Extremal Weighted Path Lengths in Random Binary Search Trees

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1 Extremal Weighted Path Legths i Radom Biary Search Trees Rafik Aguech 1 Nabil Lasmar 2 Hosam Mahmoud 3 December 18, 2008 Abstract. We cosider weighted path legths to the extremal leaves i a radom biary search tree. Whe liearly scaled, the weighted path legth to the miimal label has Dickma s ifiitely divisible distributio as a limit. By cotrast, the weighted path legth to the maximal label eeds to be cetered ad scaled to coverge to a stadard ormal variate i distributio. The exercise shows that path legths associated with differet raks exhibit differet behaviors depedig o the rak. However, the majority of the raks have a weighted path legth with average behavior similar to that of the weighted path to the maximal ode. AMS subject classificatios. Primary: 05C05, 60C05; secodary: 60F05, 68P05, 68P10, 68P20. Key words. Radom trees, path legth, recurrece, ifiitely divisible distributio, reflectio priciple. 1 Itroductio Various sorts of extremal path legths i biary search trees have bee studied, owig to their importace as iterpretatios of some aalyses of algorithm (mostly i the areas of searchig ad sortig. For example, the height 1 Faculté des Scieces de Moastir, Départemet de mathématiques, 5019 Moastir, Tuisia. rafikaguech@ipeit.ru.t 2 Istitut préparatoire aux études d igéieurs de Tuis, Départemet de mathématiques, IPEIT, Rue lelahrou-motfleury, Tuis, Tuisia. abillasmar@yahoo.fr 3 Departmet of Statistics, The George Washigto Uiversity, Washigto, D.C , U.S.A. hosam@gwu.edu 1

2 of the biary search tree is cosidered i various sources for its role as a global measure of worst case search i a radom tree (see Robso, 1979, Mahmoud ad Pittel, 1984, Pittel 1984, Devroye, , Drmota , ad Reed, At the other ed of the spectrum, the legth of the shortest root-toleaf path is cosidered as a measure of optimism for the best search time (see Pittel (1984. May of these results are surveyed i sortig textbooks such as Kuth (1998 ad Mahmoud (2000. The above-metioed extremal path legths have a commo thread: They all are the raw depth of some extremal leaf i the tree. We are cocered i this ivestigatio with weighted extremal path legths, where odes o the path have types of cotributio to the path legth other tha a mere cout of their icomig edge, such as, for example, cotributig their ow value. The path legths ivolved have other iterpretatios as quatities uderlyig certai algorithms. Some algorithm may go dow a path from the root of a biary search tree of size to the ode raked j, collectig the sum of the values ecoutered. We ivestigate i this paper the distributio of such paths i some extreme cases. Let W j ( be the value of the path legth associated with traversig the tree from its root to the ode labeled j, while aggregatig the values o the path. We shall see that W 1 (, whe appropriately scaled, has Dickma s ifiitely divisible distributio (a result that parallels i some way the Dickma distributio associated with fidig the smallest item via th oe-sided Quicksort (the so-called Quickselect algorithm; see Mahmoud, Modarres, ad Smythe (1995, ad Hwag ad Tsai (2002. By cotrast, W (, whe appropriately cetered ad scaled, coverges i distributio to a ormal variate. The exercise demostrates that there is a variety of differet distributios associated with W j ( for differet values of j. 2 Scope A biary tree is a hierarchical structure of odes each havig o childre, oe left child, oe right child, or two childre (oe left ad oe right. The odes of such a tree ca be labeled from some ordered set, say the atural umbers. The tree ca further be edowed with a search property (to support fast searchig of the items (also called keys stored i it, which imposes the restrictio o the labelig scheme that the label of ay ode is larger tha the labels i its left subtree ad o greater tha ay label i its right subtree. 2

3 For defiitios ad combiatorial properties see Mahmoud (1992, ad for applicatios i sortig see Kuth (1998 or Mahmoud (2000. A biary search tree is costructed from the permutatio (π 1,..., π of {1, 2,..., } by the followig algorithm. The first elemet of the permutatio is iserted i a empty tree, a root ode is allocated for it. A subsequet elemet π j (with j 2 is directed to the left subtree if π j < π 1, otherwise it is directed to the right subtree. I whichever subtree π j goes, it is subjected to the same isertio algorithm recursively, util it is iserted i a empty subtree, i which case a ode is allocated for it ad liked appropriately as a left (right child if its rak is less tha (at least as much as the value of the last ode o the path. Figure 1 illustrates the tree costructed from the radom permutatio (5, 8, 7, 3, 9, 1, 6, 2, 4. Several models of radomess are i commo use o biary trees. The uiform model i which all trees are equally likely has bee proposed for applicatios i formal laguages, compilers, computer algebra, etc. (see Kemp (1984. However, for the searchig ad sortig algorithms alluded to the radom permutatio model is cosidered to be more appropriate. I this model of radomess we assume that the tree is built from permutatios of {1,..., }, where a uiform probability model is imposed o the permutatios istead of the trees. Whe all! permutatios are equally likely or radom, biary search trees are ot equally likely. Several permutatios give rise to the same tree, favorig shorter ad well balaced trees rather tha scrawy ad tall shapes, which is a desirable property i searchig ad sortig algorithms (see Mahmoud (1992. The term radom tree (ad occasioally just the tree will refer to a biary search tree built from a radom permutatio. The radom permutatio model is ot really restrictive, as it covers a rather wide variety of istaces, such as whe the iput is a sample draw from ay cotiuous probability distributio, ad the costructio algorithm is cocered oly with the raks of the keys, ot their actual values. We study the weighted path legth leadig to the rightmost ad leftmost odes. For istace, i the tree of Figure 1, W 1 (9 = = 9, W 2 (9 = = 11,..., W 9 (9 = = 22. The paper is orgaized as follows. I Sectio 3 we study the weighted path legth leadig from the root to the miimal label i the tree ad show that it has Dickma s distributio (after suitable scalig. The weighted path legth leadig to the maximal label is ivestigated separately i Sectio 4 where we explore a useful reflectio priciple. It is show i Sectio 4 that the weighted path legth to the maximal label coverges i distributio to the 3

4 Figure 1: A biary search tree. ormal radom variate (after appropriate ceterig ad scalig. Sectio 5 gives some cocludig remarks where a brief derivatio of the average is give for the rest of the cases 1 < i <. 3 Weighted path to the miimal label Let L be the umber of items that appear i the left subtree, ad thus L + 1 is the value of the root. For 1, the stochastic recurrece W 1 ( = L W 1 (L represets the weighted path legth from the root to the ode labeled 1 (that is, the sum of the collectio of values o the leftmost path i the tree. Let φ (t be the characteristic fuctio of W 1 (. By coditioig the stochastic recurrece, we obtai φ (t = E[e (L+1+W 1(L it ] 1 = E[e (l+1+w1(lit ] P(L = l l=0 4

5 = 1 e kt φ k 1 (t, k=1 valid for all 1. This telescopig sum is ameable to the differecig scheme subtract a versio of the last recurrece with 1 from oe with to obtai φ (t = 1 + eit φ 1 (t, which ca be uwoud by direct iteratio to give φ (t = k 1 + e ikt. k k=1 By differetiatig this latter form oce ad twice (the evaluatig at t = 0 we obtai the mea ad the secod momet. Propositio 1 Let W 1 ( be the weighted path legth from the root to the least raked label i a biary search tree built from a radom permutatio. We the have E[W 1 (] = ; Var[W 1 (] = ( Guided by the rate of growth of the variace, we ext proceed to argue the ifiite divisibility of 1 W 1 (. We take the atural logarithm of the characteristic fuctio, ad write it i asymptotic form (as t 0 ( l(φ (t = l 1 + eikt 1 k k=1 [ e ikt 1 (( e ikt 1 2 ] = + O k k k=1 = O(t 2 e ikt 1 +. k Sice the rate of growth of the stadard deviatio is, oe expects that W 1 (/ coverges to a limit i distributio. At the level of characteristic 5 k=1

6 fuctio, this meas chagig the scale from t to t/. Let v = t/. This etails (for fixed v ( ( v l φ ( 1 = O + k=1 e ikv/ 1. k The O term coverges to 0, ad the remaiig sum approaches We thus have the covergece φ ( v The characteristic fuctio 1 0 e iuv 1 u du. ( 1 e iuv 1 exp du, as. 0 u ( 1 e iuv 1 iuv ψ X (v = exp dv 0 u is that of Dickma s ifiitely divisible radom variable X i Kolmogorov s caoical form (see Billigsley (1995; P That is, φ ( v e R ( (iv iv du exp 0 e iuv 1 u ( 1 = e iv e iuv 1 iuv exp 0 u = E[e i(1+xv ]. We have arrived at the mai result of this sectio. du du Theorem 1 Let W 1 ( be the weighted path legth from the root to the least raked label i a biary search tree built from a radom permutatio. The, W 1 ( where X is Dickma s radom variable. D 1 + X, 6

7 Remark: The limitig radom variable for 1 W 1 ( bears some similarity to the limitig radom variable for 1 C [1], the ormalized umber of comparisos required by Quickselect to fid the least item i a radom iput (of size with raks followig the radom permutatio model. It is show i Mahmoud, Modarres ad Smythe (1995 that 1 C [1] coverges i distributio to 2 + X. Thus, asymptotically, the distributio of 1 C [1] behaves like that of W 1 (. 4 Weighted path to the maximal label Let us itroduce a reflectio operatio, which may geerally be useful i this type of problems. I a biary search tree T of size, exchage the right ad left childre of every ode, startig at the root ad progressig recursively toward the leaves to obtai the reflected tree T. This reflectio cocers oly the shape of the tree, ad ot the labels. Oe ca thik of this operatio as if a two-sided mirror has bee placed o a vertical axis passig through the root, the oe sees the right subtree of T as the reflectio i the left side of the mirror of the left subtree of T, ad the left subtree of T as the reflectio i the right side of the mirror of the right subtree of T. To maitai the biary search property i T, we reisert the umbers 1,..., i a maer cosistet with the search property. For example, the reflected tree of that i Figure 1, is show i Figure 2. Note that, by the symmetry of biary search trees, T has the same probability as T. That is, there are as may permutatios of {1, 2,..., } producig T as those producig T. Observe that a key K i T correspods to the value + 1 K i the reflectio. Let the legth of the rightmost path i T be Q, ad suppose the chai of values appearig o it from the root to the rightmost ode (cotaiig is Y 1, Y 2,..., Y Q+1. Observe that the rightmost path i T becomes a leftmost oe (of the same legth i T, ad suppose that the correspodig labels i the reflectio are Y 1, Y 2,..., Y Q. +1 This coectio suggests that we ca use the distributio of the path to the miimal value, which was established i Sectio 3, for the rightmost path as follows. We have W ( = Q +1 j=1 Y j L = Q +1 j=1 ( + 1 Y j = (Q + 1( + 1 W 1 (. (1 7

8 Figure 2: The reflectio of the tree of Figure 1 We ca ow quickly develop a Gaussia law for W (, from kow results about Q. The latter variable is kow to be asymptotically ormal, satisfyig Q l l D N (0, 1; see Devroye (1988. This idicates that ceterig ad scalig the relatio (1 with asymptotic mea ad stadard deviatio of Q will yield a limit distributio. Let W := W ( l l Accordig to Theorem 1, we have L = Q l + Q l l l W 1( l. W 1 ( l a.s. 0, idicatig that the mai cotributio i W ( comes from the legth of the rightmost path. Also, accordig to the limit law of Q, Q /( l a.s. 0, ad of course ( + 1/( l a.s. 0. Hece, W D N (0, 1. 8

9 5 Coclusio The useful otatio j k will stad for the evet that the label j is ecoutered o the path to k, thus cotributig its value to the weighted path legth W k (. I view of this otatio, the weighted path legth of the ode k is W k ( = j1 {j k}. (2 j=1 The idicators ivolved are depedet, ad it is ot easy to determie limit distributios from this represetatio. Eve computatios such as the variace are dautig. But of course, the represetatio beig a sum, the average is o major obstacle. We develop this average, to use as a bechmark for what falls betwee the two extremes. Lemma 1 P (j k = 1 k j + 1. Proof. Suppose j < k. A property of biary search trees is that whe j k, all the umbers i A kj = {j, j + 1,..., k} appear after j (See Devroye ad Neiiger (2004 for a discussio via the theory of records. It suffices to cout B, the umber of permutatios favorable to the evet j k. If j appears at positio p i a favorable permutatio, there must be at least k j positios past p to receive the umbers i A kj {j}. Thus, p k j. To complete the costructio of a favorable permutatio, choose ay of these j k positios for the elemets of A kj {j}, ad permute its j k umbers over these positios i (k j! ways. Now permute the remaiig (k j+1 elemets i a urestricted way over the remaiig (k j + 1 positios. Therfore, ( p B = (k j! ( k + j 1! k j p=1 ( = (k j! ( k + j 1! k j + 1! = k j

10 The argumet for j > k is symmetrical, with j ad k exchagig roles. It follows from Lemma 1 ad the represetatio (2 that E[W k (] = k 1 j=1 j k j k + j=k+1 j j k + 1. The substitutio m = k j + 1 i the first sum together with a symmetrical oe i the secod sum gives the result i a simple form: E[W k (] = (k 1H k+1 + 3k (k + 1H k, For low- where H r is the rth harmoic umber r s=1 1/s. The form for E[W k (] is for the etire spectrum of odes. idexed odes k = o(, we have E[W k (], whereas for odes with high idex k = o(, E[W k (] l, but if k α, for 0 < α < 1, the asymptotic approximatio is E[W k (] 2α l, which idicates that the majority of the medium rage idexes lea toward the behavior of the weight of the path legth to the maximal ode, rather tha the liear order of magitude associated with the miimal ode. This may ultimately be reflected i the average distributio across all the odes. For istace, if we select a ode radomly i the tree, its idex K will be uiformly distributed o the set {1,..., }, ad cosequetly its weighted path legth has the average E[W K (] = j=1 E[W j (] l 1 0 2α dα = l. with a order of magitude just like that of the weight of the path legth to the maximal ode 10

11 Refereces [1] Billigsley, P. (1995. Probability ad Measure. Wiley, New York. [2] Devroye, L. (1986. A ote o the height of biary search trees. Joural of the ACM, 33, [3] Devroye, L. (1987. Brachig processes i the aalysis of the height of trees. Acta Iformatica, 24, [4] Devroye, L. (1988. Applicatios of the theory of records i the study of radom trees. Acta Iformatica, 26, [5] Devroye, L. ad Neiiger, R. (2004. Distaces ad figer search i radom biary search trees. SIAM Joural o Computig, 33, [6] Drmota. M. (2001. A aalytic approach to the height of biary search trees. Algorithmica 29, [7] Drmota. M. (2002. The variace of the height of biary search trees. Theoretical Computer Sciece, 270, [8] Hwag, H. ad Tsai, T. (2002. Quickselect ad Dickma fuctio. Combiatorics, Probability ad Computig, 11, [9] Kemp, R. (1984. Fudametals of the Average Case Aalysis of Particular Algorithms. Wiley-Teuber Series i Computer Sciece, Joh Wiley & Sos, New York. [10] Kuth, D. (1998. The Art of Computer Programmig, Vol. 3: Sortig ad Searchig, 2d ed. Addiso-Wesley, Readig, Massachusetts. [11] Mahmoud, H. (1992. Evolutio of Radom Search Trees. Wiley, New York. [12] Mahmoud, H. (2000. Sortig: A Distributio Theory. Wiley, New York. [13] Mahmoud, H., Modarres, R. ad Smythe, R. (1995. Aalysis of quickselect: A algorithm for order statistics. RAIRO: Theoretical Iformatics ad Its Applicatios, 29,

12 [14] Mahmoud, H. ad Pittel, B. (1984. O the most probable shape of a search tree grow from a radom permutatio. SIAM Joural o Algebraic ad Discrete Methods, 5, [15] Pittel, B. (1984. O growig radom biary trees. Joural of Mathematical Aalysis ad its Applicatios, 103, [16] Reed, B. (2003. The height of a radom biary search tree. Joural of the Associatio for Computig Machiery, 50, [17] Robso, J. (1979. The height of biary search trees. The Australia Computer Joural, 11,

Analysis of the Expected Number of Bit Comparisons Required by Quickselect

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