8 Dynamic Binary Search Trees (February 8)
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1 Evrt s or t oputrs t o. Tr ust sot, ou u sot s. Tr ust tt, ou u to or or ou o t s pp, t sp s out o otro. B, B s 7, Bro (Mr 6, 1978) A oo spot s r s o s souto to t pro. Aoous Lt s p. E Mr [Atoo Brs], Dspro (1992) 8 D Br Sr Trs (Frur 8) 8.1 Dtos I ssu tt vro s r r t t str troo or r sr trs o, sr,, root, tr o,, rt, t, prt, st, s, stor, sutr, prorr, postorr, orr, t. s s t str orts or sr or o, srt o, or t o. Otrs, s Cptr 12 o CLRS. For ts tur, osr o u r trs r vr tr o s t to r r o t tr os tu stor sr s. I prt, rprst t vs t u potrs. R tt t pt (v) o o v s ts st ro t root, ts t (v) s t st to t rtst ts sutr. T t (or pt) o t tr s ust t t o t root. T s v o v s t ur o os ts sutr. T s o t o tr s ust t tot ur o os, I usu ot. A tr t t s t ost 2 vs, so t u t o - r tr s. I t orst s, t t rqur or sr, srto, or to to t t o t tr, so r ou p t t s os to s poss. T st poss o s to v prt tr, sutr s s os to t vs s poss, ot sutrs r prt. T t o prt tr s, so t orst-s sr t s O(o ). Hovr, v strt t prt tr, ous squ o srtos /or tos ou t tr rtrr u, rv t sr t up to Θ(). To vo ts pro, to pro o t tr to t. Tr r svr tos or o ts, p o t to us, t sr tr s v rt. Eps u AVL trs, r- trs, t- trs, t trs, ou- trs, pt- trs, B-trs, trps, ro r sr trs, sp sts, 1 upsts. 2 So o ts trs support srs, srtos, tos, O(o ) orst-s t, otrs O(o ) ort t, st otrs O(o ) pt t. I ts tur, I suss to r sr tr t struturs t oo ort pror. T rst s t spot tr, sovr Ar Arsso 1989 pt 1 Y,. Sp sts r t r r sr trs. Wtvr ou s, Mr. P. 2 Ts r sst ro vrts o t Po r sr trs ou s t rst tr! [H. Brö, F. Cs, M. Dur. Ro upsts: A up-- tor t strutur. Musrpt, ttp://poto.po.u/ r/pu/upst.t.] So o ou o o to. 1
2 I Gpr Ro Rvst T so s t sp tr, sovr D Stor Bo Tr L Dtos: Go Ru Frst t s osr t sp s r strt t prt- tr, o t to pror srs tos. To t oo sr t ts, us tqu o ru. W t t rqust, ot r t o to t, ut o t tu t t. Ts rqurs sp oto to our sr ort st us r os to u srs, ut sr or r o, t sr rout ss t s t tr. Ts ps t tr or or ss, ut o t sr t s o or uto o t out o t urrt stor t tr. To r ts, so p tr o o os v r, t pp t oo ru: Go Ru Ru. As soo s t os t tr v r, ru prt tr ot o t ur os. 5 Wt ts ru p, sr ts O(o ) t t orst s, r s t ur o ur os. Sp, s t tr s t ost r os, or 2 os totr, to t ost + 1 s. Tr r svr tos or ru t tr O() t, r s t s o t tr. (Hoor!) So s to ost Θ() t t orst s, ut o v to ru; ost tos t o O(o ) t. W sp O() t ru, ut o tr Ω() tos, so t ort ost o ru t tr s O(1) pr to. (Hr I us sp vrso o t tto to. For to, r $1 t; tr tos, v ot $, s ust ou to p or r t tr ot t r os.) S so v to r t o t, t tot ort t or to s O(o ). 8.3 Isrtos: Prt Ru No suppos o t to support srs srtos. W t ot r srt os to t tr, s tt ou t uv to t sr ort. 6 So st, us otr to prt ru. W srt os or, ut vr sutr os u ou, ru t. T to o u ou ps o rtrr ostt α > 1. E o v o so stor ts t (v) t s o ts sutr v. W o o our srto ort t t oo ru: 3 A. Arsso. Gr trs. J. Aorts 30:1-28, I. Gpr R. L. Rvst. Spot trs. Pro. SODA 1993, pp , T o p s Arsso s; t or vrso o s ppr ppr T to pprs tu sr vr st rt r orts. T ort I us r s osr to Arsso s, ut ss s osr to Gpr Rvst s. 4 D. D. Stor R. Tr. S-ust r sr trs. J. ACM 32: , Atrt: W t ur o ur os s o ss t t por o to, ru t tr. Ts ru surs tt t tr s s t. 6 Atu, tr s otr t strutur tqu, rst sr Jo Bt Js S 1980, tt os t r srt t o! Ist, tr ort puts t o to r t strutur ts. T s o s tr r to trs o t t s s, tos to trs r r to tr. So ts s t r outr st o o -o tr, ou v oto o 2 -os trs o stt ss. T ort ost o srt t s O(o ). Uortut, v to oo trou up to trs to t, so t sr t os up to O(o 2 ). [J. L. Bt J. B. S. Dopos sr pros I: Stt-to- trsorto. J. Aorts 1: , 1980.] S so Hoor 3. 2
3 Prt Ru Ru. Atr srt o, up t tr upt t ts ss o t os o t sr pt. I outr o v r (v) > α v, ru ts sutr to prt tr ( O( v ) t). I s oo ts ru, t tr srto, t t o t tr s t ost α. Tus, s α s ostt, t orst-s sr t s O(o ). I t orst s, srtos rqur Θ() t t v to ru t tr tr. Hovr, t ort t or srto s o O(o ). Not surprs, t proo s tt t or opt t or tos. D t I(v) o o v to o ss t t sout r t t ss o ts to sutrs, or ro, vr s rr: I(v) = { t(v) rt(v) 1, 0 } A sp uto proo ps tt I(v) = 0 or vr o v prt tr. So t tr ru t sutr o v, v I(v) = 0. O t otr, srto to t sutr o v rts tr t(v) or rt(v), so I(v) s t ost 1. T o ss os o to t oo. L 1. Just or ru v s sutr, I(v) = Ω( v ). Bor prov ts, t s rst oo t t t ps. I I(v) = Ω( v ), t Ω( v ) s v srt t v s sutr s t st t t s rut ro srt. O t otr, ru t sutr rqurs O( v ) t. Tus, ort t ru ost ross t srtos s t st ru, v s r ostt t or srto to ts sutr. S s srt to t ost α = O(o ) sutrs, t tot ort ost o srto s O(o ). Proo: S r out to ru t sutr t v, ust v (v) > α v. Wtout oss o rt, suppos tt t o ust srt t to v s t sutr. Etr ust rut ts sutr or t v to, so so v (t(v)) α t(v). Co ts to quts t t rursv to o t, t α v < (v) (t(v)) + 1 α t(v) + 1. Atr so r, ts sps to t(v) > v /2 1/α. Co ts t t tt v = t(v) + rt(v) + 1 o so or r vs us t qut rt(v) < ( 1 1/2 1/α) v 1. F, o ts to quts us t rursv to o. I(v) t(v) rt(v) 1 > ( 2/2 1/α 1 ) v S α s ostt r t 1, t tor prtss s postv ostt. 3
4 8.4 Spot Trs F, to ot srtos tos t, spot trs us ot o t prvous tqus. W us prt ru to r- t tr tr srtos, o ru to r- t tr tr tos. E sr ts O(o ) t t orst s, t ort t or srto or to s so O(o ). Tr r s t ts t ( I o t sr), ut o s r rqur. O v o t ss, tu sp t t strutur. It s ot r to prov tt t ost o sutr (t spot) s rut ur srto. Lss ovous, v t t s ort t ous (pt or s ostt tor) o t t tr trs to to t tu tr: t s o t tr tr, t t o t tr tr, t ur o r os. Wvr srto uss t tr to o u, oput t ss o t sutrs o t sr pt, strt t t stopp t t spot, t proporto to t s o t spot sutr. S tt u t to r- t spot sutr, ts oputto rss t ru t o s ostt tor! Tus, u ost vr otr o trs, spot trs rqur o O(1) tr sp. 8.5 Rottos, Dou Rottos, Sp Aotr to or t r sr trs s ust t sp o t tr o, us oprto rotto. A rotto t o rss ts pt o rss ts prt s pt o. Rottos pror ostt t, s t o vov sp potr puto. rt t A rt rotto t t rotto t r vrss. For t rsos, to us rottos to t t. Tr r to tps o ou rottos, t ror-ostr -. A ror-ostr t o ossts o rotto t s prt oo rotto t, ot t s rto. A - t ossts o to rottos t, oppost rtos. E ou rotto rss t pt o to, vs t pt o ts prt u, rss t pt o ts rprt tr o or to, p o t tp o ou rotto. Etr tp o ou rotto pror ostt t. A rt ror-ostr t t ror-ostr t. 4
5 A - t. T str s s ot so. F, sp oprto ovs rtrr o t tr up to t root trou srs o ou rottos, poss t o s rotto t t. Sp o v rqurs t proporto to (v). (Ovous, ts s t pt or sp, s tr sp v s t root tus s pt ro!) Sp o. Irrvt sutrs r ott or rt. 8.6 Sp Trs A sp tr s r sr tr tt s pt or or ss sp. Itutv, tr ss o, ov t to t root t sp oprto. I or t: Sr: F t o ot t us t usu ort, or ts prssor or sussor t s ot prst. Sp vr o s ou. Isrt: Isrt o us t usu ort, t sp tt o. Dt: F t o to t, sp t, t t t. Ts spts t tr to to sutrs, o t s ss t, t otr t s r t. F t o t t sutr t t rst (.., t orr prssor o t or tr), sp t, o t to t rt sutr. Dt o sp tr. E sr, srto, or to ossts o ostt ur o oprtos o t or o to o, t sp t up to t root. S t o s r pr t t sp up, to t oo ort ous or sp trs s to rv oo ort ous or s sp. 5
6 Bv t or ot, t sst to o ts uss t pott to. T r o o v s s r(v) = v. I prtur, v s t root o t tr, t r(v) =. W t pott o sp tr to t su o t rs o ts os: Φ = v r(v) = v v T ort ss o sp trs os o to t oo. Hr, r(v) ots t r o v or (s or ou) rotto, r (v) ots ts r trrs. L 2. T ort ost o s rotto t v s t ost 1 + 3r (v) 3r(v), t ort ost o ou rotto t v s t ost 3r (v) 3r(v). Prov ts rqurs s ut or s ss o t rt tps o rottos, ts up ost p Stor Tr s or ppr. (T t t Ass L.) I o t rpt t r. B up t ort osts o t rottos, tt t tot ort ost o sp o v s t ost 1 + 3r (v) 3r(v), r r (v) s t r o v tr t sp. (T trt rs out tsop su.) But tr t sp, v s t root, so r (v) =, s tt t ort ost o sp s t ost 3 1 = O(o ). Tus, vr srto, to, or sr sp tr ts O(o ) ort t, s opt. Atu, sp trs r opt u stror ss. I p(v) ots t prot o sr or o v, t t ort sr t or v s O(o(1/p(v))). I vr o s qu, v p(v) = 1/ so O(o(1/p(v))) = O(o ), s or. Ev t os r t qu, tou, t opt stt r sr tr or ts st o t os s sr t o Θ(o(1/p(v))). Sp trs v us ts opt ort t sr t t o to t t strutur or t sp ort. 6
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