CSE 5311 Notes 2: Binary Search Trees

Transcription

1 S Notes : inry Serch Trees (Lst updted /0/7 0: M) ROTTIONS c d Single left rottion t (K rotting edge ) Single right rottion t (K rotting edge ) d c c d F oule right rottion t F G F G Wht two single rottions re equivlent?

2 (OTTOM-UP) R-LK TRS red-lc tree is inry serch tree whose height is logrithmic in the numer of eys stored.. very node is colored red or lc. (olors re only emined during insertion nd deletion). very lef (the sentinel) is colored lc.. oth children of red node re lc.. very simple pth from child of node X to lef hs the sme numer of lc nodes. This numer is nown s the lc-height of X (h(x)). mple: = red = lc Oservtions:. red-lc tree with n internl nodes ( eys ) hs height t most lg(n+).. If node X is not lef nd its siling is lef, then X must e red.. There my e mny wys to color inry serch tree to me it red-lc tree.. If the root is colored red, then it my e switched to lc without violting structurl properties.

3 INSRTION. Strt with unlnced insert of dt lef (oth children re the sentinel).. olor of new node is.. My violte structurl property. Leds to three cses, long with symmetric versions. The pointer points t red node whose prent might lso e red. se : = red = lc c d c d se : = red = lc d e c d e c

4 se : = red = lc c d e c - d e mple: = red = lc Insert Insert

5 Insert Insert Reset to lc mple: Insert

6 eletion Strt with one of the unlnced deletion cses:. eleted node is dt lef.. Splice round to sentinel.. olor of deleted node? Red one lc Set doule lc pointer t sentinel. etermine which of four relncing cses pplies.. eleted node is prent of one dt lef.. Splice round to dt lef. olor of deleted node? Red Not possile lc dt lef must e red. hnge its color to lc.. Node with ey-to-delete is prent of two dt nodes.. Stel ey nd dt from successor (ut not the color).. elete successor using the pproprite one of the previous two cses.

7 se : 7 = red = lc e f c d e f c d se : = red = 0 = lc = = either +color() +color() c d e f c d e f se : = red = 0 = lc = = either +color() +color() c - c d e f d - e f

8 se : 8 = red = 0 = lc = = either +color()+color() +color()+color() - +color() +color() - +color() - - +color() - +color() - +color() +color() - e - +color() f c d e f c d (t most three rottions occur while processing the deletion of one ey) mple: elete

9 If reches the root, then done. Only plce in tree where this hppens. elete elete If reches red node, then chnge color to lc nd done.

10 0 elete elete elete elete

11 VL TRS n VL tree is inry serch tree whose height is logrithmic in the numer of eys stored.. ch node stores the difference of the heights (nown s the lnce fctor) of the right nd left sutrees rooted y the children: height right - height left lnce fctor must e +, 0, - (lens right, lnced, lens left).. n insertion is implemented y:. ttching lef. Rippling chnges to lnce fctor:. Right child ripple Prent.l = 0 + nd ripple to prent Prent.l = - 0 to complete insertion Prent.l = + + nd ROTTION to complete insertion. Left child ripple Prent.l = 0 - nd ripple to prent Prent.l = + 0 to complete insertion Prent.l = - - nd ROTTION to complete insertion 0 0

12 . Rottions. Single (LL) - right rottion t Rest of Tree Rest of Tree h h h h h h Restores height of sutree to pre-insertion numer of levels RR cse is symmetric. oule (LR) F - Rest of Tree 0 Rest of Tree + G h 0 F + h - h h- h- G h h- h- Insert on either sutree Restores height of sutree to pre-insertion numer of levels RL cse is symmetric

13 eletion - Still hve RR, RL, LL, nd LR, ut two ddditionl (symmetric) cses rise. Suppose 70 is deleted from this tree. ither LL or LR my e pplied Fioncci Trees - specil cse of VL trees ehiiting two worst-cse ehviors -. Mimlly sewed. (m height is roughly log.68 n =. lg n, epected height is lg n +.). θ(log n) rottions for single deletion. (empty) (empty) 0 6 7

14 TRPS (LRS, p. ) Hyrid of ST nd min-hep ides Gives code tht is clerer thn R or VL (ut comprle to sip lists) pected height of tree is logrithmic (. lg n) Keys re used s in ST Tree lso hs min-hep property sed on ech node hving priority: Rndomized priority - generted when new ey is inserted Virtul priority - computed (when needed) using function similr to hsh function sides: the first pulished such hyrid were the crtesin trees of J. Vuillemin, Unifying Loo t t Structures,. M (), pril 9, 9-9. more complete eplntion ppers in.m. Mcreight, Priority Serch Trees, SIM J. omputing (), My 98, 7-76 nd chpter 0 of M. de erg et.l. These re lso used in the elegnt implementtion in M.. eno nd T.. Striovsy, omputing Longest ommon Sustrings in.. Hirsch, omputer Science - Theory nd pplictions, LNS 00, 008, 6-7. Insertion Insert s lef Generte rndom priority (lrge rnge to minimize duplictes) Single rottions to fi min-hep property

15 mple: Insert 6 with priority of fter rottions: eletion Find node nd chnge priority to Rotte to ring up child with lower priority. ontinue until min-hep property holds. Remove lef.

16 elete ey : 6 UGMNTING T STRUTURS Red LRS, section. on using R tree with rning informtion for order sttistics. Retrieving n element with given rn etermine the rn of n element Prolem: Mintin summry informtion to support n ggregte opertion on the smllest (or lrgest) eys in O(log n) time. mple: Prefi Sum Given ey, determine the sum of ll eys given ey (prefi sum). Solution: Store sum of ll eys in sutree t the root of the sutree Key Prefi Sum

17 To compute prefi sum for ey: 7 Initilize sum to 0 Serch for ey, modifying totl s serch progresses: Serch goes left - leve totl lone Serch goes right or ey hs een found - dd present node s ey nd left child s sum to totl Key is : ( + 0) + (0 + 6) + ( + ) = 6 Key is 0: ( + 0) + (0 + 9) = 0 Key is 6: ( + 0) + (6 + 0) = Vrition: etermine the smllest ey tht hs prefi sum specified vlue. Updtes to tree: Non-structurl (ttch/remove node) - modify node nd every ncestor Single rottion (for prefi sum) Σ Σ Σ Σ Σ Σ+Σ+ Σ Σ Σ Σ (Similr for doule rottion) Generl cse - see LRS., especilly Theorem. Intervl trees (LRS.) - more significnt ppliction Set of (closed) intervls [low, high] - low is the ey, ut duplictes re llowed ch sutree root contins the m vlue ppering in ny intervl in tht sutree ggregte opertion to support - find ny intervl tht overlps given intervl [low, high ] Modify ST serch...

18 if ptr == nil no intervl in tree overlps [low, high ] 8 if high ptr->low nd ptr->high low return ptr s n nswer if ptr->left!= nil nd ptr->left->m low ptr := ptr->left else ptr := ptr->right Updtes to tree - similr to prefi sum, ut replce dditions with mimums OPTIML INRY SRH TRS Wht is the optiml wy to orgnize sttic list for serching?. y decresing ccess proility - optiml sttic/fied ordering.. Key order - if misses will e frequent, to void serching entire list. Other onsidertions:. ccess proilities my chnge (or my e unnown).. Set of eys my chnge. These led to proposls (lter in this set of notes) for (online) dt structures whose dptive ehvior is symptoticlly close (nlyzed in Notes ) to tht of n optiml (offline) strtegy. Online - must process ech request efore the net request is reveled. Offline - given the entire sequence of requests efore ny processing. ( nows the future ) Wht is the optiml wy to orgnize sttic tree for serching? n optiml (sttic) inry serch tree is significntly more complicted to construct thn n optiml list.. ssume ccess proilities re nown: eys re K < K <! < K n pi = proility of request for Ki q i = proility of request with K i < request < K i+ q0 = proility of request < K q n = proility of request > K n

19 . ssume tht levels re numered with root t level 0. Minimize the epected numer of comprisons to complete serch: 9 n n pj ( KeyLevel( j) +) + q j MissLevel j j= j=0 ( ). mple tree: 0 K p K p K p q0 q K K p p q q q q. Solution is y dynmic progrmming: Principle of optimlity - solution is not optiml unless the sutrees re optiml. se cse - empty tree, costs nothing to serch. pi+ pj qi qj qi+ qj- c( i, j) cost of sutree with eys K i+,!,k j c( i, j) lwys includes ectly p i+,!, p j nd q i,!,q j c( i,i) = 0 se cse, no eys, just misses for q i (request etween K i nd K i+ )

20 Recurrence for finding optiml sutree: 0 c( i, j) = w( i, j) + min ( c( i, ) + c(, j) ) i< j tries every possile root ( ) for the sutree with eys Ki+,!,K j w( i, j) = p i+ +!+ p j + q i +!+ q j ccounts for dding nother proe for ll eys in sutree : Left: p i+ +!+ p + q i +!q Right: p+ +!+ pj + q +!qj Root: p K c(i,-) c(,j). Implementtion: -fmily is ll cses for c( i,i + ). -fmilies re computed in scending order from to n. Suppose n = : _0 _ c( 0,0) c( 0,) c( 0,) c( 0,) c 0, c(, ) c(, ) c(, ) c(, ) c(, ) c(,) c(,) c(,) c(,) c(,) c(, ) c(,) c(,) c(,) c, ( ) ( ) c( 0,) ompleity: O n spce is ovious. O n time from: n = ( n + ) where is the numer of roots for ech c i,i + in fmily. ( ) nd n + is the numer of c( i,i + ) cses

21 6. Trcec - esides hving the minimum vlue for ech c( i, j), it is necessry to sve the suscript for the optiml root for c( i, j) s r[i][j]. This lso leds to Knuth s improvement: ( ) must hve ey with suscript no less thn the ey ( ) nd no greter thn the ey suscript for the Theorem: The root for the optiml tree c i, j suscript for the root of the optiml tree for c i, j root of optiml tree c( i +, j). (These roots re computed in the preceding fmily.) Proof:. onsider dding p j nd qj to tree for c( i, j ). Optiml tree for c i, j t the root or use one further to the right. ( ) must eep the sme ey Ki+ Kj-. onsider dding pi+ nd q i to tree for c( i +, j). Optiml tree for c i, j ey t the root or use one further to the left. ( ) must eep the sme Ki+ Kj 7. nlysis of Knuth s improvement. ( ) cse for -fmily will vry in the numer of roots to try, ut overll time is reduced to ch c i, j O n y using telescoping sum:

22 n n n ( r[ i +] [ i + ] r[ i] [ i + ] +) = = i=0 = n = ( r[ n +] [ n] r[ 0] [ ] + n +) = n n ( n 0 + n +) = ( n +) = O n = = r[ ] [ ] r[ 0] [ ] + + r[ ] [ + ] r[ ] [ ] + + r[ ] [ + ] r[ ] [ + ] + +! + r[ n +] [ n] r[ n ] [ n ] + n=7; q[0]=0.06; p[]=0.0; q[]=0.06; p[]=0.06; q[]=0.06; p[]=0.08; q[]=0.06; p[]=0.0; q[]=0.0; p[]=0.0; q[]=0.0; p[6]=0.; q[6]=0.0; p[7]=0.; q[7]=0.0; for (i=;i<=n;i++) ey[i]=i;

23 w[0][0]= w[0][]= w[0][]=0.000 w[0][]=0.000 w[0][]= w[0][]= w[0][6]= w[0][7]= w[][]= w[][]=0.000 w[][]= w[][]= w[][]=0.000 w[][6]= w[][7]= w[][]= w[][]= w[][]= w[][]=0.000 w[][6]= w[][7]= w[][]= w[][]= w[][]=0.000 w[][6]= w[][7]= w[][]= w[][]= w[][6]= w[][7]= w[][]= w[][6]= w[][7]= w[6][6]= w[6][7]=0.000 w[7][7]= ounts - root tric without root tric 77 verge proe length is.6000 trees in prenthesized prefi c(0,0) cost c(,) cost c(,) cost c(,) cost c(,) cost c(,) cost c(6,6) cost c(7,7) cost c(0,) cost c(,) cost c(,) cost c(,) cost c(,) cost c(,6) cost c(6,7) cost c(0,) cost (,) c(,) cost (,) c(,) cost (,) c(,) cost (,) c(,6) cost (,) c(,7) cost (6,) c(0,) cost (,) c(,) cost (,) c(,) cost (,) c(,6) cost (,6) c(,7) cost (,7) c(0,) cost (,(,)) c(,) cost.0000 (,(,)) c(,6) cost.0000 ((,),6) c(,7) cost ((,),7) c(0,) cost ((,),(,)) c(,6) cost ((,),6) c(,7) cost ((,),7(6,)) c(0,6) cost ((,),(,6)) c(,7) cost.0000 ((,),7(6,)) c(0,7) cost.6000 ((,(,)),7(6,)) : c(0,) + c(,7) + w[0][7] =.7 : c(0,) + c(,7) + w[0][7] =.78 : c(0,) + c(,7) + w[0][7] =

24 ONPTS OF SLF-ORGNIZING LINR SRH Hve list dpt to give etter performnce. dvntges: Simple to code. onvenient for situtions with reltively smll # of elements to void more elorte mechnism. Useful for some user interfces. ccess istriutions for Proilistic nlysis: Uniform - Theoreticlly convenient -0 (or 90-0) Rule Zipf - n items, Pi = ihn, H n n = = Since distriution my e unnown or chnging, we re deling with Loclity (temporry hevy ccesses) vs. onvergence (otining optiml ordering) Implementtion pproches Move-to-front (good loclity) Trnspose (Slow to converge. lternting request nomly.) ount - Numer of ccesses is stored in ech record (or use LRS prolem - to reduce its) Sort records in decresing count order Move-hed-: more ggressive thn trnspose Proilistic nlysis my e pursued y Mrov (stte-trnstion) pproches or simultion SPLY TRS Self-djusting counterprt to VL nd red-lc trees dvntges - ) no lnce its, ) some help with loclity of reference, ) mortized compleity is sme s VL nd red-lc trees isdvntge - worst-cse for opertion is O(n)

25 lgorithms re sed on rottions to sply the lst node processed () to root position. Zig-Zig: y z y z. Single right rottion t z.. Single right rottion t y. (+ symmetric cse) Zig-Zg: z oule right rottion t z. y y z (+ symmetric cse) Zig: pplies ONLY t the root y Single right rottion t y. y (+ symmetric cse) Insertion: ttch new lef nd then sply to root.

26 eletion: 6. ccess node to delete, including sply to root.. ccess predecessor in left sutree nd sply to root of left sutree.. Te right sutree of nd me it the right sutree of.