THE HEIGHT OF q-binary SEARCH TREES

Transcription

1 THE HEIGHT OF q-binary SEARCH TREES MICHAEL DRMOTA AND HELMUT PRODINGER Abstract. q biary search trees are obtaied from words, equipped with the geometric distributio istead of permutatios. The average ad variace of the height are computed, based o radom words of legth, as well as a Gaussia limit law.. Itroductio The paper [8] itroduces for the first time a meaigful q model of biary search trees: istead of biary search trees, oe cosiders touramet trees, which differ oly margially from biary search trees; if oe starts from a permutatio ( π 2 π 2 π ), the oe iserts the umber i istead of the umber π i. Thus, traversig the tree i iorder, we might thi of the associated permutatio as ρσ, where goes ito the root, ad ρ resp. σ form (recursively) the left resp. right subtree. We could have called this paper The Height of q-touramet Trees; however we decided ot to do so sice biary search trees are by far better ow, both, i the commuity of theoretical computer scietists, ad combiatorialists. A ice referece for touramet trees ad icreasig trees i geeral is [2]. Now istead of cosiderig permutatios π π 2 π, we cosider words over the alphabet {, 2, 3,... }, ad (geometric) probabilities attached to the letters, i. e., the probability of letter i is pq i, with p+q =. The biary search tree is the costructed by writig a oempty word w as w = xay, where a is the smallest letter occurrig, ad x {a +, a + 2,... } ad y {a, a +,... }. The letter a goes ito the root ad x resp. y form the left resp. right subtree. The paper [8] dealt with the path legth; here we cosider the height. The height of a biary search tree (ad thus of the trees i our model) is defied to be the largest umber of odes i a path from the root to a leaf; the empty tree (related to the empty word) has height 0. We will prove that the expected height, whe cosiderig radom words of legth, is asymptotic to p; the variace will also be computed as well as a Gaussia limit law. (The letter p will always deote q i this paper.) Recall that the result for traditioal biary search trees is c log with c = 4.307, see [3]; ufortuately we do ot get that result as the limit q as it so happeed for the path legth. However, othig is wrog here, sice the limit q i p simply tells us that the average height should be less tha liear. Although this origial hope of gettig the Key words ad phrases. Biary search tree, q aalog, height.

2 2 MICHAEL DRMOTA AND HELMUT PRODINGER classical result as a corollary did ot wor out, we evertheless thi that the results preseted here are of idepedet iterest. Note also that p is the expected umber of letters i a (radom) strig of legth. I our asymmetric model, they must all lie o oe path. I more detail, we will obtai the followig results: Let H deote the height of q biary search trees with (iteral) odes (this is a radom variable, defied o words of legth ). Theorem. For every positive q < the height H of q biary search trees with (iteral) odes satifies a cetral limit theorem of the form x p Pr{H x} Φ = O( /2 ), () pq sup x R where Φ(x) deotes the ormal distributio fuctio The expected value is give by Φ(x) = 2π x E H = p + O e t2 /2 dt., (2) p i which the O-costat is uiform for 0 < q <. The variace ca be estimated by V H = pq + O q ( /2 log 2 ). (3) The O-costat i the error term of the variace is ot uiform for 0 < q <. The depedecy o q has ot bee wored out sice the order of magitude of the error term is surely ot optimal. However, although the error term for the expected value is uiform, this theorem caot be used to cover the case q where it is ow that the expected value is give by E H = c log + O (log log ) (with c = 4.307, see [3]) ad the variace is bouded: V H = O() (see [5, 6]). Ituitively, this result (ad its proof) says that the height is domiated by the umber of s i the sequece. This seems to be due to the fact that most s appear cosecutively i the sequece, producig rather sew subtrees. The cotributio from 2 s is asymptotically egligible sice there are much fewer of them, etc. Studyig the combiatorics (of words) of geometrically distributed radom variables has bee a log term project of oe of us (H. P.), ad further papers ca be foud o this author s webpage. We do ot wat to give more details here, but i all previous

3 THE HEIGHT OF q-binary SEARCH TREES 3 cases the correspodece betwee the model of words ad its limit (permutatios) led to very satisfactory results. We would lie to cite the paper [] which is of geeral iterest i this cotext. 2. Wea Covergece Lemma. Let y (x) deote the geeratig fuctio The we have y 0 (x) 0, y (0) = ad for 0. y (x) = 0 Pr{H }x. y + (x) = y (x)y (qx) + y + (qx) (4) Proof. The proof is a straightforward traslatio of the basic decompositio w = xay: If a = (described by ), the the left subtree cosists oly of letters i {2, 3,... }, described by y (qx), but the right subtree ca have ay letters, which is described by y (x). However, if a >, we get the term y + (qx) (we might the thi of all letters beig reduced by ). If we write this i the form the the limit q gives y + (x) y + (qx) ( q)x y +(x) = y 2 (x), = y (qx)y (x), the usual recursio i the istace of biary search trees. (Recall that f(x) f(qx) =: ( q)x (D q f)(x), ad D q is called the q differece operator.) Also, if we write y (x) = a,j x j, 0 j<2 the we get, by comparig coefficiets, a +,j = p q j a,i a,j i q i for j, 0 i<j ad a,0 = for all 0. Thus, Theorem may be reformulated i terms of a,j. However, i this paper we will ot mae use of this otio. Lemma 2. The geeratig fuctios y (x) ( 0, 0 x < ) are bouded above by y (x) x x (. (5) x

4 4 MICHAEL DRMOTA AND HELMUT PRODINGER Furthermore, this iequality is also true o the level of coefficiets, i. e. for ad Pr{H } = [x ]y (x) ( [x x ] x = p l q l. l l=0 (6) Note that this upper boud is a exact biomial distributio fuctio which is asymptotically ormal with mea p( ) + = p + q ad variace pq( ). Proof. Sice y (x) /( x) we get from (4) that y + (x) y (x) +. Thus, (5) follows by iductio. Note that ay step i these calculatios is also true o the level of coefficiets. Thus, Set ad Pr{H } [x x ] x (. Y (x) = x x ( x (x) = Y (x) Y (x) = p (. Sice [x ] (x) = p q, (6) follows immediately.

5 THE HEIGHT OF q-binary SEARCH TREES 5 Lemma 3. The geeratig fuctios y (x) ( 0, 0 x < ) are bouded below by y (x) x x ( x (qx)2 ( (qx) qx qx (qx) x qx ( + qx2 (qx x. Furthermore, this iequality is also true o the level of coefficiets. Proof. First we use the trivial lower boud (7) y (x) x+ x ad the upper boud (5) to obtai the iequality y + (x) = y (x)y (qx) + y + (qx) x( q)y (x) (qx)+ y (x) + (qx)+ x x x + (qx)+2 (qx)+2. Now (7) follows by iductio. As i the proof of Lemma 2 it ca be observed that (7) is also true o the level of coefficiets. I order to obtai proper error terms for the lower boud for Pr{H } which may derived from Lemma 3 we mae use of the followig lemma. Lemma 4. Let 0 < q < be give ad let F (x) be a fuctio which is aalytic for x < + ε for some ε > 0. The there exist c > 0 ad η > 0 (depedig o q) such that ( ( ( [x ]F (x) = O /2 exp c )) 2 p (8) ad [x ]F (x) ( (qx) qx ( ( = O /2 exp c )) 2 p + O(q ) (9)

6 6 MICHAEL DRMOTA AND HELMUT PRODINGER uiformly for p η as. Proof. By usig stadard saddle poit asymptotics (compare with [4]) it follows that [x ]F (x) ( = 2πi = O z =x 0 F (z) /2 x 0 ( qz z + ( 0 0, where the saddle poit x = x 0 = ( ) /q is determied by the equatio x d dx =. I particular we have x 0 = if / = p. Fially a local expasio of x 0 ( 0 0 = q p ( ( ) completes the proof of (8). The proof of (9) rus alog similar lies. However, we have to be a little bit more careful. The reaso is that the deomiator qz qz is sigular for z = 0 ad for z 2 = ( + 2q)/q 2. Nevertheless the whole part ( (qz) qz

7 THE HEIGHT OF q-binary SEARCH TREES 7 is regular for z < /q. Note that z 2 > for 0 < q < 2 ad that z 2 < for 2 < q <. Thus, if q 2 we get similarly to the above [x ]F (x) = 2πi = 2πi = O ( (qx) qx = 2πi z =x 0 z =x 0 /2 x 0 F (z) qz F (z) qz ( 0 0 z =x 0 F (z) z + 2πi z + 2πi + O(q ). ( z =x 0 z = (qz) qz z + (qz F (z) qz (qz F (z) qz z + z + Note that we just shifted the paths of itegratio i regios of aalyticity if η is chose sufficietly small such that x 0 < z 2. If q = 2 the there are polar sigularities at z 2 = after splittig the itegral. However, the residues of the fuctios ivolved (at z = ) are both of order O ( ( 2 ) = O(q ), which implies that we get the same error term as i the case q 2. Note further that (with a little bit more effort) we could have bee much more precise. However, the boud give by Lemma 4 is sufficiet for our purposes. Lemma 5. For every 0 < q < there exist c > 0 ad η > 0 (depedig o q) such that ( ( Pr{H } p l q l = O /2 exp c )) 2 p l=0 l (0) uiformly for p η ad Pr{ H p m} = O uiformly for m η as. ( ) exp c m2 () Proof. First we ote that (9) of Lemma 4 applies to the fuctio (qx) 2 (qx) qx.

8 8 MICHAEL DRMOTA AND HELMUT PRODINGER Thus we obtai [x ] (qx)2 ( (qx) qx ( ( = O /2 exp c )) 2 p + O(q ) uiformly for p η as. The remaiig terms will ow be treated i the same way as i the proof of Lemma 4: [x ] qx (qx) x qx + qx2 (qx x = qz (qz) 2πi z qz qz qz2 (qz z qz qz z z = ε + = qz ( qz2 2πi z qz qz z qz qz z + 2πi + 2πi = O z =x 0 z = z = /2 x 0 qz (qz) z qz qz qz 2 ( (qz z qz qz ( O(q ). z + z + For the first itegral we use the fact that the fuctio F (z) = qz qz2 z qz qz z qz is regular at z = ud thus bouded i a viciity of z = (compare with the proof of Lemma 4). The secod ad third itegrals are easy to estimate. We oly have to apply the trivial bouds max z = z =, ad max z = z max = O() z = = ( + O( ) qz = O().

9 THE HEIGHT OF q-binary SEARCH TREES 9 Fially observe that i the rage p η we surely have ( ( q = O /2 exp c )) 2 p which completes the proof of (0). The proof of () is ow easy. We just have to combie a proper tail estimate for the biomial distributio, i. e. ( ) p l q l = O exp c m2, l with (0). l p m We are ow able to complete the first part of Theorem. It is well ow that the distributio fuctio of the biomial distributio ca be estimated by the distributio fuctio of the ormal distributio up to a uiform error of order O( /2 ), i. e., as, sup x R p l q l Φ l x l ( x p pq ) = O( /2 ), (2) compare with [7, p. 542]. Furthermore, by (0) we get a similar result for the distributio of H : sup Pr{H x} p l q l x R l x l = O( /2 ). (3) Note that (0) provides (3) just for x with x p η. However, by () we ow that Pr{ H p η} ( ( = O exp cη 2 )). By the mootoicity of the distributio fuctio this implies that for x with x p η we also have Pr{ H p x} = O ( exp ( cη 2 )). Thus (3) follows. The first part of Theorem, i. e. (), is ow a trivial cosequece of (2) ad (3). 3. Covergece of Momets Lemmata 2 ad 3 ca be easily used to get quite tight bouds for the expected value. Note that E H x = 0 0 x y (x). (4)

10 0 MICHAEL DRMOTA AND HELMUT PRODINGER Lemma 6. The expected value of H ca be bouded by p + q E H p + O. (5) p Proof. Set The by (5) we obtai E(x) E(x) = 0 x y (x). x = x 0 x() ( x) 2, which is also true o the level of coefficiets. Thus, we have Similarly we get the upper boud. From E(x) we directly obtai x() ( x) + (qx)2 2 + qx x qx2 x E H p + q. ( ( ) qx ) qx 2 ( q) x() = ( x) + (qx) 2 2 ( x)() pqx 2 + ( x) 2 () qx2 ( x) + qx 2 2 ( x) pqx2 x() = ( x) + qx 2 () 2 ( x) ( pqx 2 ) This completes the proof of Lemma 6. E H p + O. p Ufortuately it seems that we caot prove similarly tight estimates for the variace. However, we ca obtai a o trivial result: Lemma 7. The variace of H ca be bouded by V H = pq + O q ( /2 log 2 ). (6)

11 THE HEIGHT OF q-binary SEARCH TREES Proof. If F (x) = Pr{X x} deotes the distributio fuctio of a radom variable X (of compact support) the the variace of X ca be represeted by V X = 2 EX (EX y)f (y) dy + 2 EX (y EX)( F (y)) dy. We apply this formula for the height H where we have very precise estimates for EH ad its distributio fuctio F (x) = Pr{H x}, compare with Lemma 5 ad Lemma 6. Let u η be a parameter to be defied later. By applyig Lemma 5 ad Lemma 6 we get EH u 2 (EH y)f (y) dy = O ( e cu2), ad EH EH u EH u EH (EH y)f (y) dy = pq 2 + O ( u 2 ), (y EH )( F (y)) dy = pq 2 + O ( u 2 ), 2 EH (y EH )( F (y)) dy = O ( e cu2). u Choosig u = log 2 gives the result. It should be further metioed that it is quite easy to get asymptotic relatios for all cetral momets of the form E(H E H c (pq/2 ( ) by the method as described i Lemma 7, where c 2 = (2)!/(2!) ad c 2+ = 0. Thus we have ot oly a wea covergece result for H (properly ormalized) but covergece of momets, too. Refereces [] M. Barlow, R. Pematle ad E. Peris, Diffusio limited aggregatio o a homogeeous tree, Prob. Theory ad Related Fields 07 (997), 60. [2] F. Bergero, P. Flajolet ad B. Salvy, Varieties of icreasig trees, Lecture Notes i Computer Sciece 58 (992), [3] L. Devroye ad B. Reed, O the variace of the height of radom biary search trees, SIAM J. Comput. 24 (995), [4] M. Drmota, A bivariate asymptotic expasio of coefficiets of powers of geeratig fuctios, Europ. J. Combiatorics 5 (994), [5] M. Drmota, The variace of the height of biary search trees, Theoret. Comput. Sci. 270 (2002), [6] B. Reed, The height of a radom biary search tree, J. Assoc. Comput. Mach., to appear. [7] W. Feller, A Itroductio to Probability Theory ad Its Applicatios, Vol. II, 2 d ed., J. Wiley, New Yor, 97.

12 2 MICHAEL DRMOTA AND HELMUT PRODINGER [8] H. Prodiger, A q aalogue of the path legth of biary search trees, Algorithmica 3 (200), Departmet of Geometry, Techical Uiversity of Viea, Wieder Hauptstrasse 8 0, A-040 Viea, Austria address: michael.drmota@tuwie.ac.at The Joh Kopfmacher Cetre for Applicable Aalysis ad Number Theory, School of Mathematics, Uiversity of the Witwatersrad, P. O. Wits, 2050 Johaesburg, South Africa address: helmut@staff.ms.wits.ac.za WWW-address: