Balanced binary search tree (BST) Self-organized trees. Splay trees. by HUNG Lau Yung (Simon)

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1 J S LN T U U V T U U H G? ;; r y su -. ~..... p $ og cc QP R IKJ LMN alaced bary search tree (ST (log per operato Ubalaced after soe sertos ad deletos Sky tree Self-orgazed trees Restructurg the tree accordg to the operatos State-based algorth ad eoryless algorth by HUNG Lau Yug (So CMP67K -le lgorth 4 th Noveber 3 9 EF CD 8 :@? 9> 9:7; / *+ ( %'& v} s r z{u vq x qkr stu Splay trees Iveted by Sleator ad Tara Meoryless algorth Dfferet fro state-based algorth Dyac ootoe trees WPL trees ad D-trees Store forato about the accesses to the odes Itroducto Splayg Cost of splayg Splay tree operatos pplcato of splay tree Cocluso Self-dustg Search Tree y Jorge Stolf #!" a kl ` bhg daef ab_c ]^ X_` [\ WXYZ

2 œ Ÿ ó ª ª š Ù ì àà Å Ì ÆÈ Ñ Ò Ó Ò Ò Ò Ó Ã» œ Ÿ Q Kœ žÿ ðñò ï í'î Itroducto Splayg Cost of splayg Splay tree operatos pplcato of splay tree Cocluso Splay tree Slar to Move-to-Root algorth [RIN LlEN ad IN MUNR 978] oth of the ove the accessed ode to the root Dfferet fro sple exchage algorth Rotate the accessed ode th ts paret Self-dustg Search Tree y Jorge Stolf Š Œ Ž Œ ˆ Š ƒ é æçý ßå áþâã ÞßÜà ÚÛ ÕÜÝ ØÙ ÔÕÖ Þ êë ÉÐ Ï Æ Å ÍÎÈ ÉQÊ Ë ÄKÅ ÆÇÈ Splayg rules Each rule cossts of a teplate ad a result tree oth teplate ad result are bary trees cotag the sae uber of odes Each rule s ether a teral rule or o-teral rule otto-up splayg 3 dfferet splayg rules Splay tree ll operatos are splayg Stll keep syetrc order Reduce the heght of a ST Good perforace for skeed data accesses -le algorth Restructurg after each access µ À ÁÂ ½¾ ¼» µ¹º µ ³ ±² ³ «þ ÿ ÿ ý ûü öýþ ùú õö ø ÿ

E 4 $ + '!" v tu s pqr o Ä LM JK I FGH ³ žÿ š œ V ª RR L_ CD @?

$ccess a ^ V P Z[ \] X QW Y SPTU PQNR GN Splay tree deo Ž Œ Š7 Rule 3 (ZIG-ZG$

3 E 4 $ + '!" v tu s pqr o Ä LM JK I FGH ³ žÿ š œ V ª RR L_ ;: Rule (ZIG rule teral rule (Ths rule s appled at ost oce depedg the access path s odd or eve ccess b ccess a 3 +'' % / -. & (%* %&# # $ l k f h g efd bc `7a Rule (ZIG-ZIG rule o-teral rule ccess c ccess a ^ V P Z[ \] X QW Y SPTU PQNR GN Splay tree deo Ž Œ Š7 Rule 3 (ZIG-ZG rule o-teral rule ccess c ccess c ˆ z { ƒy }z~ z{x qxy ÂÃ Á À» ¾½ ¼ º» ¹ µ7 Exaple of splayg (access a ² ª ± «

4 ç C ßÞ Ý Õ ÑÑ / È.&"" *+-!" u v v ù õõ Mb éêëì ayuu TZ STQU åæ ã 4 > ;: (x : eght of a ode x the tree a fxed but arbtrary value T : deotes the hole splay tree T x : deotes the tree rooted at x y Tx : eas y s a ode the tree Tx sze(x Rak(xlog(sze(x Potetal fucto ( y Tx (y E F for splay tree x T rak( x G H Itroducto Splayg Cost of splayg Splay tree operatos pplcato of splay tree Cocluso Self-dustg Search Tree y Jorge Stolf Ï ÛÜ ÙÚ ØÎ ÐÖÕ ÒÏÓÔ ÏÐÍÑ ËÌ ÆÍÎ ÉÊ ÅÆÇ (!'& # $% st r f c o d q op kl gh fd e cd Exaple of potetal of a tree ad all odes have the sae eghts hch are /6 e.g. the rak(c log(/ Potetal ( 3*log(/6+log(/6 +log(3/6+log(6/ Not terested the cost of each operato Iterested the cost of a sequece of accesses e use aortzed cost Theoretcal cost of a gve sequece of operatos Cosder the chage of potetal of the tree Potetal fucto easure of a data structure cofgurato or state ó ýþ ÿ û üò úù öó ø óñõ ïð êñò íî S_` ]^ [\R Y VSWX MN P JQR IJKL

5 } ˆ ƒ úù øðìì êëì./ 7 6 Ó ; ; Ø Ø Ü Û Û Ú ß Ù Ù > ÁÀ ³³ ² ±² ³ ( ' Ÿ œž š þ û ü ÿ þü ý ûü Let h be the heght of the teplate e.g. h ccess bot - ortzed cost s defed by actual operato cost + chage of potetal Let be the aortzed cost of applyg a splayg rule (here e oly cosder the o-teral rules frst ed + Potetal of the tree after applyg the splayg rule Let ed Potetal of the tree before applyg the splayg rule > + Cost for replacg the teplate th the result tree Ž Œ Š ƒ ˆ } ~ y xyz{ êö õ òóé ëñð íêîï å æç áé àáâã ÑÒ Ð Å Â Í Ã Ï ÌÍÎ ÉÊË ÆÇÈ ÅÃ Ä ÂÃ 9: 8-5 * * +* h+ odes the teplate because the teplate ust be a path ecause rak + (top + rak - (top - e oly cosder h odes ÄÖ h(rak? (top Rak + (top + : the axu rak after applyg the rule Rak - - : the u rak before applyg the rule rak ly the raks of the odes the teplate ll be chaged ples ÔÕ Ö y : odes the teplate T : the teplate tree R : the result tree y R rak rak + (y: the rak of ode y after applyg the rule y ÜÞÝ rak - (y: the rak of ode y before applyg the rule y T rak y ±½¾»¼ ¹º ±µ «ª #$ %&! "

$_` ÂÃ h Ë a Ä g Ê l Ï l ZY X P LL ½¼» ³ Œ ïð ø ñ ü š ýþ œ š ÿ œ œ š š ÿ ž Ÿ œ } yy êé op à ÜÜ ÐÑÒÓ k ^ f [ \ ë f bcd ^ \ ] [+\ ÍÎ Ì Á É ¾ È É ÅÆÇ Á À ¾+ Case : the result of the rule s a path To$

6 _` ÂÃ h Ë a Ä g Ê l Ï l ZY X P LL ½¼» ³ Œ ïð ø ñ ü š ýþ œ š ÿ œ œ š š ÿ ž Ÿ œ } yy êé op à ÜÜ ÐÑÒÓ k ^ f [ \ ë f bcd ^ \ ] [+\ ÍÎ Ì Á É ¾ È É ÅÆÇ Á À ¾+ Case : the result of the rule s a path To derve a better upper boud cosder to cases Case : the result of the rule s ot a KQP MJN JK FG HI HL DE TU R SI J VW ³ ±² ª ««µ ¹º Š ˆ Ž Š ˆ +ˆ úû ù î ö ë ì õ ö òó î ì í ë+ì Cosder the sze of top + bot + ad bot - (ote that bot + ad bot - are dsoted the result tree sze (sze rak (top (top (top sze 4sze rak sze sze rak by equalty (a b 4ab fter takg log ecause y ad y are the chldre of top + ples sze (top sze (y sze (y (sze (top (sze (y sze (y ( sze (top 4sze (ysze (y rak (top rak (y rak (y h(rak (top - rak The last step s because the result tree cotas y ad y therefore they are overestate by the rak + (top + by equalty (a b fter takg log 4ab ƒ v x~ } z{ x st uv uy qr Ú å æç â ãù Ûáà ÝÚÞß ÚÛ Ö ÑØÙ ØÜ ÔÕ

7 *+ ± 5 ¹ ² 4 9 Å Æ Â Á ¾ À Æ ¾ À Å À h Á ; ; : : ½? > > KJ M I N G F E C N D D E C M ½ F ½ $ # «ª ž ž æç v ï o u î ó z ˆ ƒ } ˆ ~ ~ } ó h z { Œ Š Š õ ö õ õ Œ ú ù Sh g _ [ [ áà ß Ó Ó 78 6 % & 3 -/. & ( %'&»¼ º 3 µ ³/ ' Ä À bot rak top ÂÃ -rak The above equalty s oly for applyg oe splayg rule If e repeat to apply the splayg rule utl the accessed ode becoes the root The e ust su up the above equato for applyg dfferet splayg rules to fd the upper boud for aortzed cost per access ll ters the teredate trees ll be cacelled out I ÄÖ h(rak (top rak t overestates rak + + by rak + (top + f e add rak + (top + -rak - - to the rght had sde ad cobe th rak (top rak rak We get L E h E bot rak GH top -rak "! œ Ÿœ œ š šž xy l t s3tr p/q l k ' ñò ð å í â ã ì3íë é/ê å ã â'ã So e oly cocer the chage of potetal for the tal tree ad the result tree No let access_ode be the aortzed cost for splayg a ode to the root ecause the heght of the splayg teplate s at ost access_ode Rak of the root 3 rak r Rak of the tal access ode -rak ööø 3log(sze(r/sze( (log(sze(r/sze( ú If the access path s odd the rule eed to be appled Cobe case ad case e have bot bot rak top Sce aortzed cost ( after applyg oe splayg rule + ples h rak top Ž -rak -rak [SHK SURMNIN996] Y c d ef a bx Z`_ \Y]^ YZ UV P WX W[ ST R PQ Ñ Û Ü ÝÞ Ù ÚÐ ÒØ ÔÑÕÖ ÑÒ ÍÎ È ÏÐ ÏÓ ËÌ Ê ÇÈÉ

8 xy & u v z %! } Š Š * ˆ + ü ÿ þ rs q e e YZ[ \ KL T J H I «M S P ³» X ½ ¾ ¼ ¾ ½ ¼ ¼ 3 F E / ( ' # "$# ƒ t u ~$ { tu The largest potetal drop ( - s here ( s the eght of ode ad So e have t total (3log(sze(r/( + + log(w/( (3log(W/( + + log(w/( W log W/ Œ Ž ( sze(rw because r s the root The total operato cost (t total over accesses Recall that aortzed costactual operato cost + chage of potetal so e have t + : the aortzed cost for the th access t :the actual operato cost for the th access -: the potetal before the th access : the potetal after the th access - ûüý c op k l b d fcgh cd _` Z a ae b ]^ VW U J R G H Q$R N GH ¹º µ ª «$µ ± ² ª«alace Theore: [SLETR ad TRJN985] the total access cost (t total s log ¾ So total operato cost (t total over operatos s: t total t ( + Φ - Φ - : the uber of odes the tree : the uber of accesses t t t total total total + Φ - Φ (3log(sze(r/sze( + + Φ (3log(sze(r/( + + Φ - Φ The tal sze of the th accessed ode - Φ ecause sze( s at least ( 7 6 8> :7; Ÿ š Ÿ œž š

9 ÝÞ EF æ Ü Ú Û ß G å â R T R S S Ø Ø Ï Ë &'( Â > 6 v "! ú ö ö þ ÿ Y êëì í e aa X UVW é ç Ü Ù Ú ã$ à á ÙÚ PQ D L N M KLJ HI D C Proof of Statc ptalty Theore: ssg a eght of q(/ to ode the Proof of balace theore: ssg a eght of / to each ode the tree the t total (3log(W/( + + (3log(/(q( / + + log(/(q(/ (3log(/q( + + log(/q( q((3log(/q( + + log(/q( q((log(/q( + ( log(w/( W(q(/ (3log(/q( q((3log(/q( ecause each ode s accessed at least oce so the last ter ca be dropped t total (3log(W/( + + (3log(/(/ + + log(/(/ * 3log( + + *log( (( + log( + log(w/( ecause ( (/ ad W ÊÐÏ ÌÉÍÎ ÀÁ :; 8 9/ ÉÊ ÅÆ À Ç ÇË È ÃÄ. - './ *+ É ÕÖ Ó Ô Ñ Ò È stu r oqp Itroducto Splayg Cost of splayg Splay tree operatos pplcato of splay tree Cocluso Self-dustg Search Tree y Jorge Stolf Statc ptalty Theore: [SLETR ad TRJN985] If every te s accessed at least oce the the total access cost (t total s q : the uber of odes the tree : the uber of accesses q(: the uber of accesses to ode ad log /q q #%$ ü ý ó õûú øù _ kl g h^ `fe b_cd õ ðñ ë ò òö ó îï _`]a [\ V]^ YZ

Ÿ ª «ª } ˆ ö õ í éé ÝÞßà ÕÖ 3 4 9 6 6 Wt 5 8 7 4 Ú Ù > > ; Ø Ú Ù ÇÈ Æ ¾ ºº!

to represet the total eghts of a tree t.

ode to the root r ortzed cost (: 3 log sze sze or Cut the left lk or rght lk of ode W 3log t If ode ot

Splay - or + to the root ortzed cost (: ªW 3log t xyz{ Œ Š ƒ ˆ ƒ ~ y } Ž ñò ï ðæ îí êçëì çåé ã Þåæ áâ ç ó

* & - # +* %'&( "$# ortzed Cost ( of splt operato: If s the tree t the Cost for splayg ode to the root of

ode - or + to the root of t W t - 9 : Cost for cut the left or rght lk of ode Cost for cut the left or

10 Ÿ ª «ª } ˆ ö õ í éé ÝÞßà ÕÖ Wt Ú Ù > > ; Ø Ú Ù ÇÈ Æ ¾ ºº! ± ž œ š š š þ û ùÿ ýþÿ ÿ úûü øù Splt(t: splt the tree t at the ode Splay ode t I the ths secto e use W t to represet the total eghts of a tree t. ccess (t : access the ode the tree t If ode s the tree t Frst go do the tree ad locate the ode Splay the ode to the root r ortzed cost (: 3 log sze sze or Cut the left lk or rght lk of ode W 3log t If ode ot the tree t Frst go do the tree ad locate a ode - or + here - s the ode precedg ode + s the ode follog Splay - or + to the root ortzed cost (: ªW 3log t xyz{ Œ Š ƒ ˆ ƒ ~ y } Ž ñò ï ðæ îí êçëì çåé ã Þåæ áâ ç ó Ó Ô Ò Ï Ì Ñ ÊÐ ÎÏÐ Ð ËÌÍ ÉÊ /. * & - # +* %'&( "$# ortzed Cost ( of splt operato: If s the tree t the Cost for splayg ode to the root of t If s ot the tree t the (Recall that - s the ode precedg + s the ode follog 3log 3log Cost for splayg ode - or + to the root of t W t - 9 : Cost for cut the left or rght lk of ode Cost for cut the left or rght lk of ode - or + Jo(t t : o a tree t th aother tree t all the eleets t are greater tha t Splay the largest te to the root of t 3log ÙW t ortzed Cost of o operato: Cost for splayg the ode t ÚÜÛ Make t be the rght chld of Cost for akg t be the rght chld of ÂÃ ÄÅ À Á ¹ ¾» ¼½ ¹ º µ ²³

$_+`a a \']^ Z$[ ¼½» µ º ³¹ + ¹ ¹ 'µ ²$³ Delete(t: delete ode fro the tree t ccess$ the ode Isert(t: sert a ode to the tree t Splt the tree t at ode Jo tree t ad tree

be left subtree ad t to be the rght subtree of?

Ý'Þß Û$Ü ortzed cost ( of delete operato: ëw t 3log ìîí 3log ëw t - - ìîí ortzed

the to subree t ad t [SLETR ad TRJN 985] Cost for cut the lks the teredate tree The

-( The chage potetal for the tree before the sert of (but after the splt step ad

11 fg ¾ h À XY W KK ± š ç é ì ë ì ë ê ì ë ì ë ƒ y uu Ú Ú kl Ù Ñ ÍÍ ÁÂÃÄ de c ` ] b [a _+`a a \']^ Z$[ ¼½» µ º ³¹ + ¹ ¹ 'µ ²$³ Delete(t: delete ode fro the tree t ccess the ode Isert(t: sert a ode to the tree t Splt the tree t at ode Jo tree t ad tree t to for tree t Cut the left lk ad rght lk of ad delete Isert the ode by akg t to be left subtree ad t to be the rght subtree of?@ ST Q RH JP LIMN IJGK CD I UV «ª Ÿ ž Ÿ œ Ž Š Œ +Š ' ˆ $ åæ á Þ ã Üâ à+áâ â Ý'Þß Û$Ü ortzed cost ( of delete operato: ëw t 3log ìîí 3log ëw t - - ìîí ortzed Cost ( of sert operato: W T - 3log - W log T Cost for access ode tree t Cost for o the to subree t ad t [SLETR ad TRJN 985] Cost for cut the lks the teredate tree The cost for splttg the tree t because tree t does ot cota so the total eght of t (W T -( The chage potetal for the tree before the sert of (but after the splt step ad the after the sert of Soe costat cost for lkg the to trees the fal step s }~ { r tzy vsx stqu op qr Ë ÕÖ Ø Ó ÔÊ ÌÒÑ ÎËÏÐ ËÌÉÍ ÇÈ ÂÉÊ ÅÆ

12 `f ] \ de cb a ` ø ZF Y Q MM CD 5; / 9: ˆ - $ ƒ { kl gh ^4b ]^4_ [\ \ Exaple of spalyg a lexcographc splay tree Splay at a Itroducto Splayg Cost of splayg Splay tree operatos pplcato of splay tree Cocluso Self-dustg Search Tree y Jorge Stolf ÿ ù ûû ú ÿ üùýþ ùú û õö ð ø ó ïðñò K WX UV S TJ LRQ NKP KLIM GH IJ EF > > / / 4 ˆ 4Š Whe a strg s accessed e ust splay the strg character by character Most frequet accessed strg ll be ear the root The tree store the prefx for the strg so ths reduce the redudacy Lexcographc search tree Crcle ad square are odes square s teral ode of a strg The sold les for the bary trees ad they represet dfferet strgs The dashed les represet a cotuous strg The graph cossts the ords: at as bat bats bog boy day he ( *+ & ' % $!"# u } ~t v { xuyz uvs qr lst op

13 ¼ ³ ² º» ¹ À Á Â Â þÿ ý õ ññ åæç ã - / / É Ü Û Ó ÏÏ %& ÃÄÅÆ $ ½¾ 4 ³ 4µ ±² ² Splayg 3 dfferet rules Reduce the tree heght Move the frequetly accessed ode ear the root The aortzed ad actual cost of splayg Use of potetal fucto alace ad statc optalty theore ther applcato data copresso Dyac Huffa codg [D.W. JNES988] Dgtal search tree Good copresso schee for age data ª«Ÿ ž œ žÿ š ùú øî ðöõ òïó ïðíñ ëì æíî éê ï ûü àáâ ß ÝÞ. ( + * ' ( Splay tree operato ccess Jo Splt Isert Delete pplcato of splay tree Lexcographc splay tree Itroducto Splayg Cost of splayg Splay tree operatos pplcato of splay tree Cocluso - Ed - Self-dustg Search Tree y Jorge Stolf Í Ø ÙÚ Õ ÖÌ ÎÔÓ ÐÍÑÒ ÍÎËÏ ÉÊ ÄËÌ ÇÈ "#!

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