trgt = ft; if (Hp[trgt] < v) { Hp[ho] = Hp[trgt]; ho = trgt; } s rk; } rturn ho; } Buiding Hp

10 // Buiding Hp Adding th itms on t tim is O(n og n) in th worst cs I promisd O(n) for tody Buidhp psudocod privt void uidhp() { for ( int i = currntsiz/; i > ; i-- ) prcotdown( i ); } runtim: Working on Hps BuidHp: Foyd s Mthod Wht r th two proprtis of hp? Structur Proprty Ordr Proprty How do w work on hps? Fix th structur Fix th ordr BuidHp: Foyd s Mthod BuidHp: Foyd s Mthod Add mnts ritrriy to form compt tr. Prtnd it s hp nd fix th hp-ordr proprty!

11 // BuidHp: Foyd s Mthod Finy runtim: BuidHp: Foyd s Mthod Mor Priority Quu Oprtions dcrsky givn pointr to n ojct in th quu, rduc its priority vu Soution: chng priority nd incrsky givn pointr to n ojct in th quu, incrs its priority vu Why do w nd pointr? Why not simpy dt vu? Soution: chng priority nd BuidHp: Foyd s Mthod Mor Priority Quu Oprtions Rmov(ojPtr) givn pointr to n ojct in th quu, rmov th ojct from th quu Soution: st priority to ngtiv infinity, prcot up to root nd dtmin FindMx

12 // Fcts out Hps Osrvtions: Finding chid/prnt indx is mutipy/divid y two Oprtions jump widy through th hp Ech prcot stp ooks t ony two nw nods Insrts r t st s common s dtmins Tris hwww.mthcs.mory.du/~chung/fcourss//sy us/ook/powrpoint/tris.ppt Ritis: Division/mutipiction y powrs of two r quy fst Looking t ony two nw pics of dt: d for cch! With hug dt sts, disk ccsss domint miz nimiz z i nimiz mi z nimiz z Tris Cycs to ccss: CPU Cch Mmory Disk Prprocssing Strings Prprocssing th pttrn spds up pttrn mtching quris Aftr prprocssing th pttrn, KMP s gorithm prforms pttrn mtching in tim proportion to th txt siz If th txt is rg, immut nd srchd for oftn (.g., works y Shkspr), w my wnt to prprocss th txt instd of th pttrn A tri is compct dt structur for rprsnting st of strings, such s th words in txt A tris supports pttrn mtching quris in tim proportion to th pttrn siz Tris A Soution: d-hps Ech nod hs d chidrn Sti rprsnt y rry Good choics for d: (choos powr of two for fficincy) fit on st of chidrn in cch in fit on st of chidrn on mmory pg/disk ock Stndrd Tris Th stndrd tri for st of strings S is n ordrd tr such tht: Ech nod ut th root is d with chrctr Th chidrn of nod r phticy ordrd Th pths from th xtrn nods to th root yid th strings of S Exmp: stndrd tri for th st of strings S = { r,, id, u, uy, s, stock, stop } r i d u y Tris k s c t o p

13 // Anysis of Stndrd Tris A stndrd tri uss O(n) spc nd supports srchs, insrtions nd dtions in tim O(dm), whr: n tot siz of th strings in S m siz of th string prmtr of th oprtion d siz of th pht r i d u y Tris k s c t o p Compct Rprsnttion Compct rprsnttion of comprssd tri for n rry of strings: Stors t th nods rngs of indics instd of sustrings Uss O(s) spc, whr s is th numr of strings in th rry Srvs s n uxiiry indx structur,, S[] = s S[] = S[] = S[] =,,,, r s s t o c k S[] = u S[] = h r S[] = S[] = u y i d S[] = S[] = s t o p,,,,,,,,,,,,,,,,,,,,,,,,,, Tris Word Mtching with Tri insrt th words of th txt into tri Ech f is ssocitd w/ on prticur word f stors indics whr ssocitd word gins ( s strts t indx &, f for s stors thos indics) s r? s s t o c k! s u? u y s t o c k! i d s t o c k! i d s t o c k! h r t h? s t o p! Suffix Tri Th suffix tri of string X is th comprssd tri of th suffixs of X m i n i m i z i u h s t i mi nimiz z r d, y, r c p k Tris,,, o miz nimiz z nimiz z Tris Comprssd Tris A comprssd tri hs intrn nods of dgr t st two It is otind from stndrd tri y comprssing chins of r rdundnt nods x. th i nd d in id r rdundnt cus thy signify th sm word id s u y s ck to p Anysis of Suffix Tris Compct rprsnttion of th suffix tri for string X of siz n from n pht of siz d Uss O(n) spc Supports ritrry pttrn mtching quris in X in O(dm) tim, whr m is th siz of th pttrn Cn constructd in O(n) tim m i n i z m i i u t d y o,,,,, r c p,,,,, Tris k Tris

14 // Encoding Tri () A cod is mpping of ch chrctr of n pht to inry cod-word A prfix cod is inry cod such tht no cod-word is th prfix of nothr cod-word An ncoding tri rprsnts prfix cod Ech f stors chrctr Th cod word of chrctr is givn y th pth from th root to th f storing th chrctr ( for ft chid nd for right chid c d Tris c d Huffmn s Agorithm Givn string X, Huffmn s gorithm construct prfix cod th minimizs th siz of th ncoding of X It runs in tim O(n + d og d), whr n is th siz of X nd d is th numr of distinct chrctrs of X A hp-sd priority quu is usd s n uxiiry structur Agorithm HuffmnEncoding(X) Input string X of siz n Output optim ncoding tri for X C distinctchrctrs(x) computfrquncis(c, X) Q nw mpty hp for c C T nw sing-nod tr storing c Q.insrt(gtFrquncy(c), T) whi Q.siz() > f Q.min() T Q.rmovMin() f Q.min() T Q.rmovMin() T join(t, T ) Q.insrt(f + f, T) rturn Q.rmovMin() Tris Encoding Tri () Givn txt string X, w wnt to find prfix cod for th chrctrs of X tht yids sm ncoding for X Frqunt chrctrs shoud hv short cod-words Rr chrctrs shoud hv ong cod-words Exmp X = rcdr T ncods X into its T ncods X into its T T X =rcdr Frquncis c d r c d r Exmp c d r c d r c d r r c d Tris c d r c d r Tris Th End On Mor Oprtion Mrg two hps Add th itms from on into nothr? O(n og n) Strt ovr nd uid it from scrtch? O(n)

15 // CSE : Dt Structurs Priority Quus Lftist Hps & Skw Hps Dfinition: Nu Pth Lngth nu pth ngth (np) of nod x = th numr of nods twn x nd nu in its sutr OR np(x) = min distnc to dscndnt with or chidrn np(nu) = - np(f) = np(sing-chid nod) =??? Equivnt dfinitions:?. np(x) is th hight of rgst compt sutr rootd t x. np(x) = + min{np(ft(x)), np(right(x))} Nw Hp Oprtion: Mrg Givn two hps, mrg thm into on hp first ttmpt: insrt ch mnt of th smr hp into th rgr. runtim: scond ttmpt: conctnt inry hps rrys nd run uidhp. runtim: Lftist Hp Proprtis Hp-ordr proprty prnt s priority vu is to chidrns priority vus rsut: minimum mnt is t th root Lftist proprty For vry nod x, np(ft(x)) np(right(x)) rsut: tr is t st s hvy on th ft s th right Lftist Hps Ar Ths Lftist? Id: Focus hp mintnnc work in on sm prt of th hp Lftist hps:. Most nods r on th ft. A th mrging work is don on th right Evry sutr of ftist tr is ftist!

16 // Right Pth in Lftist Tr is Short (#) Cim: Th right pth is s short s ny in th tr. Proof: (By contrdiction) Pick shortr pth: D < D Sy it divrgs from right pth t x np(l) D - cus of th pth of ngth D - to nu np(r) D - cus vry nod on right pth is ftist D L x R D Mrg two hps (sic id) Put th smr root s th nw root, Hng its ft sutr on th ft. Rcursivy mrg its right sutr nd th othr tr. Lftist proprty t x viotd! Right Pth in Lftist Tr is Short (#) Cim: If th right pth hs r nods, thn th tr hs t st r - nods. Proof: (By induction) Bs cs : r=. Tr hs t st - = nod Inductiv stp : ssum tru for r < r. Prov for tr with right pth t st r.. Right sutr: right pth of r- nods r- - right sutr nods (y induction). Lft sutr: so right pth of ngth t st r- (y prvious sid) r- - ft sutr nods (y induction) Tot tr siz: ( r- -) + ( r- -) + = r - Mrging Two Lftist Hps mrg(t,t ) rturns on ftist hp contining mnts of th two (distinct) ftist hps T nd T mrg T L R T < L mrg R L R L R Why do w hv th ftist proprty? Bcus it gurnts tht: th right pth is ry short comprd to th numr of nods in th tr A ftist tr of N nods, hs right pth of t most g (N+) nods Mrg Continud If np(r ) > np(l ) L R R = Mrg(R, T ) R L Id prform work on th right pth runtim:

17 // Lt s do n xmp, ut first Othr Hp Oprtions insrt? dtmin? Swing Up th Exmp??? Don? Oprtions on Lftist Hps mrg with two trs of tot siz n: O(og n) insrt with hp siz n: O(og n) prtnd nod is siz ftist hp insrt y mrging origin hp with on nod hp mrg Finy dtmin with hp siz n: O(og n) rmov nd rturn root mrg ft nd right sutrs mrg mrg Lftst Mrg Exmp? mrg? mrg (spci cs) Good Bd Lftist Hps: Summry

18 // Rndom Dfinition: Amortizd Tim m or tizd tim: Running tim imit rsuting from writing off xpnsiv runs of n gorithm ovr mutip chp runs of th gorithm, usuy rsuting in owr ovr running tim thn If indictd M oprtions y th tk worst tot O(M possi og N) cs. tim, mortizd tim pr oprtion is O(og N) Diffrnc from vrg tim: mrg Exmp mrg mrg Skw Hps Proms with ftist hps xtr storg for np xtr compxity/ogic to mintin nd chck np right sid is oftn hvy nd rquirs switch Soution: skw hps indy djusting vrsion of ftist hps mrg wys switchs chidrn whn fixing right pth mortizd tim for: mrg, insrt, dtmin = O(og n) howvr, worst cs tim for thr = O(n) Skw Hp Cod void mrg(hp, hp) { cs { hp == NULL: rturn hp; hp == NULL: rturn hp; hp.findmin() < hp.findmin(): tmp = hp.right; hp.right = hp.ft; hp.ft = mrg(hp, tmp); rturn hp; othrwis: rturn mrg(hp, hp); } } mrg T Mrging Two Skw Hps L R T L R < mrg R L R Ony on stp pr itrtion, with chidrn wys switchd L Runtim Anysis: Worst-cs nd Amortizd No worst cs gurnt on right pth ngth! A oprtions ry on mrg worst cs compxity of ops = Proy won t gt to mortizd nysis in this cours, ut s Chptr if curious. Rsut: M mrgs tk tim M og n mortizd compxity of ops =

19 // Binry Hps Compring Hps Lftist Hps Th Binomi Tr, B h B h hs hight h nd xcty h nods B h is formd y mking B h- chid of nothr B h- Root hs xcty h chidrn Numr of nods t dpth d is inomi coff. Hnc th nm; w wi not us this st proprty h d d-hps Skw Hps B B B B Sti room for improvmnt! (Whr?) Dt Structurs Binomi Quus Binomi Quu with n mnts Binomi Q with n mnts hs uniqu structur rprsnttion in trms of inomi trs! Writ n in inry: n = (s ) = (s ) B B No B B Yt Anothr Dt Structur: Binomi Quus Structur proprty Forst of inomi trs with t most on tr of ny hight Wht s forst? Wht s inomi tr? Ordr proprty Ech inomi tr hs th hp-ordr proprty Proprtis of Binomi Quu At most on inomi tr of ny hight n nods inry rprsnttion is of siz? dpst tr hs hight? numr of trs is? Dfin: hight(forst F) = mx tr T in F { hight(t) } Binomi Q with n nods hs hight Θ(og n)

20 // Oprtions on Binomi Quu Wi gin dfin mrg s th s oprtion insrt, dtmin, uidbinomiq wi us mrg Exmp: Binomi Quu Mrg H: H: - Cn w do incrsky fficinty? dcrsky? Wht out findmin? Mrging Two Binomi Quus Essntiy ik dding two inry numrs!. Comin th two forsts. For k from to mxhight {. m tot numr of B k s in th two BQs. if m=: continu; # of s c. if m=: continu; + = d. if m=: comin th two B k s to form + = B k+ + = +c. if m=: rtin on B k nd ++c = +c comin th othr two to form B k+ } Cim: Whn this procss nds, th forst hs t most on tr of ny hight Exmp: Binomi Quu Mrg H: H: - Exmp: Binomi Quu Mrg H: H: Exmp: Binomi Quu Mrg H: H: - -

21 // Exmp: Binomi Quu Mrg H: H: Insrt in Binomi Quu Insrt(x): Simir to ftist or skw hp - runtim Worst cs compxity: sm s mrg O( ) Avrg cs compxity: O() Why?? Hint: Think of dding to Exmp: Binomi Quu Mrg H: H: dtmin in Binomi Quu Simir to ftist nd skw hps. - Compxity of Mrg Constnt tim for ch hight Mx numr of hights is: og n BQ dtmin: Exmp worst cs running tim = Θ( ) find nd dt smst root mrg BQ (without BQ th shdd prt) nd BQ

22 // dtmin: Exmp Rsut: runtim:

CPE702 Algorithm Analysis and Design Week 11 String Processing

CPE702 Algorithm Analysis and Design Week 11 String Processing CPE702 Agorithm Anaysis and Dsign Wk 11 String Procssing Prut Boonma prut@ng.cmu.ac.th Dpartmnt of Computr Enginring Facuty of Enginring, Chiang Mai Univrsity Basd on Sids by M.T. Goodrich and R. Tamassia