Algorithms. Algorithms 5.2 TRIES. R-way tries ternary search tries character-based operations ROBERT SEDGEWICK KEVIN WAYNE

Transcription

1 Agoritm ROBERT SEDGEWICK KEVIN WAYNE 5.2 TRIES Agoritm F O U R T H E D I T I O N R-way tri trnary arc tri caractr-bad opration ROBERT SEDGEWICK KEVIN WAYNE ttp://ag4.c.princton.du

2 Summary of t prformanc of ymbo-tab impmntation Ordr of growt of t frquncy of opration. impmntation typica ca arc inrt dt ordrd opration opration on ky rd-back BST og N og N og N y comparto() a tab no qua() acod() undr uniform aing aumption Q. Can w do bttr? A. Y, if w can avoid xamining t ntir ky, a wit tring orting. 2

3 String ymbo tab baic API String ymbo tab. Symbo tab pciaizd to tring ky. pubic ca StringST<Vau> StringST() crat an mpty ymbo tab void put(string ky, Vau va) put ky-vau pair into t ymbo tab Vau gt(string ky) rturn vau paird wit givn ky void dt(string ky) dt ky and corrponding vau Goa. Fatr tan aing, mor fxib tan BST. 3

4 String ymbo tab impmntation cot ummary caractr acc (typica ca) ddup impmntation arc it arc mi inrt pac (rfrnc) moby.txt actor.txt rd-back BST L + c g 2 N c g 2 N c g 2 N 4N aing (inar probing) L L L 4N to 16N Paramtr N = numbr of tring L = ngt of tring R = radix fi iz word ditinct moby.txt 1.2 MB 210 K 32 K actor.txt 82 MB 11.4 M 900 K Cang. Efficint prformanc for tring ky. 4

5 5.2 TRIES Agoritm R-way tri trnary arc tri caractr-bad opration ROBERT SEDGEWICK KEVIN WAYNE ttp://ag4.c.princton.du

6 Tri Tri. [from rtriva, but pronouncd "try"] Stor caractr in nod (not ky). Eac nod a R cidrn, on for ac poib caractr. For now, w do not draw nu ink. root ink to tri for a ky tat tart wit for now w do not draw nu ink ink to tri for a ky tat tart wit b t y 4 ky vau by 4 a 6 0 o 5 a r vau for in nod corrponding to at ky caractr or 7 t 5 3 6

7 Sarc in a tri Foow ink corrponding to ac caractr in t ky. Sarc it: nod wr arc nd a a non-nu vau. Sarc mi: rac nu ink or nod wr arc nd a nu vau. gt("") b t y 4 a 6 0 o 5 r rturn vau aociatd wit at ky caractr (rturn 3) 7

8 Sarc in a tri Foow ink corrponding to ac caractr in t ky. Sarc it: nod wr arc nd a a non-nu vau. Sarc mi: rac nu ink or nod wr arc nd a nu vau. gt("") b t y 4 a 6 0 o 5 r arc may trminatd at an intrmdiat nod (rturn 0)

9 Sarc in a tri Foow ink corrponding to ac caractr in t ky. Sarc it: nod wr arc nd a a non-nu vau. Sarc mi: rac nu ink or nod wr arc nd a nu vau. gt("") b t y 4 a 6 0 o 5 r no vau aociatd wit at ky caractr (rturn nu) 9

10 Sarc in a tri Foow ink corrponding to ac caractr in t ky. Sarc it: nod wr arc nd a a non-nu vau. Sarc mi: rac nu ink or nod wr arc nd a nu vau. gt("tr") b t y 4 a 6 0 o 5 r no ink to t (rturn nu) 10

11 Foow ink corrponding to ac caractr in t ky. Encountr a nu ink: crat nw nod. Encountr t at caractr of t ky: t vau in tat nod. 11 Inrtion into a tri r put("or", 7) a b y o t

12 Tri contruction dmo tri

13 Tri contruction dmo tri b t y 4 a 6 0 o 5 r 1 7 3

14 Tri rprntation: Java impmntation Nod. A vau, pu rfrnc to R nod. privat tatic ca Nod { privat Objct vau; privat Nod[] nxt = nw Nod[R]; } u Objct intad of Vau inc no gnric array cration in Java a 2 0 caractr ar impicity dfind by ink indx a 2 0 nitr ky nor caractr ar xpicity tord 1 1 ac nod a an array of ink and a vau Tri rprntation 14

15 R-way tri: Java impmntation pubic ca TriST<Vau> { privat tatic fina int R = 256; privat Nod root = nw Nod(); xtndd ASCII privat tatic ca Nod { /* prviou id */ } pubic void put(string ky, Vau va) { root = put(root, ky, va, 0); } privat Nod put(nod x, String ky, Vau va, int d) { } if (x == nu) x = nw Nod(); if (d == ky.ngt()) { x.va = va; rturn x; } car c = ky.carat(d); x.nxt[c] = put(x.nxt[c], ky, va, d+1); rturn x; 15

16 R-way tri: Java impmntation (continud) pubic booan contain(string ky) { rturn gt(ky)!= nu; } pubic Vau gt(string ky) { } Nod x = gt(root, ky, 0); if (x == nu) rturn nu; rturn (Vau) x.va; cat ndd privat Nod gt(nod x, String ky, int d) { } if (x == nu) rturn nu; if (d == ky.ngt()) rturn x; car c = ky.carat(d); rturn gt(x.nxt[c], ky, d+1); } 16

17 Tri prformanc Sarc it. Nd to xamin a L caractr for quaity. Sarc mi. Coud av mimatc on firt caractr. Typica ca: xamin ony a fw caractr (ubinar). Spac. R nu ink at ac af. (but ubinar pac poib if many ort tring ar common prfix) aractr ar impicity dfind by ink indx a ac nod a an array of in and a vau Bottom in. Fat arc it and vn fatr arc mi, but wat pac. 17

18 To dt a ky-vau pair: Find t nod corrponding to ky and t vau to nu. If nod a nu vau and a nu ink, rmov tat nod (and rcur) Dtion in an R-way tri r a b y o t dt("") t vau to nu

19 Dtion in an R-way tri To dt a ky-vau pair: Find t nod corrponding to ky and t vau to nu. If nod a nu vau and a nu ink, rmov tat nod (and rcur). dt("") b t y 4 a 6 0 o 5 r 1 7 nu vau and ink (dt nod) 19

20 String ymbo tab impmntation cot ummary caractr acc (typica ca) ddup impmntation arc it arc mi inrt pac (rfrnc) moby.txt actor.txt rd-back BST L + c g 2 N c g 2 N c g 2 N 4N aing (inar probing) L L L 4N to 16N R-way tri L og R N L (R+1) N 1.12 out of mmory R-way tri. Mtod of coic for ma R. Too muc mmory for arg R. Cang. U mmory,.g., 65,536-way tri for Unicod! 20

21 5.2 TRIES Agoritm R-way tri trnary arc tri caractr-bad opration ROBERT SEDGEWICK KEVIN WAYNE ttp://ag4.c.princton.du

22 Trnary arc tri Stor caractr and vau in nod (not ky). Eac nod a 3 cidrn: mar (ft), qua (midd), argr (rigt). Fat Agoritm for Sorting and Sarcing String Jon L. Bnty* Robrt Sdgwick# Abtract W prnt tortica agoritm for orting and arcing mutiky data, and driv from tm practica C impmntation for appication in wic ky ar caractr tring. T orting agoritm bnd Quickort and radix ort; it i comptitiv wit t bt known C ort cod. T arcing agoritm bnd tri and binary arc tr; it i fatr tan aing and otr commony ud arc mtod. T baic ida bind t agoritm dat back at at to t 1960 but tir practica tat i comptitiv wit t mot fficint tring orting program known. T cond program i a ymbo tab impmntation tat i fatr tan aing, wic i commony rgardd a t fatt ymbo tab impmntation. T ymbo tab impmntation i muc mor pac-fficint tan mutiway tr, and upport mor advancd arc. In many appication program, ort u a Quickort impmntation bad on an abtract compar opration, and arc u aing or binary arc tr. T do 22

23 Trnary arc tri Stor caractr and vau in nod (not ky). Eac nod a 3 cidrn: mar (ft), qua (midd), argr (rigt). a b t ink to TST for a ky tat tart wit a ttr bfor a b ink to TST for a ky tat tart wit t r y 4 u r y 4 u 12 a o r a o r r 7 ac nod a tr ink 11 r 7 15 y y 13 TST rprntation of a tri 23

24 Sarc it in a TST gt("a") b t y a o r rturn vau aociatd wit at ky caractr

25 Sarc mi in a TST gt("tr") b t y a o r no ink to t (rturn nu) 25

26 Trnary arc tri contruction dmo trnary arc tri 26

27 Trnary arc tri contruction dmo trnary arc tri b t y a o r

28 Sarc in a TST Foow ink corrponding to ac caractr in t ky. If, tak ft ink; if gratr, tak rigt ink. If qua, tak t midd ink and mov to t nxt ky caractr. Sarc it. Nod wr arc nd a a non-nu vau. Sarc mi. Rac a nu ink or nod wr arc nd a nu vau. gt("a") mimatc: tak ft or rigt ink, do not mov to nxt car b a matc: tak midd ink, mov to nxt car t r y 4 u a 10 o r 8 r rturn vau aociatd wit at ky caractr y 13 TST arc xamp 28

29 26-way tri v. TST 26-way tri. 26 nu ink in ac af. TST. 3 nu ink in ac af. 26-way tri (1035 nu ink, not own) TST (155 nu ink) now for tip ik dim tag jot ob nob ky ut ac bt mn gg fw jay ow joy rap gig w wa cab wad caw cu f tap ago tar jam dug and 29

30 TST rprntation in Java A TST nod i fiv fid: A vau. A caractr c. A rfrnc to a ft TST. A rfrnc to a midd TST. A rfrnc to a rigt TST. privat ca Nod { privat Vau va; privat car c; privat Nod ft, mid, rigt; } tandard array of ink (R = 26) u ink for ky tat tart wit Tri nod rprntation trnary arc tr (TST) ink for ky tat tart wit u u 30

31 TST: Java impmntation pubic ca TST<Vau> { privat Nod root; privat ca Nod { /* prviou id */ } pubic void put(string ky, Vau va) { root = put(root, ky, va, 0); } privat Nod put(nod x, String ky, Vau va, int d) { car c = ky.carat(d); if (x == nu) { x = nw Nod(); x.c = c; } if (c < x.c) x.ft = put(x.ft, ky, va, d); if (c > x.c) x.rigt = put(x.rigt, ky, va, d); if (d < ky.ngt() - 1) x.mid = put(x.mid, ky, va, d+1); x.va = va; rturn x; } 31

32 TST: Java impmntation (continud) pubic booan contain(string ky) { rturn gt(ky)!= nu; } pubic Vau gt(string ky) { Nod x = gt(root, ky, 0); if (x == nu) rturn nu; rturn x.va; } } privat Nod gt(nod x, String ky, int d) { if (x == nu) rturn nu; car c = ky.carat(d); if (c < x.c) rturn gt(x.ft, ky, d); if (c > x.c) rturn gt(x.rigt, ky, d); if (d < ky.ngt() - 1) rturn gt(x.mid, ky, d+1); rturn x; } 32

33 String ymbo tab impmntation cot ummary caractr acc (typica ca) ddup impmntation arc it arc mi inrt pac (rfrnc) moby.txt actor.txt rd-back BST L + c g 2 N c g 2 N c g 2 N 4 N aing (inar probing) L L L 4 N to 16 N R-way tri L og R N L (R + 1) N 1.12 out of mmory TST L + n N n N L + n N 4 N Rmark. Can buid baancd TST via rotation to aciv L + og N wort-ca guarant. Bottom in. TST i a fat a aing (for tring ky), pac fficint. 33

34 TST wit R 2 brancing at root Hybrid of R-way tri and TST. Do R 2 -way brancing at root. Eac of R 2 root nod point to a TST. array of 26 2 root aa ab ac zy zz TST TST TST TST TST Q. Wat about on- and two-ttr word? 34

35 String ymbo tab impmntation cot ummary caractr acc (typica ca) ddup impmntation arc it arc mi inrt pac (rfrnc) moby.txt actor.txt rd-back BST L + c g 2 N c g 2 N c g 2 N 4 N aing (inar probing) L L L 4 N to 16 N R-way tri L og R N L (R + 1) N 1.12 out of mmory TST L + n N n N L + n N 4 N TST wit R 2 L + n N n N L + n N 4 N + R Bottom in. Fatr tan aing for our bncmark cint. 35

36 TST v. aing Haing. Nd to xamin ntir ky. Sarc it and mi cot about t am. Prformanc ri on a function. Do not upport ordrd ymbo tab opration. TST. Work ony for tring (or digita ky). Ony xamin jut noug ky caractr. Sarc mi may invov ony a fw caractr. Support ordrd ymbo tab opration (pu otr!). Bottom in. TST ar: Fatr tan aing (pciay for arc mi). Mor fxib tan rd-back BST. [tay tund] 36

37 5.2 TRIES Agoritm R-way tri trnary arc tri caractr-bad opration ROBERT SEDGEWICK KEVIN WAYNE ttp://ag4.c.princton.du

38 String ymbo tab API Caractr-bad opration. T tring ymbo tab API upport vra ufu caractr-bad opration. ky vau by 4 a or 7 t 5 Prfix matc. Ky wit prfix :,, and or. Widcard matc. Ky tat matc.: and t. Longt prfix. Ky tat i t ongt prfix of ort:. 38

39 String ymbo tab API pubic ca StringST<Vau> StringST() crat a ymbo tab wit tring ky void put(string ky, Vau va) put ky-vau pair into t ymbo tab Vau gt(string ky) vau paird wit ky void dt(string ky) dt ky and corrponding vau Itrab<String> ky() a ky Itrab<String> kywitprfix(string ) ky aving a a prfix Itrab<String> kytatmatc(string ) ky tat matc (wr. i a widcard) String ongtprfixof(string ) ongt ky tat i a prfix of Rmark. Can ao add otr ordrd ST mtod,.g., foor() and rank(). 39

40 Warmup: ordrd itration To itrat troug a ky in ortd ordr: Do inordr travra of tri; add ky ncountrd to a quu. Maintain qunc of caractr on pat from root to nod. kywitprfix(""); ky() ky b by a o or or t t t q by b y 4 by a a 6 by a by a by a by a or by a or t o r 7 t 5 Cocting t ky in a tri (trac) 40

41 Ordrd itration: Java impmntation To itrat troug a ky in ortd ordr: Do inordr travra of tri; add ky ncountrd to a quu. Maintain qunc of caractr on pat from root to nod. pubic Itrab<String> ky() { } Quu<String> quu = nw Quu<String>(); coct(root, "", quu); rturn quu; qunc of caractr on pat from root to x privat void coct(nod x, String prfix, Quu<String> q) { } if (x == nu) rturn; if (x.va!= nu) q.nquu(prfix); for (car c = 0; c < R; c++) coct(x.nxt[c], prfix + c, q); 41

42 Prfix matc Find a ky in a ymbo tab tarting wit a givn prfix. Ex. Autocompt in a c pon, arc bar, txt ditor, or. Ur typ caractr on at a tim. Sytm rport a matcing tring. 42

43 Prfix matc in an R-way tri Find a ky in a ymbo tab tarting wit a givn prfix. kywitprfix(""); b y 4 a 6 find ubtri for a ky bginning wit "" o r 7 t 5 b y 4 a o r 7 t 5 o coct ky in tat ubtri Prfix matc in a tri pubic Itrab<String> kywitprfix(string prfix) { } Quu<String> quu = nw Quu<String>(); Nod x = gt(root, prfix, 0); coct(x, prfix, quu); rturn quu; root of ubtri for a tring bginning wit givn prfix ky o or or quu or 43

44 Longt prfix Find ongt ky in ymbo tab tat i a prfix of qury tring. Ex. To nd packt toward dtination IP addr, routr coo IP addr in routing tab tat i ongt prfix matc. "128" " " rprntd a 32-bit binary numbr for IPv4 (intad of tring) " " " " " " " " " " ongtprfixof(" ") = " " ongtprfixof(" ") = " " ongtprfixof(" ") = "128" " " " " Not. Not t am a foor: foor(" ") = " " 44

45 Longt prfix in an R-way tri Find ongt ky in ymbo tab tat i a prfix of qury tring. Sarc for qury tring. Kp track of ongt ky ncountrd. "" "" "ort" a arc nd at nd of tring vau i not nu rturn a arc nd at nd of tring vau i nu rturn (at ky on pat) Poibiiti for ongtprfixof() Of() a arc nd at nu ink rturn (at ky on pat) 45

46 Longt prfix in an R-way tri: Java impmntation Find ongt ky in ymbo tab tat i a prfix of qury tring. Sarc for qury tring. Kp track of ongt ky ncountrd. pubic String ongtprfixof(string qury) { int ngt = arc(root, qury, 0, 0); rturn qury.ubtring(0, ngt); } privat int arc(nod x, String qury, int d, int ngt) { } if (x == nu) rturn ngt; if (x.va!= nu) ngt = d; if (d == qury.ngt()) rturn ngt; car c = qury.carat(d); rturn arc(x.nxt[c], qury, d+1, ngt); 46

47 T9 txting Goa. Typ txt mag on a pon kypad. Muti-tap input. Entr a ttr by rpatdy pring a ky unti t dird ttr appar. T9 txt input. "a muc fatr and mor fun way to ntr txt" Find a word tat corrpond to givn qunc of numbr. Pr 0 to a comption option. Ex. o Muti-tap: T9: Q. How to impmnt? 47

48 Patricia tri Patricia tri. [Practica Agoritm to Rtriv Information Codd in Apanumric] Rmov on-way brancing. Eac nod rprnt a qunc of caractr. Impmntation: on tp byond ti cour. put("", 1); put("fi", 2); tandard tri no on-way brancing Appication. Databa arc. P2P ntwork arc. IP routing tab: find ongt prfix matc. Comprd quad-tr for N-body imuation. Efficinty toring and qurying XML documnt. 1 f intrna on-way brancing fi 1 2 i xtrna on-way brancing Ao known a: crit-bit tr, radix tr. 2 48

49 Suffix tr Suffix tr. Patricia tri of uffix of a tring. Linar-tim contruction: byond ti cour. uffix tr for BANANAS BANANAS A NA S NA S S NAS NAS S Appication. Linar-tim: ongt rpatd ubtring, ongt common ubtring, ongt paindromic ubtring, ubtring arc, tandm rpat,. Computationa bioogy databa (BLAST, FASTA). 49

50 String ymbo tab ummary A ucc tory in agoritm dign and anayi. Rd-back BST. Prformanc guarant: og N ky compar. Support ordrd ymbo tab API. Ha tab. Prformanc guarant: contant numbr of prob. Rquir good a function for ky typ. Tri. R-way, TST. Prformanc guarant: og N caractr accd. Support caractr-bad opration. Bottom in. You can gt at anyting by xamining bit (!!!) 50