Algorithms. Algorithms 5.2 TRIES. R-way tries ternary search tries character-based operations ROBERT SEDGEWICK KEVIN WAYNE
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1 Agoritm ROBERT SEDGEWICK KEVIN WAYNE 5.2 TRIES Agoritm F O U R T H E D I T I O N R-wy tri trnry rc tri crctr-bd oprtion ROBERT SEDGEWICK KEVIN WAYNE ttp://g4.c.princton.du Lt updtd on 11/12/15 7:45 AM
2 Summry of t prformnc of ymbo-tb impmnttion Ordr of growt of t frquncy of oprtion. impmnttion Q. Cn w do bttr? typic c rc inrt dt ordrd oprtion oprtion on ky rd-bck BST og N og N og N comprto() tb qu() Cod() undr uniform ing umption A. Y, if w cn void xmining t ntir ky, wit tring orting. 2
3 String ymbo tb bic API String ymbo tb. Symbo tb pciizd to tring ky. pubic c StringST<Vu> StringST() crt n mpty ymbo tb void put(string ky, Vu v) put ky-vu pir into t ymbo tb Vu gt(string ky) rturn vu pird wit givn ky void dt(string ky) dt ky nd corrponding vu Go. Ftr tn ing, mor fxib tn BST. 3
4 String ymbo tb impmnttion cot ummry crctr cc (typic c) ddup impmnttion rc it rc mi inrt pc (rfrnc) moby.txt ctor.txt rd-bck BST L + c g 2 N c g 2 N c g 2 N 4 N ing (inr probing) L L L 4 N to 16 N Prmtr N = numbr of tring L = ngt of tring fi iz word ditinct moby.txt 1.2 MB 210 K 32 K ctor.txt 82 MB 11.4 M 900 K Cng. Efficint prformnc for tring ky. 4
5 5.2 TRIES Agoritm R-wy tri trnry rc tri crctr-bd oprtion ROBERT SEDGEWICK KEVIN WAYNE ttp://g4.c.princton.du
6 Tri 6
7 Tri Tri. [from rtriv, but pronouncd "try"] Stor crctr in nod (not ky). Ec nod R cidrn, on for c poib crctr. (for now, w do not drw nu ink) root ink to tri for ky tt trt wit ink to tri for ky tt trt wit b t y 4 ky vu by o r vu for in nod corrponding to t crctr in ky or 7 t 5 3 7
8 Src in tri Foow ink corrponding to c crctr in t ky. Src it: nod wr rc nd non-nu vu. Src mi: rc nu ink or nod wr rc nd nu vu. gt("") b t y o 5 r rturn vu in nod corrponding to t crctr in ky (rturn 3) 8
9 Src in tri Foow ink corrponding to c crctr in t ky. Src it: nod wr rc nd non-nu vu. Src mi: rc nu ink or nod wr rc nd nu vu. gt("") b t y o 5 r rc my trmintd t n intrmdit nod (rturn 0)
10 Src in tri Foow ink corrponding to c crctr in t ky. Src it: nod wr rc nd non-nu vu. Src mi: rc nu ink or nod wr rc nd nu vu. gt("") b t y o 5 r no vu ocitd nod corrponding to t crctr in ky (rturn nu) 10
11 Src in tri Foow ink corrponding to c crctr in t ky. Src it: nod wr rc nd non-nu vu. Src mi: rc nu ink or nod wr rc nd nu vu. gt("tr") b t y o 5 r no ink to t (rturn nu) 11
12 Foow ink corrponding to c crctr in t ky. Encountr nu ink: crt nw nod. Encountr t t crctr of t ky: t vu in tt nod. 12 Inrtion into tri r put("or", 7) b y o t
13 Tri contruction dmo tri b t y o 5 r 1 7 3
14 Tri contruction dmo tri
15 Tri contruction dmo put("", 0) 0 ky i qunc of crctr from root to vu vu i in nod corrponding to t crctr
16 Tri contruction dmo tri 0
17 Tri contruction dmo tri 0
18 Tri contruction dmo put("", 1) 0 1
19 Tri contruction dmo tri 0 1
20 Tri contruction dmo tri 0 1
21 Tri contruction dmo put("", 2) 2 0 1
22 Tri contruction dmo tri 2 0 1
23 Tri contruction dmo put("", 3)
24 Tri contruction dmo tri
25 Tri contruction dmo put("by", 4) b y
26 Tri contruction dmo tri b y
27 Tri contruction dmo put("t", 5) b t y
28 Tri contruction dmo tri b t y
29 Tri contruction dmo put("", 6) b t y ovrwrit od vu wit nw vu 1 3
30 Tri contruction dmo tri b t y
31 Tri contruction dmo tri b t y
32 Tri contruction dmo put("or", 7) b t y o 5 r 1 7 3
33 Tri contruction dmo tri b t y o 5 r 1 7 3
34 Tri rprnttion: Jv impmnttion Nod. A vu, pu rfrnc to R nod. privt ttic c Nod { } privt Objct v; // no gnric rry crtion privt Nod[] nxt = nw Nod[R]; crctr r impicity dfind by ink indx Tri rprnttion c nod n rry of ink nd vu Rmrk. Nitr ky nor crctr r tord xpicity
35 R-wy tri: Jv impmnttion pubic c TriST<Vu> { privt ttic fin int R = 256; privt Nod root = nw Nod(); privt ttic c Nod { /* prviou id */ } xtndd ASCII pubic void put(string ky, Vu v) { root = put(root, ky, v, 0); } privt Nod put(nod x, String ky, Vu v, int d) { if (x == nu) x = nw Nod(); if (d == ky.ngt()) { x.v = v; rturn x; } cr c = ky.crat(d); x.nxt[c] = put(x.nxt[c], ky, v, d+1); rturn x; } 35
36 R-wy tri: Jv impmnttion (continud) pubic Vu gt(string ky) { } Nod x = gt(root, ky, 0); if (x == nu) rturn nu; rturn (Vu) x.v; // ct ndd privt Nod gt(nod x, String ky, int d) { } if (x == nu) rturn nu; if (d == ky.ngt()) rturn x; cr c = ky.crat(d); rturn gt(x.nxt[c], ky, d+1); } 36
37 Tri prformnc Src it. Nd to xmin L crctr for quity. Src mi. Coud v mimtc on firt crctr. Typic c: xmin ony fw crctr (ubinr). Spc. R nu ink t c f. (but ubinr pc poib if mny tring r ong common prfix) rctr r impicity dfind by ink indx 2 c nod n rry of in nd vu Bottom in. Ft rc it nd vn ftr rc mi, but wt pc
38 To dt ky-vu pir: Find t nod corrponding to ky nd t vu to nu. If nod nu vu nd nu ink, rmov tt nod (nd rcur) Dtion in n R-wy tri r b y o t dt("") t vu to nu
39 Dtion in n R-wy tri To dt ky-vu pir: Find t nod corrponding to ky nd t vu to nu. If nod nu vu nd nu ink, rmov tt nod (nd rcur). dt("") b t y o 5 r 1 7 nu vu nd ink (dt nod) 39
40 String ymbo tb impmnttion cot ummry impmnttion R-wy tri. rc it Mtod of coic for m R. Work w for mdium R. Too muc mmory for rg R. crctr cc (typic c) rc mi inrt pc (rfrnc) Cng. U mmory,.g., wy tri for Unicod! moby.txt ddup ctor.txt rd-bck BST L + c g 2 N c g 2 N c g 2 N 4 N ing (inr probing) L L L 4 N to 16 N R-wy tri L og R N R + L (R+1) N 1.12 out of mmory 40
41 5.2 TRIES Agoritm R-wy tri trnry rc tri crctr-bd oprtion ROBERT SEDGEWICK KEVIN WAYNE ttp://g4.c.princton.du
42 Trnry rc tri Stor crctr nd vu in nod (not ky). Ec nod 3 cidrn: mr (ft), qu (midd), rgr (rigt). Ft Agoritm for Sorting nd Srcing String Jon L. Bnty* Robrt Sdgwick# Abtrct W prnt tortic goritm for orting nd rcing mutiky dt, nd driv from tm prctic C impmnttion for ppiction in wic ky r crctr tring. T orting goritm bnd Quickort nd rdix ort; it i comptitiv wit t bt known C ort cod. T rcing goritm bnd tri nd binry rc tr; it i ftr tn ing nd otr commony ud rc mtod. T bic id bind t go- tt i comptitiv wit t mot fficint tring orting progrm known. T cond progrm i ymbo tb impmnttion tt i ftr tn ing, wic i commony rgrdd t ftt ymbo tb impmnttion. T ymbo tb impmnttion i muc mor pc-fficint tn mutiwy tr, nd upport mor dvncd rc. In mny ppiction progrm, ort u Quickort impmnttion bd on n btrct compr oprtion, 42
43 Stor crctr nd vu in nod (not ky). Ec nod 3 cidrn: mr (ft), qu (midd), rgr (rigt). 43 Trnry rc tri c nod 3 ink ink to TST for ky tt trt wit crctr tn ink to TST for ky tt trt wit b y o r t r 1 r y u 1 r b y o r r y u t btrct tri TST
44 Src it in TST gt("") b t y o r rturn vu in nod corrponding to t crctr in ky
45 Src mi in TST gt("tr") b t y o r no ink to t (rturn nu) 45
46 Src in TST Foow ink corrponding to c crctr in t ky. If, tk ft ink; if grtr, tk rigt ink. If qu, tk t midd ink nd mov to t nxt ky crctr. Src it. Nod wr rc nd non-nu vu. Src mi. Rc nu ink or nod wr rc nd nu vu. gt("") mimtc: tk ft or rigt ink, do not mov to nxt cr b mtc: tk midd ink, mov to nxt cr t r y 4 u o r 8 r rturn vu ocitd wit t ky crctr y 13 46
47 Trnry rc tri contruction dmo trnry rc tri b t y o r
48 Trnry rc tri contruction dmo trnry rc tri 48
49 Trnry rc tri contruction dmo put("", 0) 0 ky i qunc of crctr from root to vu uing midd ink vu i in nod corrponding to t crctr 49
50 Trnry rc tri contruction dmo put("", 0) 0 50
51 Trnry rc tri contruction dmo put("", 1)
52 Trnry rc tri contruction dmo trnry rc tri
53 Trnry rc tri contruction dmo put("", 2)
54 Trnry rc tri contruction dmo trnry rc tri
55 Trnry rc tri contruction dmo put("", 3)
56 Trnry rc tri contruction dmo trnry rc tri
57 Trnry rc tri contruction dmo put("by", 4) b y
58 Trnry rc tri contruction dmo trnry rc tri b y
59 Trnry rc tri contruction dmo put("t", 5) b t y
60 Trnry rc tri contruction dmo trnry rc tri b t y
61 Trnry rc tri contruction dmo put("", 6) b t y ovrwrit od vu wit nw vu
62 Trnry rc tri contruction dmo trnry rc tri b t y
63 Trnry rc tri contruction dmo put("or", 7) b t y o r
64 Trnry rc tri contruction dmo trnry rc tri b t y o r
65 Trnry rc tri contruction dmo trnry rc tri b t y o r
66 26-wy tri v. TST 26-wy tri. 26 nu ink in c f. TST. 3 nu ink in c f. 26-wy tri (1035 nu ink, not own) TST (155 nu ink) now for tip ik dim tg jot ob nob ky ut c bt mn gg fw jy ow joy rp gig w w cb wd cw cu f tp go tr jm dug nd 66
67 TST rprnttion in Jv A TST nod i fiv fid: A vu. A crctr c. A rfrnc to ft TST. A rfrnc to midd TST. A rfrnc to rigt TST. privt c Nod { privt Vu v; privt cr c; privt Nod ft, mid, rigt; } tndrd rry of ink (R = 26) trnry rc tr (TST) u ink for ky tt trt wit Tri nod rprnttion ink for ky tt trt wit u u 67
68 TST: Jv impmnttion pubic c TST<Vu> { privt Nod root; privt c Nod { /* prviou id */ } pubic Vu gt(string ky) { Nod x = gt(root, ky, 0); if (x == nu) rturn nu; rturn x.v; } privt Nod gt(nod x, String ky, int d) { if (x == nu) rturn nu; cr c = ky.crat(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; } } pubic void put(string Ky, Vu v) { /* imir, book or bookit */ } 68
69 String ymbo tb impmnttion cot ummry impmnttion rc it Rmrk. Cn buid bncd TST vi rottion to civ L + og N wort-c gurnt. crctr cc (typic c) rc mi inrt pc (rfrnc) moby.txt ddup Bottom in. TST i ft ing (for tring ky), pc fficint. ctor.txt rd-bck BST L + c g 2 N c g 2 N c g 2 N 4 N ing (inr probing) L L L 4 N to 16 N R-wy tri L og R N R + L (R+1) N 1.12 out of mmory TST L + n N n N L + n N 4 N
70 TST wit R 2 brncing t root Hybrid of R-wy tri nd TST. Do R 2 -wy brncing t root. Ec of R 2 root nod point to TST. rry of 26 2 root b c zy zz TST TST TST TST TST Q. Wt bout on- nd two-ttr word? 70
71 String ymbo tb impmnttion cot ummry crctr cc (typic c) ddup impmnttion rc it rc mi inrt pc (rfrnc) moby.txt ctor.txt rd-bck BST L + c g 2 N c g 2 N c g 2 N 4N ing (inr probing) L L L 4N to 16N R-wy tri L og R N R + 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. Ftr tn ing for our bncmrk cint. 71
72 TST v. ing Hing. Nd to xmin ntir ky. Src it nd mi cot bout t m. Prformnc ri on function. Do not upport ordrd ymbo tb oprtion. TST. Work ony for tring (or digit) ky. Src mi my invov ony fw crctr. Support ordrd ymbo tb oprtion (pu xtr!). Bottom in. TST r: Ftr tn ing (pciy for rc mi). Mor fxib tn rd-bck BST. [ty tund] 72
73 5.2 TRIES Agoritm R-wy tri trnry rc tri crctr-bd oprtion ROBERT SEDGEWICK KEVIN WAYNE ttp://g4.c.princton.du
74 String ymbo tb API Crctr-bd oprtion. T tring ymbo tb API upport vr ufu crctr-bd oprtion. ky vu by or 7 t 5 Prfix mtc. Ky wit prfix :,, nd or. Widcrd mtc. Ky tt mtc.: nd t. Longt prfix. Ky tt i t ongt prfix of ort:. 74
75 String ymbo tb API pubic c StringST<Vu> StringST() crt ymbo tb wit tring ky void put(string ky, Vu v) put ky-vu pir into t ymbo tb Vu gt(string ky) vu pird wit ky void dt(string ky) dt ky nd corrponding vu Itrb<String> ky() ky Itrb<String> kywitprfix(string ) ky ving prfix Itrb<String> kyttmtc(string ) ky tt mtc (wr. i widcrd) String ongtprfixof(string ) ongt ky tt i prfix of Rmrk. Cn o dd otr ordrd ST mtod,.g., foor() nd rnk(). 75
76 Wrmup: ordrd itrtion To itrt troug ky in ortd ordr: Do inordr trvr of tri; dd ky ncountrd to quu. Mintin qunc of crctr on pt from root to nod. kywitprfix(""); ky() ky b by o or or t t t q by by by by b y by by or by or t o r 7 t 5 Cocting t ky in tri (trc) 76
77 Ordrd itrtion: Jv impmnttion To itrt troug ky in ortd ordr: Do inordr trvr of tri; dd ky ncountrd to quu. Mintin qunc of crctr on pt from root to nod. pubic Itrb<String> ky() { } Quu<String> quu = nw Quu<String>(); coct(root, "", quu); rturn quu; qunc of crctr on pt from root to x privt void coct(nod x, String prfix, Quu<String> quu) { } if (x == nu) rturn; if (x.v!= nu) quu.nquu(prfix); for (cr c = 0; c < R; c++) coct(x.nxt[c], prfix + c, quu); or u StringBuidr 77
78 Prfix mtc Find ky in ymbo tb trting wit givn prfix. Ex. Autocompt in c pon, rc br, txt ditor, or. Ur typ crctr on t tim. Sytm rport mtcing tring. 78
79 Prfix mtc in n R-wy tri Find ky in ymbo tb trting wit givn prfix. kywitprfix(""); b y 4 6 find ubtri for ky bginning wit "" o r 7 t 5 b y o r 7 t 5 coct ky in tt ubtri Prfix mtc in tri pubic Itrb<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 tring bginning wit givn prfix ky o or or quu or 79
80 Longt prfix Find ongt ky in ymbo tb tt i prfix of qury tring. Ex. To nd pckt towrd dtintion IP ddr, routr coo IP ddr in routing tb tt i ongt prfix mtc. "128" " " " " " " " " " " " " " " " " rprntd 32-bit binry numbr for IPv4 (intd of tring) ongtprfixof(" ") = " " ongtprfixof(" ") = " " ongtprfixof(" ") = "128" Not. Not t m foor: foor(" ") = " " 80
81 Longt prfix in n R-wy tri Find ongt ky in ymbo tb tt i prfix of qury tring. Src for qury tring. Kp trck of ongt ky ncountrd. "" "" "ort" rc nd t nd of tring vu i not nu rturn rc nd t nd of tring vu i nu rturn (t ky on pt) Poibiiti for ongtprfixof() xof() rc nd t nu ink rturn (t ky on pt) 81
82 Longt prfix in n R-wy tri: Jv impmnttion Find ongt ky in ymbo tb tt i prfix of qury tring. Src for qury tring. Kp trck of ongt ky ncountrd. pubic String ongtprfixof(string qury) { int ngt = rc(root, qury, 0, 0); rturn qury.ubtring(0, ngt); } privt int rc(nod x, String qury, int d, int ngt) { } if (x == nu) rturn ngt; if (x.v!= nu) ngt = d; if (d == qury.ngt()) rturn ngt; cr c = qury.crat(d); rturn rc(x.nxt[c], qury, d+1, ngt); 82
83 T9 txting (prdictiv txting) Go. Typ txt mg on pon kypd. Muti-tp input. Entr ttr by rptdy pring ky. Ex. good: T9 txt input. " muc ftr nd mor fun wy to ntr txt" Find word tt corrpond to givn qunc of numbr. 4663: good, om, gon, oof. txtonym Pr * to ct nxt option. Pr 0 to comption option. Sytm dpt to ur' tndnci. ttp:// 83
84 Ptrici tri Ptrici tri. [Prctic Agoritm to Rtriv Informtion Codd in Apnumric] Rmov on-wy brncing. Ec nod rprnt qunc of crctr. Impmnttion: on tp byond ti cour. Appiction. put("", 1); put("fi", 2); tndrd tri no on-wy brncing fi 1 2 Dtb rc. P2P ntwork rc. IP routing tb: find ongt prfix mtc. Comprd qud-tr for N-body imution. Efficinty toring nd qurying XML documnt. 1 f intrn on-wy brncing i xtrn on-wy brncing Ao known : crit-bit tr, rdix tr. 2 84
85 Suffix tr Suffix tr. Ptrici tri of uffix of tring. Linr-tim contruction: w byond cop of ti cour. uffix tr for BANANAS Appiction. BANANAS A NA S NAS NA S S NAS Linr-tim: ongt rptd ubtring, ongt common ubtring, ongt pindromic ubtring, ubtring rc, tndm rpt,. Computtion bioogy dtb (BLAST, FASTA). S 85
86 String ymbo tb ummry A ucc tory in goritm dign nd nyi. Rd-bck BST. Prformnc gurnt: og N ky compr. Support ordrd ymbo tb API. H tb. Prformnc gurnt: contnt numbr of prob. Rquir good function for ky typ. Tri. R-wy, TST. Prformnc gurnt: og N crctr ccd. Support crctr-bd oprtion. 86
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