Applied Mathematics Letters. Asymptotic distribution of two-protected nodes in random binary search trees

Size: px

Start display at page:

Download "Applied Mathematics Letters. Asymptotic distribution of two-protected nodes in random binary search trees"

Alfred Goodwin
5 years ago
Views:

Applied Mathematics Letters 25 (2012) 2218 2222 Cotets lists available at SciVerse ScieceDirect Applied Mathematics Letters joral homepage: wwwelseviercom/locate/aml Asymptotic distribtio of

Uiversity, West Lafayette, IN 47907, USA a r t i c l e i f o a b s t r a c t Article history: Received 27 Jaary 2012 Accepted 6 Je 2012 Dedicated to the memory of Philippe Flajolet We derive exact

1 Applied Mathematics Letters 25 (2012) Cotets lists available at SciVerse ScieceDirect Applied Mathematics Letters joral homepage: wwwelseviercom/locate/aml Asymptotic distribtio of two-protected odes i radom biary search trees Hosam M Mahmod a, Mar Daiel Ward b, a Departmet of Statistics, The George Washigto Uiversity, Washigto, DC 20052, USA b Departmet of Statistics, Prde Uiversity, West Lafayette, IN 47907, USA a r t i c l e i f o a b s t r a c t Article history: Received 27 Jaary 2012 Accepted 6 Je 2012 Dedicated to the memory of Philippe Flajolet We derive exact momets of the mber of 2-protected odes i biary search trees grow from radom permtatios Frthermore, we show that a properly ormalized versio of this tree parameter coverges to a Gassia limit 2012 Elsevier Ltd All rights reserved Keywords: Biary search trees Radom strctre Combiatorial probability Asymptotic aalysis 1 Itrodctio The stdy of 2-protected odes i classes of radom trees is i the voge Cheo ad Shapiro [1] ivestigate the average mber of 2-protected odes i labeled, ordered trees ad i ary biary trees (those with 0, 1, or 2 childre per ode) Masor [2] cosiders the average mber of 2-protected odes i -ary trees Recetly, D ad Prodiger [3] have aalyzed the average of this parameter i radom digital trees, with a iform probability model I this article, we cosider the mber of 2-protected odes i a radom biary search tree (BST) These are biary trees, lie those i the 2-ary case of Masor [2], bt differ i their derlyig probability distribtio Those i [2] are iformly distribted, ie, all trees of the same size (mber of odes) are eqally liely I cotrast, the BST grows from a radom permtatio that idces a BST probability model, which is oiform The BST model is of prime importace i compter sciece as it represets the bacboe of some fdametal algorithms, sch as Qicsort (see Kth [4] or Mahmod [5]), ad are basic efficiet data strctres i their ow right (see Mahmod [6]) The BST grows from a iformly radom permtatio (π 1, π 2,, π ), of {1, 2,, }, as follows I the compter sciece jargo, elemets of the permtatio are ofte called eys The first ey π 1 goes ito the root ode of a tree, with distigished left ad right sbtrees (which are empty as of yet) The secod ey is gided to the left sbtree, if it is smaller tha the root ey (ie, if π 2 < π 1 ), where it is iserted i a ode ad lied as a left child of the root; otherwise (ie, if π 2 > π 1 ) the secod ey goes ito the right sbtree, where it is iserted i a ode ad lied as a right child of the root Sbseqet eys go to the left or right sbtrees, accordig to whether they are smaller tha the root ey or ot, where they are iserted recrsively i the sbtree by the same algorithm Note that whe the permtatios of {1, 2,, } are eqally liely, they give rise to a oiform probability distribtio o the shapes of BST We call sch distribtio the BST probability model Correspodig athor addresses: hosam@gwed (HM Mahmod), mdw@prdeed (MD Ward) /$ see frot matter 2012 Elsevier Ltd All rights reserved doi:101016/jaml

2 HM Mahmod, MD Ward / Applied Mathematics Letters 25 (2012) Fig 1 Example of a biary search tree correspodig to the permtatio (5, 9, 6, 4, 7, 2, 3, 1, 8); 2-protected odes i bold This BST probability model is deemed more relevat to compter sciece applicatios tha the iform model o biary trees as it coforms more closely to the atre of data arisig i sortig ad searchig applicatios For istace, data samples of size tae from ay arbitrary cotios distribtio have ras that are almost srely (sice ties occr with probability 0) a radom permtatio o {1, 2,, } Sch real-mbered data ca be assimilated by their ras to bild a biary tree with the aforemetioed BST distribtio A ode with o descedats i a BST is a leaf A ode i a BST is said to be a 2-protected ode if its distace (measred i mber of edges) to the earest descedat leaf is at least 2 Fig 1 shows a BST of size 9 grow from the permtatio (5, 9, 6, 4, 7, 2, 3, 1, 8) The odes represeted by bold circles are 2-protected I this ote, we ivestigate the mber of 2-protected odes i a BST Or program does ot stop at the derivatio of mea, bt coties to fid asymptotic distribtios 2 Momets of the mber of 2-protected odes Let the mber of 2-protected odes i a radom BST of size be X I the tree show i Fig 1, X 9 = 4 Let U be the size i the left sbtree of the root, ad so 1 U is the size of the right sbtree I view of the BST probability model, the root is eqally liely to be ay of the mbers i the set {1, 2,, } Ths, U is iformly distribted o the set {0,, 1}, ad so symmetrically is 1 U Let R be the evet that the root ode is ot 2-protected Evet R occrs if: the root is a leaf itself ( = 1), or both childre of the root are leaves (possible whe = 3), or the root has exactly oe child that is a leaf For 1, we have a stochastic recrrece for X It is the combied mber of 2-protected odes i the two sbtrees of the root, pls 1 (to accot for the root beig 2-protected) less R occrs Ths, we have a eqality i distribtio, amely, X D = X U + X 1 U R (Note: the tilded radom variable X 1 U is coditioally idepedet of X U (give U )) The variables X 0, X 1, X 2 are always 0 We are sig a idicator otatio, ie, 1 R = 1, if R occrs, ad 0 otherwise Ths, for the momet geeratig fctio φ (t) := E[e X t ] of X, we have φ (t) = E e (φ U +φ 1 U +1 1 R )t Whe 4, we see that R oly occrs if U = 1 (ie, the left child of the root is a leaf) or 1 U = 1 (ie, the right child of the root is a leaf) (For 4, both childre of the root caot simltaeosly be leaves) Sice X U ad X 1 U are coditioally idepedet (give U ), a recrrece eses by coditioig o U Namely, for 4, we have φ (t) = et = et 0 1 1, 2 φ (t) φ 1 (t) + 2 φ 2(t) 1 φ (t) φ 1 (t) + 2 φ 2(t)(1 e t ) (1) Differetiatig r times with respect to t, the settig t = 0, gives a recrsio for E[X r ] As r icreases, the recrrece eqatios qicly become more complicated, a pheomeo commoly called the combiatorial explosio It is sfficiet for or prpose to get a exact soltio for the recrrece relatios for the first two momets, ad from there we shall maage to get a shortct to the higher asymptotic momets

3 2220 HM Mahmod, MD Ward / Applied Mathematics Letters 25 (2012) For r = 1 ad 4, we obtai a recrrece for the mea, amely, E[X ] = 2 1 E[X ] The recrrece for E[X ] ca be solved by stadard methods sch as differecig, for example If we deote Eq (2) as S(), the for 5, we see S() S( 1) has telescopig sms that disappear, ad the resltig liear recrrece ca be easily solved, with bodary coditios E[X 0 ] = E[X 1 ] = E[X 2 ] = 0, E[X 3 ] = 2/3, ad E[X 4 ] = 5/6 The bodary coditio = 4 agrees with the geeral form, ad we get a simple soltio for the mea Theorem 21 Let X deote the mber of 2-protected odes i a biary search tree grow from a iformly chose radom permtatio of {1,, } The we have E[X ] = 11 19, for 4 For r = 2, we se (1) to develop a recrrece for the secod momet: E[X 2 ] = 2 1 E[X 2 ] E[X ] E[X ] E[X 1 ] 4 E[X 2] + 2, valid for 4 With E[X ] ow determied, we ca solve the recrrece for E[X 2 ] Solvig the recrrece for the secod momet is ot qite as simple as solvig that for the first momet For istace, differecig does ot shave off the sms A more straightforward strategy is to gess a soltio, the prove it by idctio This procedre yields the followig reslt Theorem 22 Let X deote the mber of 2-protected odes i a biary search tree grow from a iformly chose radom permtatio of {1,, } The we have E[X 2 ] = , for 8, 100 ad the variace follows: Var[X ] = E[X 2 ] (E[X ]) 2 = , for 8 Note the exact cacellatio of the qadratic terms, leavig oly a liear variace, which gives a chace for asymptotic ormality to hold, as it fits icely ito the two momets ad a recrrece paradigm give by Pittel [7] Momets of arbitrarily high degree ca be fod similarly For istace, we have E[X 3 ] = , for 12, E[X 4 ] = , for 16, E[X 5 ] = , for 20, etc, ad i geeral, we cojectre the followig Cojectre 21 Let X deote the mber of 2-protected odes i a biary search tree grow from a iformly chose radom permtatio of {1,, } For each fixed iteger 1, there exists a polyomial p () of degree, the leadig term of which is (11/), sch that E[X ] = p (), for all 4 (2) 3 Asymptotic ormality The mai reslt of this ote is the followig Theorem 31 Let X be the mber of 2-protected odes i a radom biary search tree grow from a iformly chose radom permtatio of {1,, } The X, properly ormalized, coverges i distribtio, amely, X 11 D N 0, 29

4 HM Mahmod, MD Ward / Applied Mathematics Letters 25 (2012) Proof Let X = 1/2 (X 11 ), ad X 11 (t) = E exp t t = exp 11 t, be its momet geeratig fctio The recrrece (1) ca be ormalized i the form exp 11 exp exp 11 = 1 exp 11 φ X 1 exp exp 11 1 exp, which we ca write as exp exp 11 1 () = + 2 φ 2 X exp exp 11( 1 ) I view of Pittel s paradigm [7], a limit () (the momet geeratig fctio of a limitig radom variable X ) exists, as Passage to the limit i the latter relatio yields 1 1 () = lim c 1 + lim O( 3/2 ) Pt / = x, to represet the last relatio as ( 1)/ () = lim ( x, ) ( x 1, ) x,, x, =0 where x, = x, x 1, is the differece operator, ad the smmatio idex x, moves p i icremets of size 1/ By the sal iterpretatio of Riema itegrals, we fially write () = 1 y=0 ( y ) ( 1 y ) dy This itegral fctioal eqatio has the fctio e c2 2 /2 as a soltio This fctio is the momet geeratig fctio of the ormal N (0, c 2 ) radom variable By Lévy s cotiity theorem we get the desired covergece i distribtio: X 11 D N (0, c 2 ), for a appropriate vale of c 2 Of corse, it mst be 29, the coefficiet of the leadig asymptotic term i the variace 4 Exteded biary search trees BSTs are ofte exteded by addig special exteral odes as childre A sfficiet mber of these exteral odes are spplied to each origial ode (ow thoght of as iteral) to mae its otdegree eqal to two I this variat, the 2-protected odes are cshioed from the exteral odes by at least oe iteral ode As a example, see Fig 2, i which we have added the exteral odes to the tree i Fig 1, ad we have agai oted the 2-protected odes for this modified model i bold If X deotes the mber of 2-protected odes i exteded biary trees, the we have E[X ] = 1 3 2, for 2, 3 ad Var[X ] = 2 + 2, for 4

5 2222 HM Mahmod, MD Ward / Applied Mathematics Letters 25 (2012) Fig 2 Example of a exteded biary search tree for the permtatio (5, 9, 6, 4, 7, 2, 3, 1, 8); 2-protected odes i bold The correspodig cetral limit reslt is X 1 3 D N 0, 2 These reslts ca be obtaied by very similar methods as those we applied to the exteded BST However, most of these reslts for exteded BSTs are already implied i the pblished literatre For istace, i the exteded BST the 2-protected odes are the odes of otdegree 2 i the tree before it got exteded The exact average of these appears i [8] The asymptotic distribtio appears i [9], where he ses a m-depedet cetral limit theorem for statioary radom variables, de to Hoeffdig ad Robbis [10] Mahmod [11] gives a accot of a proof based o modelig by Pólya r models The oly thig ew here is the exact variace, which we get via the exact secod momet, E[X 2 ] = , for 4 Aother ew aspect is that we ca agai se or recrsive methods to develop exact higher momets for the mber of 2-protected odes i a exteded BST, eg, E[X 3 ] = , for 6, 315 E[X 4 ] = , for 8, 675 E[X 5 ] = , for 10, etc I geeral, we cojectre the followig Cojectre 41 Let X deote the mber of 2-protected odes i a exteded biary search tree grow from a iformly chose radom permtatio of {1,, } For each fixed iteger 1, there exists a polyomial p () of degree, the leadig term of which is 1/3, sch that E[X ] = p (), for all 2 Acowledgmets This research was doe while the first athor was visitig Prde Uiversity The spport the first athor received from Prde s Departmet of Statistics is greatly appreciated The secod athor was spported by NSF Sciece & Techology Ceter for Sciece of Iformatio Grat CCF Refereces [1] G-S Cheo, LW Shapiro, Protected poits i ordered trees, Applied Mathematics Letters 21 (2008) [2] T Masor, Protected poits i -ary trees, Applied Mathematics Letters 24 (2011) [3] RR D, H Prodiger, Notes o protected odes i digital search trees, Applied Mathematics Letters 25 (2012) [4] DE Kth, The Art of Compter Programmig, secod ed, i: Sortig ad Searchig, vol 3, Addiso-Wesley, Readig, MA, 1998, Origially pblished i 1973 [5] HM Mahmod, Sortig: A Distribtio Theory, Wiley, New Yor, NY, 2000 [6] HM Mahmod, Evoltio of Radom Search Trees, Wiley, New Yor, NY, 1992 [7] B Pittel, Normal covergece problem? two momets ad a recrrece may be the cles, The Aals of Applied Probability 9 (1999) [8] HM Mahmod, The expected distribtio of degrees i radom biary search trees, The Compter Joral 29 (1986) [9] L Devroye, Limit laws for local coters i radom biary search trees, Radom Strctres ad Algorithms 2 (1991) [10] W Hoeffdig, H Robbis, The cetral limit theorem for depedet radom variables, De Mathematical Joral 15 (1948) [11] HM Mahmod, Pólya Ur Models, Chapma, Orlado, FL, 2008

Numerical Methods for Finding Multiple Solutions of a Superlinear Problem

Numerical Methods for Finding Multiple Solutions of a Superlinear Problem ISSN 1746-7659, Eglad, UK Joral of Iformatio ad Comptig Sciece Vol 2, No 1, 27, pp 27- Nmerical Methods for Fidig Mltiple Soltios of a Sperliear Problem G A Afrozi +, S Mahdavi, Z Naghizadeh Departmet