On Performance Deviation of Binary Search Tree Searches from the Optimal Search Tree Search Structures

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1 INTERNATIONAL JOURNAL OF COMPUTERS AND COMMUNICATIONS Issue, Volume, 7 O Performace Deviatio of Biary Search Tree Searches from the Optimal Search Tree Search Structures Ahmed Tarek Abstract Biary Search Trees are a frequetly used data structure for rapid access to the stored data. Data structures like arrays, vectors ad liked lists are limited by the trade-off betwee the ability to perform a fast search ad resize easily. They are a alterative that is both dyamic i size ad easily searchable. Due to efficiecy reaso, complete ad early complete biary search trees are of particular sigificace. This paper addresses the performace aalysis ad measuremet, collectively kow as the Performace i biary search tree search applicatios. Performace measuremet is equally sigificat asides from the performace aalysis to lear more about the deviatio from optimality. To estimate this deviatio, ew performace criteria for the biary search trees are preseted. A multi-key search algorithm is proposed ad the related aalysis followed. The algorithm is capable of searchig for multiple key elemets i the same executio, sacrificig some optimality i the timig cosideratio. This helps i pruig a subtree structure out of a give biary search tree for further processig. Keywords Complete Biary Search Tree, Nearly Complete Biary Search Tree, Performace Criteria, Sparsity Factor, Desity Factor, Multi-Key Search, Search-tree Pruig. I. INTRODUCTION Efficiet access to the stored data is a maistream reaso for the choice of a good data structure (DS). To provide efficiet access, the DS may eed to store additioal iformatio kow as the overhead. Therefore, a major objective of a DS is to keep the overhead miimum while allowig maximum access to the stored data. This paper is cocered with the aalysis of biary search trees (BSTs) as data structures of choice with several performace criteria. The deviatio from the optimality for usig the BSTs are demostrated usig performace measuremet results. BSTs ad the related applicatios are studied extesively i the literature. Amog the most otable cotributios, [] has studied the height, size performace of a class of BSTs i dictioary applicatio. I [], a applicatio of the BSTs i Neural Networks is preseted. This research paper deals with the geeral performace i BST search applicatios. A ew algorithm i searchig for multiple umber of odes i the same executio is also proposed. The multiple key BST search helps prue a subtree structure from a existig BST for further processig. Performace measuremet of the proposed multi-key BST search algorithm is also preseted. The results i this paper are both theoretical ad applied i ature. The performace graphs are obtaied usig the commo high level laguage compilers ruig o multiple Mauscript received September 3, 7; accepted February 6, 8; revised April 9, 8. This work was supported by the Califoria Uiversity of Pesylvaia i Pesylvaia, USA. Ahmed Tarek is associated with the Departmet of Math ad Computer Sciece at Califoria Uiversity of Pesylvaia, 5 Uiversity Aveue, Califoria, Pesylvaia 549, USA (phoe: (74) ; fax: (74) ; tarek@cup.edu) platforms. A umber of performace criteria are addressed. Fially, future research directios are outlied. The remaider of this paper is structured as follows. I Sectio II, terms ad otatios used i this paper are itroduced. Some ew cocepts are also defied. Sectio III cosiders performace of the BSTs usig the criteria itroduced i sectio II. Sectio IV itroduces the Multiple Key BST Search algorithm. This sectio also icorporates the related aalysis. Sectio V is based o the search-based performace of the BSTs. Sectio V I addresses the practical performace issues. Sectio V II outlies future research aveues. II. TERMINOLOGY AND NOTATIONS Followig otatios are used all throughout this paper. : Total umber of odes. T r : A biary search tree, which is abbreviated as, BST. l: Number of leaves. i : Iteral (iterior) ode cout. e : Number of exteral odes. h: Height of the BST. C i : Cost for a successful search i a BST. C e : Cost for a usuccessful search. I: Iteral path legth. E: Exteral path legth. sf: Sparsity factor. df: Desity Factor. L: Loss i capacity factor. Special terms ad cocepts are preseted by combiig meaigful idices with the correspodig otatio. Some useful defiitios are preseted ext. Deviatio i Height, h dev : The deviatio i height, h dev is the deviatio of the actual height, h from the optimal possible height, h o. This is expressed i % as follows: h dev = h ho h o %. Sparsity Factor: Justifies the relative sparsity of a actual BST i compariso to a full, ad complete BST with the same height, h. Mathematically, Sparsity Factor, sf = max m ax %. Here, max = maximum possible umber of records that may be accommodated i a complete BST with the actual height, h = ( (h+) ), ad = the actual umber of records curretly preset. Desity Factor: This determies the relative desity of a actual BST i compariso to a liear slim BST havig the same height, h. This is defied mathematically as, df = mi mi %. Here, mi = the miimum umber of records i a slim BST with the actual height, h = (h + ). III. PERFORMANCE WITH THE SPARSITY AND THE DENSITY FACTORS It is always desired that a costructed BST be as dese as possible approachig the complete or the early complete 4 Mauscript received May 6, 7: Revised received Aug. 3, 7

2 INTERNATIONAL JOURNAL OF COMPUTERS AND COMMUNICATIONS Issue, Volume, 7 BST structure, ad as less sparse as is feasible. Followig result holds true i this cotext. Theorem : The maximum height deviatio from a liear sparse BST to a early complete bushy BST havig odes is: h diffmax = ( log (+)), ad the correspodig miimum possible height deviatio is: h diffmi = ( ) log, ad the differece betwee these two extreme deviatios is: log ( (+) ). Proof: For a liear skiy tree, there will be exactly ode at each level. Sice the ode coutig starts at the root record with level, therefore, (h s + ) =. This provides, h s = ( ). Suppose that there are k records at the last level h. I that evet, ( h ) + k =, this meas, h ( ) + k =. Therefore, h = ( k), this provides, h = ( + k). Hece, h = log ( + k). For the miimum deviatio i height, there is oly record at level h. Therefore, k =, ad h bmax = log. Hece, h diffmi = (h s h bmax ) = ( ) log. For the maximum height deviatio, there are h records at level h. Therefore, k = h, ad h = ( + h ), which provides, h+ = ( + ). This yields, (h + ) = log ( + ), or h = log ( + ). Therefore, h diffmax = (h s h bmi ) = ( ) (log (+) ) = ( log (+)). Therefore, fially, h diffmax h diffmi = ( log ( + )) - (( ) log ) = ( + log ( + ) + log ) = (log () log ( + ) + log ) = log () log ( + ) = log + = log ( (+) ). The differece, ( max - mi ) defies the maximum deviatio i the umber of records with a actual height, h. Hece, mdev = max - mi = ( (h+) - (h+)) = ( (h+) h ). The deviatio i height, h dev is defied as the deviatio of the actual height, h from the optimal height, h o. This is expressed as % of h o. Mathematically: h dev = h ho h o %. (a) BST height, h umber of odes, h a & h o differece i actual ad optimal heights, d (b) umber of odes, h a - h o plot Fig.. The actual ad the optimal heights of the geerated BSTs ad their differeces are plotted agaist the umber of records,. (The lower curve i Fig. (a) represets the optimal height). The plot i Fig. (a) shows the height deviatio of the actually geerated BST from the optimal oe. For the correspodig optimal BSTs, the height does ot chage from = 6 to =. If h opt = 8, the maximum umber of records that it may cotai is, 8+ = 5. Whereas, if h opt = 9, the maximum umber of records it may cotai is = 9+ =, 3. Therefore, for ay value of ragig from 6 to,, the optimal height is 9. The sparsity factor is defied as, sf = max max %. Therefore, sf is required to be as small as possible. Sice, max will be fixed for a particular value of h, the smaller the value of ( max ), will be closer to max, ad the tree will approach that of the optimal cofiguratio. This meas that the sparsity will decrease. Though sparsity factor sf is suppose to decrease with the icreasig value of, but almost costat sf is a idicator of the relatively steady BST structures. Costat values of sf idicate that the actual umber of odes, is relatively steady i compariso to the expoetial growth of, max = ( (h+) ) with the chagig values of h. The desity factor, df = ( mi) mi %. For a costat value of h, the higher the desity factor, df, will become relatively larger ad larger i compariso to mi, ad the tree will grow relatively deser. IV. MULTI-KEY BST SEARCH ALGORITHM Usig Multi-key Search, it is possible to idetify or more subtrees i the origial BST that starts at a particular record ad eds at aother oe. Sice such subtrees are just parts of the origial BST, operatios o these may be substatially faster tha origially costructig those subtrees from the scratch. Usig the proposed algorithm, it is possible at first to idetify the subtree structure, ad the applyig the memory move operatios, it is also possible to create a BST out of the subtree for further cosideratio. Algorithm fid record Purpose: This algorithm fids a record i the geerated BST. Require: ame supplied ad this ode as iputs. if ame supplied.compareto(this ode.ame)== the retur this ode if ame supplied.compareto(this ode.ame) < the if this ode.getleftchild() is ot NULL the retur fid record (ame supplied, this ode.getleftchild()) {recursive call to fid record} retur NULL if this ode.getrightchild() is ot NULL the retur fid record (ame supplied, this ode.getrightchild()) retur NULL The -key biary search tree search algorithm makes use of the classical -key versio. Algorithm fid record key Purpose: This algorithm performs -key biary search tree search. The supplied parameters are: array ames[], curret ode verified this ode. fid record key fids out two matchig odes if available for the array ames[] ad retur those as array search[]. Require: ames[].compareto(ames[]) < 4

3 INTERNATIONAL JOURNAL OF COMPUTERS AND COMMUNICATIONS Issue, Volume, 7 Esure: a array of correct records or NULLs are retured if ames[].compareto(this ode.ame) < the if this ode.getleftchild() is ot NULL the search[] fid record (ames[], this ode.getleftchild()) search[] fid record (ames[], this ode.getleftchild()) {Make calls to fid record o the left subtree} search[] NULL search[] NULL retur search[] if ames[].compareto(this ode.ame) > the if this ode.getrightchild() is ot NULL the search[] fid record (ames[], this ode.getrightchild()) search[] fid record (ames[], this ode.getrightchild()) {Make calls to fid record o the right subtree} search[] NULL search[] NULL retur search[] if ames[].compareto(this ode.ame) < ad ames[].compareto(this ode.ame) > the if this ode.getrightchild() is ot NULL ad this ode.getleftchild() is ot NULL the search[] fid record (ames[], this ode.getleftchild()) search[] fid record (ames[], this ode.getrightchild()) {Make calls to fid record o two subtrees} if this ode.getrightchild() is ot NULL the search[] NULL search[] fid record (ames[], this ode.getrightchild()) if this ode.getleftchild() is ot NULL the search[] fid record (ames[], this ode.getleftchild()) search[] NULL search[] NULL search[] NULL retur search[] if ames[].compareto(this ode.ame) == the if this ode.getrightchild() is ot NULL the search[] this ode search[] fid record (ames[], this ode.getrightchild()) search[] this ode search[] NULL retur search[] if ames[].compareto(this ode.ame) == the if this ode.getleftchild() is ot NULL the search[] this ode search[] fid record (ames[], this ode.getleftchild()) search[] this ode search[] NULL retur search[] retur search[] A. Multi-key BST Search Aalysis For clarity, cosider the -key BST search i this aalysis. Suppose that the -st key is located at height h ad at k positio coutig from the left-most record at height h. Similarly, suppose that the d key is at height h ad at k positio coutig from the left-most record at height h. Followig are the possible scearios with this -key BST Search. The st key is o the left subtree, ad the d key is o the right subtree of the root record. I this case, the subtree startig at k ad edig at k icludes the root ode. Both k ad k are o the left subtree. The the subtree startig at k ad edig at k will ot iclude the root, ad cotais oly a part of the left subtree. Both k ad k are o the right subtree. The the subtree startig at k ad edig at k will ot iclude the root, ad cotais a part of the right subtree. k is the root ode ad k is o right subtree. The subtree spaig from k to k icludes a portio of the right subtree icludig the root. k is o the left subtree ad k is at the root. I this case, the subtree from k to k cotais a portio of the left subtree, which icludes the root. Followig aalysis is based o the assumptio that the BST is complete up to a height of maximum{h, h }. Suppose that the height of the BST is h ad the total umber of records is. Therefore, h h, ad h h. For the followig aalysis, the left ad the right subtrees coutig from the root record are cosidered as two separate subtrees, sice all calculatios begi at the root ad proceed either through the left subtree or alog the right subtree of the root. Suppose h < h, ad both key ad key are o the left subtree. The for key, it is complete up to the level (h ), ad there are k records at level h coutig from the left-most ode at the same height. Similarly, for key, it is complete up to the level (h ), ad there are k records at level h coutig from the leftmost ode. Sice, h < h, therefore, the total umber of odes i betwee key ad key, which are o the left subtree is, = ( h ) + k ( h ) k = (h + h + + h h ) + (k k ) = ( h h + + k k ) = h ( h h ) + (k k ). 43

4 INTERNATIONAL JOURNAL OF COMPUTERS AND COMMUNICATIONS Issue, Volume, 7 Let h = h, ad both key ad key are o the left subtree. Sice the assumptio is that the keys are orgaized i ascedig order ad therefore, k < k. Hece, = (k k ). Suppose that h < h, ad k is o the left subtree ad k is o the right subtree. I this case, = ( h + k ) + ( h + h h ) + (k h ). Sice the tree is complete up to the level h, therefore of the total odes up to the h level lies o the left subtree, ad the rest are o the right subtree, ad k ad k are couted startig from the left-most ode o the left subtree. Hece, = (h ) + (h ) + k + k - (h ) = ( h + k + k ) = ( h + k + k ). Suppose that h > h, ad k is o the left subtree ad k is o the right subtree. I this case, = (h ) + (h ) + k +k - (h ) = ( h +k +k ) = ( h + k + k ). Suppose that h is equal to h, ad k is o the left subtree, ad k is o the right subtree. Here, = (h ) + (h ) + k + k - (h ) = ( h + k + k ) = ( h + k + k ). h < h, ad both key ad key are o the right subtree. For key, it is complete up to the level (h ), ad there are k records at level h coutig from the left. Similarly, for key, it is complete up to the level (h ), ad there are k records at level h coutig from the left. Sice h < h, therefore, the total umber of records i betwee key ad key, which are o the right subtree is, = ( h ) + (k (h )) ( h ) (k (h )) = ( h + h + h h +k k ) = ( h h )+(k k ). h = h, ad both key ad key are o the right subtree. Sice the assumptio is that the keys are orgaized i ascedig order, therefore, k < k. Hece, = k (h ) (k (h )) = k k. key is at the root ad key is o the right subtree. I that evet, key is at level ad the oly possible record. Assumig key is at level h, ad the BST is complete up to the level h, = + ( h )+(k ))) (iclusive) = + (h ) (h + k ) (h = k. key is o the left subtree ad key is the root ode. I that evet, key is at level ad the oly possible record. Assumig key is at level h, ad the BST is complete up to the level h, = + ( h )+k = + k + (h ) = k + h. Followig the aalysis of -key BST search, for the 3-key BST search, followig are the possible sceario: The first keys k ad k exist o the left subtree, ad the 3rd key, k 3 exists o the right subtree. I that evet, the subtree startig at k ad edig at k 3 (iclusive) icludes the root ode. The st key, k exists o the left subtree, ad the d ad the 3rd keys, k ad k 3, respectively, are o the right subtree. I that case, the subtree startig at k ad edig at k (iclusive) icludes the root, whereas the subtree startig at k ad edig at k 3 icludes oly a part of the right subtree. All 3 keys, k, k ad k 3 exist o the right subtree. I that evet, the subtree startig at k ad edig at k 3 (iclusive) does ot iclude the root. All 3 keys k, k ad k 3 exist o the left subtree. I this case also, the subtree startig at k ad edig at k 3 does ot iclude the root. k is o the left subtree, k is at the root ad k 3 is o the right subtree. I this case, the subtree startig at k ad edig at k 3 icludes the root (iclusive). k is at the root, k ad k 3 are o the right subtree. I this case also, the subtree startig at k ad edig at k 3 (iclusive) icludes the root ad a part of the right subtree oly. k ad k are o the left subtree ad k 3 is at the root. I that case, the subtree from k to k 3 (iclusive) icludes the root ode. B. Special Cosideratios st key is o the left subtree at level h h, ad the d key is at level h h o the right subtree coutig from the root ode. I this case, the subtree startig at k ad edig at k icludes the root. If k is the leftmost record ad k is the right-most record, the the total umber of records startig at k ad edig at k, = ( h )+ ( h ) = (h+ ) ( ) + (h+ ) ) = (h+ )+ (h+ ) = (h+ + h+ ) (sice from the assumptio, all the levels up to h ad h are complete o the left ad the right subtrees, respectively) = ( h + h ) = h + h = Maximum umber of records possible betwee k ad k (iclusive). If k is the right-most record o the left subtree at level h, ad k is the left-most record o the right subtree at level h. Assumig that all levels up to (h ) are complete o the left subtree, ad all the levels up to (h ) are complete o the right subtree as well. The = ( h ) + ) + ( h ) + ) = (h + ) + + ( h + ) ( ) + = h + + h + = h + h + = (h + h + ) = Miimum umber of odes (iclusive) possible betwee k ad k. Therefore, followig relatioship holds true for k i the left subtree ad k o the right oe: (h + h +) ( h + h ). Hece, ( max mi ) = ( h + h ) (h + h + ) = h + h. Let k is the left-most ode at level h i the left subtree ad k is the right-most ode at level h i the left subtree as well. Assumig all the itermediate levels icludig the levels at h ad h are complete, the total umber of odes (iclusive), = (h h ) = ( h + h h ) - ( h ) = ( (h + ) ( ) ) ( (h ) ( ) = h h + = ( h h ), which gives us the expressio for the maximum umber of records o the same subtree. Hece, 44

5 INTERNATIONAL JOURNAL OF COMPUTERS AND COMMUNICATIONS Issue, Volume, 7 ( h h ). Let k be the right-most ode at level h, ad k is the left-most ode at level h o the same subtree. The h < h. Hece, the miimum umber of odes i betwee k ad k (iclusive) is = + (h h ) + = + ( h + h h ) - ( h ) = + ( h + ( ) ) ( (h+ ) ( ) = + (h ) (h+ ) = ( + h h ). Therefore, ( + h h ). If both the keys are o the same subtree, the the followig relatio holds true: ( + h h ) ( h h ). Hece, ( max mi ) = ( h h ) ( + h h ) = h + h. If key is at the root, ad key is o the right subtree, the k is at level, ad the oly possible ode. Assumig k is at level h, there are istaces possible. I oe istace, k is the right-most ode of the right subtree at level h. I that evet, max = + ( h ) (iclusive) = + (h+ ) = h. O the other had, if k is the left-most ode, ad the oly record available at level h, mi = + ( h ) + = + (h ) = + h = + h. With k at the root, ad k o the right subtree, followig is the restrictio: + h h. Also, ( max mi ) = ( h h ) = ( h ). If key is o the left subtree, ad key is at the root, the k is at level, ad the oly possible ode. Hece, max = + ( h ) (iclusive) = + (h+ ) = + h = h. Oce agai, mi = + ( h ) + (iclusive) = + (h ) = + h. Therefore, ( max mi ) = h h = h h = h. Next cosider a very special case of pruig the right subtree of the st key ad the left subtree of the d key i the two key BST Search together with the other odes from key to key. Further assume that key is o the left subtree ad key is o the right subtree. If key is at level h ad key is at level h with h h ad h h, where h is height of the BST, the the umber of odes i the right subtree of the st key = (h h ) = (h h +) = h h. Similarly, the total umber of odes i the left subtree of the d key = (h h ) = (h h +) = h h. These umbers may be computed by imagiig a BST subtree with root at the right child of the st key ad aother subtree havig the root at the left child of the d key. Off-course for this aalysis to hold true, the BST is required to be complete. If h < h, ad k is o the left subtree ad k is o the right subtree, = ( h + k + k ). Hece, total umber of odes i the prued subtree = + h h + h h = ( h + k + k ) + h h + h h Similarly, if h > h, total umber of odes i the prued subtree = ( h + k + k ) + h h + h h. Also if h = h, total umber of odes i the prued subtree = ( h + k + k ) + h h + h h. V. MULTI-KEY SEARCH PERFORMANCE I this sectio, the results pertaiig to multiple key-based searches i BSTs are cosidered. Theorem : The average cost of a usuccessful search is give by, C e = log e ( + ), ad the average cost C i for a successful search is = (log e -). Proof: Let E be the expected exteral path legth, ad I be the expected iteral path legth for a BST with odes. Therefore, followig recurrece relatio holds true: E o =, E = ( + ) + i= (E i + E i ). Here, ( + ) is to accout for the root ode cost for each exteral ode (there are ( + ) of them). i= (E i + E i ) accouts for the average exteral path legth of the left ad the right subtrees over with a arbitrary ith elemet at the i= E i). Hece, rewritig the root. But i= E i = expressio for E, E = (+)+ i= E i. The E = + i = E i, which provides: ( )E ( ) = i= E i. Now expadig the expressio for E, E = ( + ) + E + i= E i. From the previous expressio for E, i= E i = E. Hece, E = ( + ) + E + [( )E ( )]. From this last expressio, E = + (+) E. Expadig the recurrece relatioship for E, E = + (+) + (+) ( ) E = + (+) + (+) ( ) E = + (+) + (+) ( ) + (+) ( ) E 3 = + (+) i= i +(+)E = + (+) (+) i= i (+) (+) = (+) (+) i= i (as E = ) = (+)H +. Here, H + is the ( + )th Harmoic Fuctio. Usig the properties of Harmoic Fuctios, H + log e ( + ) + γ, where γ = Euler s costat.577. Hece, E ( + )(log e ( + E (+) ) + γ). The average cost for a usuccessful search is, = (log e ( + ) + γ) log e ( + ) (see []). The highest order term i this expressio is: log e ( + ). Therefore, C e log e. Agai, E = I +. This provides, I = (+)(log e (+)+γ). The average cost of a successful search is, I (+ )log e(+) (log e ( + ) + log e( + ) ). As,. Hece, C i log e (+). The highest order term i this expressio is, log e ( + ). As, log e ( + ) log e. Fially, C i (log e ). Hece, C i log e also. (a) o. of comparisos (+)(log e(+) o. of records, S() & U() curves differece i o. of comparisos, c-diff (b) o. of records, U() S() plot Fig.. Average umber of comparisos for successful (S()) ad usuccessful searches (U()) are plotted agaist the total umber of records,. (The lower curve i Fig. (a) represets the curve for successful searches). From the stadard DS literature, average umber of comparisos for a successful search, S() =.39log. Average umber of comparisos for a usuccessful search, U() = 45

6 INTERNATIONAL JOURNAL OF COMPUTERS AND COMMUNICATIONS Issue, Volume, 7.77log S(). Both S() ad U() depeds o log. The miimum value of is, mi =, ad the maximum value of is, max =,. For example, whe =, S() = 4.6, ad U() = 9. S(). Hece, the curve for average usuccessful search, U() follows almost exact patter to that of the successful search, S(). Agai, the differece betwee U() ad S() is, d() = U() S() =.77log -.39log =.38log S(). Therefore, the differece curve, d() i Fig. (b) has almost exactly the same patter as that of S(). VI. PERFORMANCE ISSUES IN PRACTICE Lemma 3: A m-key biary search tree search algorithm may be applied to ay BST cotaiig records, where >= m. Proof: A proof by cotradictio is adopted. Suppose that < m. Therefore, the total umber of keys to search for i the BST becomes greater tha the umber of records withi the BST. I the best possible case, differet keys may be idetified at the record positios, leavig (m-) keys udecided durig the computatio, for which, o records to look for may be available. This violates the objective of the m-key BST search, which is to idetify the BST records correspodig to the m- keys withi the BST. Hece, m, ad at most, m =. Theorem 4: A m-key BST search requires cosiderig (m+) differet cases i idetifyig the records correspodig to the m keys iside the BST. Here, m. Proof: Followig is a proof by mathematical iductio. Base Case: For the base case, m=. For P(), it is the classical, sigle key BST search. It cosiders 3-differet cases. These are: () key elemet = root value, () key elemet > root value, ad (3) key elemet < root value. Hece, ( + ) = 3 differet cases are beig cosidered. Iductio: Suppose that the k-key search algorithm requires cosiderig (k+) differet cases. Here, k. It is required to show that: [P() P(k)] P(k+), which is provig that for (k+) differet keys, ((k+) + ) = k+3 differet cases are required to be cosidered. For the (k + )th key, two more cases are required i additio to the (k +) cases for the first k keys. For the sorted keys, (k + )th key is the largest ad the last key withi the list. Therefore, it requires cosiderig oly additioal cases. First of all, verifyig whether the root value is equal to the (k + )th key value. If so, the (k + )th key is foud at the root, ad it is ecessary to use the k-key steps of the BST search to locate the first k-keys. Secodly, it is required to verify whether the (k + )th key is larger, ad the kth key is smaller tha the root ode. I that evet, cofie search for the (k + )th key to the right half of the BST usig a classical BST search, ad use the steps of the k-key BST search for the first k keys. Rest of the cases are idetical to the k-key versio except that it is required to cosider (k+) keys istead of k keys. Hece, altogether, for the (k+) key versio, we require cosiderig (k + + ) = (k + ) + differet cases. Coclusio: The theorem is true for m =. Assumig that the theorem holds true for m = k, it is proved that it also holds true for m = (k + ) differet keys. As it is true for m =, it is also true for m =. As it holds true for m =, it is also true for m = 3, ad so o. Hece, the theorem holds true for ay m with m. Average display time, Tavg i secods No. of records, Fig. 3. Average time to display the sorted tree, T i secods is plotted agaist the umber of records, iside the tree with data acquired usig the Borlad s C++ 5. compiler. Fig. 3 shows time, T required to display the odes i ascedig order usig i-order traversal of the BST, which is plotted agaist the size,. The curve is obtaied usig Borlad s C++ 5. compiler. The total time, T required to display the odes cotais two compoets. Oe is the fixed timig overhead, T o required by the program to ru o a platform. The other oe is the total time, T d to display the BST etries. Therefore, T = T o + T d. Iitially, from = to =, the time required to display the odes T d is isigificat compared to the timig overhead, T o. Therefore, T o domiates over T d, as proouced by the relatively flat ature of the curve withi this rage. For = to =,, T d becomes relatively much higher compared to T o, ad T d domiates over T o. Therefore, withi this rage, the total time cosumed depeds more o T d tha oly o T o, which may be realized by the icreasig ature of the curve. Fig. 4 shows the -key, the -key ad the 3-key BST iorder traversal times plotted agaist for usig the Java JDK 5. compiler ruig o Petium 4 machie. The total time take rapidly icreases with the icreasig umber of keys as the time to display the combiatios is approximately i the order of, O( m ). Here, is the umber of elemets ad m is the umber of keys. As m, m remais relatively costat with respect to, ad the curves display polyomial characteristic as expected. VII. CONCLUSION I this paper, some ew results o BST performace, ad the deviatio of the geerated BST structures from the correspodig optimal structures are also preseted. To search a array with speed, the array eeds to be sorted. Isertig ew odes i a ordered array is a pai. Agai, isertig ew odes i a liked list is easy; but searchig for a existig ode i a liked list is difficult. BSTs are data structures that combies the good properties of arrays ad liked lists, but has 46

7 INTERNATIONAL JOURNAL OF COMPUTERS AND COMMUNICATIONS Issue, Volume, 7 6 time, T i secods to search & display all key combiatios umber of odes, Fig. 4. Total time T total to search ad display all possible key combiatios for -key, -key, ad 3-key BST searches are plotted agaist the total umber of odes,. Topmost curve is for the 3-keys cosidered together. The middle oe is for possible -key combiatios, ad the lower most curve is for the casual -key BST search. Su Java s JDK 5 compiler was used i performace measuremet. oe of the bad parts. Hece, complete ad early complete BSTs draw special attetio to the data structure researchers. Efficiecy of a BST applicatio depeds o the depth of the tree, h. Therefore, a good umber of the results are based o the height, h-based performace of the BSTs. I the best possible sceario, the optimal depth is, h best = log (). If the keys are added at radom to a gradually growig BST structure, it results i a BST with a average height, h avg. For a average BST, h avg.39log (). Therefore, h avg is oly about 4% higher tha the best possible. I future, a dyamic programmig model for geeratig a optimal BST structure with the miimal iteral ad the exteral path legths will be developed, ad the related performace issues will be addressed. REFERENCES [] Ahmed Tarek, Height Size Performace of Complete ad Nearly Complete Biary Search Trees i Dictioary Applicatios, WSEAS Trasactios o Computers, 3 (7), 8, pp , ISSN: [] P. Scarfe ad E. Lidsay, Dyamic Memory Allocatio for CMAC usig Biary Search Trees, Proceedigs of the 8th WSEAS Iteratioal Coferece o Neural Networks, 7, pp