On Binary Search Tree Recursions with Monomials as Toll Functions

Transcription

1 O Biary Search Tree Recursios with Moomials as Toll Fuctios Ralph Neiiger Istitut für Mathematische Stochastik Albert-Ludwigs-Uiversität Freiburg Eckerstr Freiburg Germay April 4, 2001 Abstract We cosider distributioal recursios which appear i the study of radom biary search trees with moomials as toll fuctios. This exteds classical parameters as the iteral path legth i biary search trees. As our mai results we derive asymptotic expasios for the momets of the radom variables uder cosideratio as well as limit laws ad properties of the desities of the limit distributios. The aalysis is based o the cotractio method. AMS subject classificatios. Primary: 60F05; secodary: 60E05, 60E10. Key words. Radom biary search tree; Weak covergece; Cotractio method; Aalysis of algorithms; Fixed-poit equatio; Probability desity. 1 Itroductio We cosider a sequece X of radom variables with distributios give by X 0 recursio 0 ad the X d X U + X 1 U + t, 1, 1 with X, X, U beig idepedet, X beig distributed as X, ad U a uiform [0, 1] distributed radom variable. The symbol d deotes equality of distributios. Throughout this work we assume moomials t α as toll fuctios with α R ad α > 1. For the special choice t 1 the X are distributed as the iteral path legth i radom biary search trees. By a well-kow equivalece this is also the umber of key comparisos eeded by Hoare s sortig algorithm Quicksort to sort a list of radomly permuted items. I the cotext of radom search trees it is a commo pheomeo that differet parameters of the same tree satisfy distributioal recursios of type 1 which oly differ i the toll fuctio t. Typically, the brachig factor of the tree is reflected i the umber of idepedet copies of the parameter o the right side of the equatio here i 1 these are the two sequeces X ad X, the splittig procedure settles the radom idices of these sequeces, ad the special parameter 1

2 uder cosideratio determies the toll fuctio; see, e.g., Devroye [2] for a list of radom search trees fittig i this scheme. The aim of this ote is twofold. First we study the asymptotic behavior of the momets ad distributios of X for our toll fuctios α. The ivestigatio of 1 with o-stadard toll fuctios was recetly started by Paholzer ad Prodiger [6] who cosidered the harmoic toll fuctio t H : i1 1/i. Their study was motivated by the occurrece of a logarithmic toll fuctio i Graber ad Prodiger [4]. It is our secod itetio to add a further example to the list of applicatios due to the cotractio method which is applied i our aalysis. The cotractio method was itroduced by Rösler [8] for the distributioal aalysis of the Quicksort algorithm, i.e. our recursio 1 with t 1. This method was further developed idepedetly i Rösler [9] ad Rachev ad Rüschedorf [7], ad later o i Rösler [10]. A survey of the method icludig the major applicatios is give i Rösler ad Rüschedorf [11]. Characteristic for recursio 1 from the poit of view of the cotractio method is that mea ad stadard deviatio of X are of the same order of magitude. As log as we make use of the miimal L 2 -metric l 2 this implies that oly kowledge of the leadig term i the expasio of the mea is required i order to derive weak covergece for the scaled versios of X. This is i cotrast to the Quicksort case α 1 where mea ad stadard deviatio are of differet orders of magitude ad the kowledge of the secod term i the expasio of the mea is ecessary; see [5] for a discussio of this problem i the cotext of the iteral path legth i radom split trees. Note that the limit distributios for the problems cosidered i this work are determied by a type of fixed-poit equatio which has ot so far appeared i other applicatios. We proceed as follows: I the secod sectio we derive the domiat term i the expasio of the mea of X. The third sectio gives the limit law for X by the approach of the cotractio method. I the fourth sectio first order expasios for the variace ad higher momets of X ad iformatio o the Laplace trasform as well as tail estimates are derived. I the last sectio it is proved by argumets of Fill ad Jaso [3] that the limit distributio has a desity which belogs to the class of rapidly decreasig C fuctios. We deote by l 2 the miimal L 2 -metric actig o the space of probability distributios with fiite secod momet see [1]. Covergece i the l 2 -metric is equivalet to weak covergece plus covergece of the secod momets. We write also l 2 X, Y : l 2 LX, LY for radom variables X, Y with laws LX, LY. 2 Expectatios I our subsequet distributioal aalysis it turs out that the kowledge of the domiat term i the expasio of the mea is sufficiet i order to obtai a limit law for X. This leadig term ca be explored by well-kow elemetary methods. We deote a : E X. The radom idices i 1 are uiformly distributed o {0,..., 1}. Thus, 1 implies a α a i, 1, i0 with iitializig value a 0 0. This implies for a α a i ad 1a 1 1 α a i. i0 i0 2

3 Subtractig these two relatios ad usig the expasio 1 α+1 α+1 α + 1 α + O α 1 2 we deduce a + 1a 1 α + 1 α + O α 1. This implies a + 1 a 1 + α + 1 α 1 + O α α + 1 i α 1 + O i α i i α + 1 α 1 α 1 + o α 1 + O α 2 α + 1 α 1 α + o α. For resolvig the sum i 3 we used the estimate i1 i α 1 i i α 2 i + 1 i1 [ α 2 ] 1 i α 1 + O α 2 i1 1 α 1 + o1 α 1 + O α 2, where the Riema itegral 1 0 xα 2 dx is coverget due to our geeral assumptio α > 1. Usig more terms i the expasio 2 may give a refied asymptotic expasio for a. For example, for α 2, 3, 4 we get the exact expressios a 3 2 6H H for α 2, a H H, for α 3, a H H, for α 4. Usig a expasio of H leads to asymptotic expressios for the a. For our further probabilistic aalysis we will oly eed the first order growth of a : Lemma 2.1 The mea of the sequece X give i 1 with t α, α > 1, satisfies 3 Limit Laws E X a α + 1 α 1 α as. 4 We will show later i Theorem 4.2 that the variace Var X admits a expasio Var X v 2α, 3

4 with some costat v > 0 depedig o α. Therefore mea ad stadard deviatio are of the same order of magitude. Thus, i order to derive a limit law for X we could scale by Y : X α or Z : X E X α 5 ad expect that weak limits Y, Z of Y ad Z respectively satisfy E Y α + 1/α 1 ad E Z 0. For techical reasos we will use both sequeces Z, Y i our aalysis. Our origial recursio 1 modifies for the scaled quatities to d U Z α 1 U Z U + + a 1 U + a 1 U + α a α U + α + 1 α 1 α 1 U Z U + U α + 1 U α α Z 1 U α Z 1 U o1, 7 α 1 where the expasio 4 is used ad agai Z, Z, U are idepedet, Z is distributed as Z, ad U is uiform [0, 1] distributed. The o1 depeds o radomess but the covergece is uiform. From this modified recursio oe ca guess a limitig form by lookig for stabilizatio for. This suggests that a limit Z of Z should satisfy the fixed-poit equatio Z d U α Z + 1 U α Z + α + 1 U α + 1 U α 2 α 1 α 1, 8 with Z, Z, U beig idepedet, Z, Z idetically distributed ad U uiformly o [0, 1] distributed. The traslated versio Y Z + α + 1/α 1 the solves the simpler fixed-poit equatio Y d U α Y + 1 U α Y + 1, 9 with relatios aalogous to 8. Accordig to the idea of the cotractio method the limits Z of Z ad Y of Y should be characterized as the uique solutios of 8, 9 respectively subject to the costraits E Z 0 ad Var Z <, ad for the traslated case E Y α + 1/α 1 ad Var Y <. For the proof of the uiqueess of such solutios ad the weak covergece we ca appeal to geeral theorems [9, 10], due to the stadard form of our recursio. Theorem 3.1 Let X be give by 1 with t α, α > 1. The fixed-poit equatio 8 has a uique distributioal solutio Z subject to E Z 0 ad Var Z < ad it holds the limit law X E X l 2 α, Z 0 as. Proof: For the uiqueess of the fixed-poit we apply Theorem 3 i [9]. The T 1, T 2, C occurrig there are give here by T 1 : U α, T 2 : 1 U α, C : α + 1 U α + 1 U α 2 α 1 α 1. 4

5 It is E 2 i1 T 2 i E C α + 1 α 1 2 2α + 1 < 1, E C2 <, ad 1 α α α 1 0. Thus the coditios of Rösler s theorem are satisfied ad it follows that 8 has a uique distributioal fixed-poit i the space of cetered probability distributios with fiite secod momet. For the l 2 -covergece we apply Theorem 3 i [10]. The Z1, Z 2, T 1, T 2, C occurrig there are give here by α Z1 U, Z2 1 U, T1 U, ad T 2 α 1 U, C 1 α a U + a 1 U + α a. 10 We check the coditios of the theorem: That E C 0 holds follows by takig expectatios i 6 ad otig that the Z i, Zi there are cetered. For ay 1 N we have 2 E i1 [ 1 {Z i 1 }T i 2] 1 j P U j + P 1 U j j0 2P U < as, which is coditio 21 i the cited theorem. Furthermore, it holds 2α l 2 2LC, T, LC, T E C C 2 + E T1 T E T2 T 2 2 α 2 E [o1 2 ] as, where o1 is the uiformly covergig o1 i 7. Now, Rösler s theorem implies covergece i the l 2 -metric. 4 Higher momets ad Laplace trasforms Similarly to Theorem 3.1, l 2 -covergece of Y to Y holds, where Y is the uique distributioal fixed-poit i 9 subject to E Y α + 1/α 1 ad Var Y <. Covergece i the l 2 -metric iduces covergece of the secod momets. This implies Var Y Var Y ad Var X Var α Y VarY 2α. 5

6 The leadig costat Var Y ca be obtaied form the fixed-poit equatio 9. We ca also pump higher order momets of Y from the fixed-poit equatio. This implies asymptotic expasios for the momets of X as soo as we kow that covergece of the momets of higher order of Y holds. This ca be show by aalyzig the Laplace trasforms of Z ad Z. For this we apply the tools developed i Lemma 4.1 ad Theorem 4.2 i [8]. Theorem 4.1 The scaled sequece Z give i 5 ad the fixed-poit Z of Theorem 3.1 satisfy for all λ R E expλz E expλz < as. Proof: I place of the radom variable U i the proof of Lemma 4.1 i [8] we use V : 2α 2α U 1 U + 1. The with C give by 10 it holds N : 1 V < 0, 11 sup E V < 0, 12 N C <. 13 sup N The proof of 12 follows from E V < 0 for all N ad from the covergece of the meas, E V E [U 2α + 1 U 2α 1] 2/2α < 0. Relatio 13 follows from the represetatio of C give i 7. Now, usig we ca coclude as i Lemma 4.1 ad Theorem 4.2 i [8] which leads to our assertio. The covergece of the Laplace trasform implies covergece of momets of arbitrary order. We ca also deduce tail estimates from this covergece. Obviously, we do oly have a right tail. Usig Markov s iequality ad E X α + 1/α 1 α + d with d o α we derive PX b P exp λ X E X b α exp λ α α + 1 α 1 + d α b E expλz exp λ α α + 1 α 1 + d α c α,λ exp λ b α for all positive sequeces b with a costat c α,λ > 0. Now, we give the first order asymptotic expasio for the higher momets of X : Theorem 4.2 Let X be give by the recursio 1 with t α, α > 1. The for all k 0 it holds E X k µ k kα, 6

7 with µ 0 1, µ 1 α + 1/α 1, ad µ k kα + 1 kα 1 r+s+tk r,s<k k r, s, t Bαr + 1, αs + 1µ r µ s, k 2, where B, deotes the Euleria beta-itegral. I particular the variace of X satisfies Var X αα + 12 Bα, α + 2α 2 2α 1 2α 1α 1 2 2α. Proof: The covergece of arbitrary momets of Y implies [ E X k E α Y k] E Y k kα E Y k kα, thus our expasio holds for µ k E Y k. This yields the values µ 0 1, µ 1 α + 1/α 1. Higher momets of Y ca be derived straightforwardly from the fixed-poit equatio 9. By the biomial formula it is the summatio idices r, s, t beig oegative itegers µ k E Y k E k U rα 1 U sα Y r Y s r+s+tk r+s+tk k r, s, t 2 kα + 1 µ k + r, s, t Brα + 1, sα + 1µ r µ s r+s+tk r,s<k k Brα + 1, sα + 1µ r µ s. r, s, t Resolvig for µ k leads to the recursio give i the theorem. The formula for the variace follows from Var Y µ 2 α + 1/α Desities I this sectio we provide iformatio o the desities of the limit distributios followig a approach of Fill ad Jaso [3] for the aalysis of the Quicksort limit distributio. Fill ad Jaso aalyze decay properties of the Fourier trasform of a distributioal fixed-poit i order to prove the existece, differetiability properties, ad bouds of a desity ad its derivatives. This aalysis ca be carried over to the family of distributios Y give by the fixed-poit equatio 9. The pure existece of a desity could also be derived by the approach of Ta ad Hadjicostas [12]. Let φt : E expity be the characteristic fuctio of the fixed-poit Y of 9. It is φt expitα + 1/α 1 E expitz with Z the limit distributio of Theorem 3.1, thus φt E expitz. The fixed-poit equatio 9 traslates ito This implies i particular 1 φt e it φu α tφ1 u α t du. 0 φt 1 0 φu α t φ1 u α t du. 14 7

8 We defie h y,y u : u α y + 1 u α y + 1 for u [0, 1] ad y, y R. The fixed-poit equatio 9 takes the the form Y h Y,Y U i distributio. The approach of Fill ad Jaso cosists of derivig first a decay rate for the characteristic fuctio of h y,y U for all y, y R usig a method of va der Corput. This boud carries over to the characteristic fuctio of Y by mixig over the distributio of Y. The the boud ca be improved by successive substitutio ito 14. This leads to itegrability properties of the characteristic fuctio which imply the existece ad further properties of a desity of the fixed-poit. I cotrast to the Quicksort limit distributio the fixed-poit Y give by 9 does ot have the whole real lie as support. Sice Y is the limit of o-egative radom variables we obtai Y 0 almost surely. Pluggig this iformatio ito 9 we obtai Y 1 almost surely. By iductio ad U α + 1 U α 2 1 α we icrease this boud to Y j0 21 α j for all N, thus almost surely. Y L α : 2α 1 2 α 1 1 Lemma 5.1 It holds φt 32/B α 1/2 t 1/2 for all t R with { 2 B α : 3 α αα 1L α for 1 < α 2 or α 3, αα 1L α for 2 < α < 3. Proof: It is for u [0, 1] thus for all y, y L α we obtai h h y,y u αα 1 [ u α 2 y + 1 u α 2 y ], y,y u αα 1L α mi u [0,1] { u α u α 2} B α for all u [0, 1]. Now, the argumet of Lemma 2.3 i [3] implies for all y, y L α 32 1/2 E expith y,y U t 1/2, t R. B α Note that the optimal choice of γ i the cited proof is here 2/B α 1/2. Sice LY has o mass o, L α we obtai by coditioig φt L α for all t R, where σ deotes the distributio of Y. E expith y,y U dσydσy 32 1/2 t 1/2 L α B α This boud ca be improved to superpolyomial decay of φ by successive substitutio ito 14: Theorem 5.2 For every real p 0 there is a smallest costat 0 < c p < such that the characteristic fuctio φ of Y satisfies φt c p t p for all t R. 15 8

9 The costats c p satisfy c 1/2 32/B α 1/2, c 2p Γ2 1 αp Γ2 2αp c2 p for 0 < p < 1 α, 16 c p+1/α 2 αp+1 αp αp 1 c1+1/αp p for p > 1 α. 17 Proof: First we show that if 15 holds for a 0 < p < 1/α with c p < the 15 holds also with p replaced by 2p, where the estimate 16 is valid: By 14 we obtai φt 1 0 c 2 p u α t p 1 u α t p du c 2 p t 2p B1 αp, 1 αp Γ2 1 αp Γ2 2αp c2 p t 2p. Next, if 15 holds for a p > 1/α with c p < the 15 holds as well with p replaced by p + 1/α with 17 beig valid: It is 1 { } { } cp φt mi u α t p, 1 c p mi 1 u α t p, 1 du. 0 Adaptig the estimates of Fill ad Jaso we cosider first t 2 α c 1/p p itegratio ito the regio [c 1/αp p 2.6 i [3] t 1/α, 1 c 1/αp p φt 2 αp+1 αp αp 1 c1+1/αp p ad split the domai of t 1/α ] ad its complemet. This implies cf. Lemma t p+1/α for t 2 α c 1/p p. For 0 < t < 2 α c 1/p p the right had side is at least oe ad egative t are covered by φ t φt. Now, the proof is completed as follows: The assertio 15 trivially holds for p 0 with c 0 1 ad, by Lemma 5.1, for p 1/2 with c 1/2 estimated i the Theorem. If α > 2 the we iterate 17 startig with p 1/2 ad obtai 15 for all p 1/2 + j/α, j N. Sice c 1/q q c 1/p p for all 0 < q p this gives the assertio for all p 0. If 1 < α < 2 we apply 16 with p 1/2 ad obtai the assertio with p 1. The we iterate 17 as i the case α > 2. Fially, for α 2 the assertio is true for p 1/2 thus as well for p 1/3. We apply 16 with p 1/3 ad obtai the assertio for p 2/3. The we ca iterate 17 startig with p 2/3. As discussed i [3] our Theorems 4.1 ad 5.2 together imply that φ belogs to the class of rapidly decreasig C fuctios, which is preserved uder Fourier trasform. Therefore, we obtai aalogous decay properties for the desity of the fixed-poit Y ad its traslated versio Z: Theorem 5.3 The limit radom variable Z of Theorem 3.1 has a ifiitely differetiable desity fuctio f. For all p 0 ad iteger k 0 there is a costat C p,k such that its k-th derivative f k satisfies f k x C p,k x p for all x R. Explicit bouds o the supremum orm of f k ca as well be established usig Theorem 5.2 ad a Fourier iversio formula. 9

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