ADAPTIVE SORTING WITH AVL TREES

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1 307 ADAPTIVE SORTING WITH AVL TREES Am Elmasy Compute Sciece Depatmet Alexadia Uivesity Alexadia, Egypt Abstact A ew adaptive sotig algoithm is itoduced. The ew implemetatio elies o usig the taditioal AVL tees, ad has the same pefomace limitatios. Moe pecisely, the umbe of compaisos pefomed by ou algoithm, o a iput sequece of legth that has I ivesios, is at most 1.44 lg I + O() 1. Ou algoithm us i time O( log I ) ad is pactically efficiet ad easy to implemet. 1. Itoductio A adaptive sotig algoithm is a sotig algoithm that beefits fom the pesotedess i the iput sequece. I the liteatue thee ae plety of adaptive sotig algoithms. Oe of the commoly ecogized measues of pesotedess is the umbe of ivesios i the iput sequece [12]. The umbe of ivesios is the umbe of pais of iput items i the wog ode. Fo a iput sequece X, the umbe of ivesios of X, Iv(X), is defied Iv(X) = {(i, j) 1 i < j ad x i > x j }. A adaptive sotig algoithm is optimal with espect to the umbe of ivesios whe it us i O( log Iv(X) ) [9]. Ufotuately, most of the kow theoetically optimal adaptive sotig algoithms ae ot pactical ad ot easy to implemet [9, 18, 3, 17, 15]. The umbe of compaisos is cosideed oe of the mai aalytical measues to compae diffeet sotig algoithms. The umbe of compaisos pefomed by a Iv-optimal sotig algoithm is at most c lg Iv(X) + O() compaisos, fo some costat c 1. Amog the adaptive sotig algoithms, Splitsot [13] ad Adaptive Heapsot [14] guaatee c = 2.5. Fige tees [9, 17], though ot pactical, guaatee c = 2. Tiomialsot [6] guaatees c = Recetly, Elmasy ad

2 308 Fedma [7] itoduced a adaptive sotig algoithm with c = 1, ad hece achievig the ifomatio theoetic lowe boud fo the umbe of compaisos. The task of achievig the optimal umbe of compaisos is theefoe accomplished, still with the pacticality issue beig ope. The algoithm i [7] uses ea optimal tees [1], which is a pactically complicated stuctue that ivolves a lage maiteace ovehead. The othe opeatios pefomed by the algoithm i [7] (splits, combies, coalescig ad eductio opeatios) cotibute with aothe ovehead facto, makig the algoithm ot fully pactical. Namely, the time boud fo the opeatios, othe tha the compaisos, pefomed by the algoithm i [7] is Θ( log Iv(X) ). Amog the adaptive sotig algoithms Splitsot [13], Adaptive Heapsot [14] ad Tiomialsot [6] ae the most pomisig fom the pactical poit of view. As a cosequece of the dyamic fige theoem fo splay tees (see Cole [5]), the splay tees of Sleato ad Taja [21] povide a simplified substitute fo fige tees that achieves the same asymptotic u-time. Moffat et al. [19] pefomed expeimets showig that Splaysot is efficiet i pactice. We itoduce a ew adaptive sotig algoithm that uses AVL tees [2]. Ou ew algoithm guaatees c = 1.44 i the wost case, while it achieves a value of c vey close to 1 (optimal) fom the pactical poit of view [10, 11]. This esult is a diect cosequece of the atue of the well-kow seach tees kow as AVL tees. The wost-case behavio of the AVL tees is achieved whe the tee is a Fiboacci tee, a case that aely pops-up i pactice. The cotibutio of this pape is to itoduce a pactically efficiet adaptive sotig algoithm, ad to show that apat fom the compaisos, the othe opeatios pefomed by this algoithm take liea time; a fact that does ot hold fo othe efficiet adaptive sotig algoithms. Fo example: Tiomialsot, Adaptive Heapsot ad Splitsot would pefom a o-liea umbe of moves. We expect ou ew algoithm to be efficiet, fast i pactice, ad easy to implemet. The space utilized by ou algoithm is O(). Othe methods that use AVL tees to implemet adaptive sotig algoithms iclude the algoithm of Mehlho [18], ad the fige tees of Tsakalidis [23]. These two algoithms equie augmetig the AVL tees with exta ifomatio that make the implemetatio o-pactical, with a lage costat fo the umbe of compaisos. Seveal authos have poposed othe measues of pesotedess ad poposed optimal algoithms with espect to these measues [8, 4, 14, 15]. Maila [16] fomalized the cocept of pesotedess. He studied seveal measues of pesotedess ad itoduced the cocept of optimality with espect to these measues. Petesso ad Moffat [20] elated all

3 309 of the vaious kow measues i a patial ode ad established ew defiitios with espect to the optimality of adaptive sotig algoithms. 2. The algoithm Coside the followig method fo isetig y ito a soted sequece, x 1 < x 2 < < x 1. Fo a specified value, we fist pefom a liea seach amog the items x, x 2, x / to detemie the iteval of legth amog x 1 < x 2 < < x 1 ito which y falls. Next, we pefom a biay seach withi the esultig iteval of legth to detemie the pecise locatio fo y. If y eds up i positio i, the i + lg + O(1) compaisos suffice fo this isetio. Usig the stategy of successively isetig the items x 1,, x i evese ode (ito a iitially empty list), let i j be the fial positio of elemet j afte the jth isetio. The total umbe of compaisos equied to sot would i j be bouded by Iv(X) 1 j + lg + O(1)) = + lg + O(). If we use = Iv(X), the equied boud o the umbe of compaisos follows. Ufotuately, we caot use this value of, sice the umbe of ivesios is ot kow befoehad. Istead, we choose to be a dyamic quatity that is maitaied as isetios take place; is iitially chose to be 1 = 1, ad duig the kth isetio, k > 1, is give by k = 1 k 1 1 j<k i j. The quatity k is at least 1. Fo completeess, we give the poof of the followig lemma, which is i [7]. Lemma 1 Ou isetio sot algoithm pefoms at most lg Iv(X) + O() compaisos to sot a iput X of legth. Poof. Defie E(k), the excess umbe of compaisos pefomed duig the fist k isetios, to be the actual umbe pefomed mius k lg( 1 k 1 j k i j ). We demostate that E(k) = O() whe k =. We poceed to estimate E(k + 1) E(k). Let 1 deote the aveage, k+1 1 j k+1 i j, ad let deote the coespodig quatity, 1 k 1 j k i j. The E(k + 1) E(k) = lg + i k+1 (k + 1) lg + k lg + O(1), = i k+1 + (k + 1)(lg lg ) + O(1). (1) Now wite = (k + i k+1 )/(k + 1) = (k/(k + 1)) g, whee g = 1 + i k+1 /(k ). Substitutig ito (1) this expessio fo, we obtai E(k + 1) E(k) = i k+1 + (k + 1) lg k + 1 k (k + 1) lg g + O(1).

4 310 The tem (k+1) lg k+1 k is O(1), leavig us to estimate i k+1 / (k+1) lg g. We have two cases: (i) i k+1 k ad (ii) i k+1 > k Fo case (i), usig the fact that lg(1 + x) x fo 0 x 1, we fid that i k+1 / (k + 1) lg g 0 (sice lg g i k+1 /(k ) fo this case). Fo case (ii), we boud i k+1 / (k + 1) lg g fom above usig i k+1. But the coditio fo case (ii), amely i k+1 > k = 1 j k i j, implies that the sum of these i k+1 estimates (ove those k fo which case (ii) applies) is at most twice the last such tem, which is bouded by. Sice E(1) = 0, we coclude that E() = O(). To covet the above costuctio to a implemetable algoithm with total uig time O( log Iv(X) ), we utilize the cosideable feedom available i the choice of the k values i the above costuctio, while pesevig the esult of the pecedig Lemma. Let α 1 be a abitay costat. If we eplace ou choice fo k i the above algoithm by ay quatity s k satisfyig k s k α k, the the above lemma still holds; the cost i k /s k + lg s k + O(1) of a sigle isetio caot gow by moe tha O(1) as s k deviates fom its iitial value k while emaiig i the idicated age. Efficiet Implemetatio At each isetio poit, the peviously iseted items ae ogaized ito a list of cosecutive bads fom left to ight. Evey bad has 1, 2, o 3 AVL tees. Each of ou AVL tees is ogaized as a seach tee with the data items stoed oly i the leaves of the tee while the iteal odes cotai idexig ifomatio. A ak value is assiged to evey bad. The tees of a bad with ak h will have heights equal to h, except fo at most oe tee that may have height equal to h 1. We call a tee whose height is oe less tha the ak of its bad a shot tee. We call a bad that has a shot tee a s-bad. These coditios ae efeed to as the ak coditios. At ay stage of the algoithm, the aks of the bads fom a iceasig cosecutive sequece m, m + 1, m + 2,... fom left to ight, with the value of m chagig though the algoithm. This is efeed to as the mootoicity coditio. With evey isetio, the elevat tee is fist idetified by employig a liea seach though the list of tees fom left to ight. Afte each isetio, the bad list may equie eogaizatio, though o a elatively ifequet basis. The details of this implemetatio is facilitated by defiig the followig opeatios:

5 Split: A AVL tee of height h ca be split i costat time ito two tees, oe of height h 1 ad the othe of height h 1 o h 2, by emovig the oot ode of the give tee. 2. Combie: Two AVL tees, oe of height h 1 ad the othe of height h 1 o h 2, ca be combied i costat time to fom a AVL tee of height h by addig a ew oot ode. The data values of the left tee ae ot lage tha those of the ight tee. 3. Fid lagest: The value of the lagest membe of a give tee ca be accessed i costat time. A poite to the lagest value is maitaied i costat time afte each of the othe opeatios. 4. Tee-isetio: A isetio of a ew value ito a AVL tee of height h ca be pefomed with at most h compaisos, ad i time O(h) [2]. Coside the cost of the sigle opeatio: isetig y ito a soted sequece S = x 1 < x 2 < < x 1. If S is ogaized, as metioed above, i a list of tees, ad y belogs to the ith tee, which is of height h, the the isetio equies o moe tha i + h compaisos. As a esult of a isetio, the height of the tees may icease ad the ak coditios ae to be maitaied. Such a case aises whe the height of a tee, i a bad of ak h, becomes h + 1. This tee is split ito two tees. If, as a esult of this split, we ow have two shot tees i this bad, these two tees ae combied. (If these two tees ae ot adjacet, the heights of the tees i this bad must have eithe the patte h 1, h, h 1 o h 1, h, h, h 1. I eithe case, we split the middle tees, whose heights ae h, the combie evey adjacet pai of the tees. This accouts fo at most 3 splits ad 3 combies.) Othewise, if the umbe of tees of this bad becomes 4, the two ight tees ae combied ad the combied tee is moved to become the left tee of the ext highe bad. This opeatio is efeed to as a pomote opeatio. This combie/pomote may be epeated seveal times though cosecutive bads. We call such a pocess a popagatig pomotio. Besides efocig the ak coditios, we maitai the additioal coditio that just pio to the kth isetio the ak m of the leftmost bad satisfies m = log θ k + 1, (2) whee θ is a paamete of the algoithm whose value will be aalyzed ad detemied late. The value of θ should satisfy 1 < θ 2. If, as a esult of a isetio, k gows such that the cuet value of m is ow equal to log θ k (ote that k may gow by at most 1),

6 312 the a coalescig opeatio is pefomed. The pupose of the coalescig opeatio is to make the ak of the leftmost bad equal to m + 1. The tees of the leftmost iteval ae combied to fom 1 o 2 tees that ae pomoted to the ext bad whose ak is m + 1 (if thee wee 3 tees i the leftmost bad, 2 of them ae combied icludig the shot tee if it exists). If thee was oly oe shot tee of ak m 1 that is pomoted, the leftmost two tees of the bad whose ak is m + 1 will have heights m 1, m o m 1, m + 1. I the fist case, these two tees ae combied. I the secod case, the tee with height m + 1 is fist split poducig a patte of heights that is eithe m 1,m,m o m 1,m 1,m o m 1,m,m 1. Fo the fist two pattes, the leftmost two tees ae combied. Fo the secod patte, the combied tee is futhe combied with the tee to its ight. Fo the thid patte, the tee, whose ak is m, is split ad each of the two esultig adjacet pais is combied (fo a total of at most 2 splits ad 2 combies). If, as a esult of the pomotio, two shot tees of ak m exist i the bad whose ak is m + 1, these two tees ae combied (as above). If the umbe of tees of this bad becomes 4 o 5, a popagatig pomotio is pefomed ad epeated as ecessay though cosecutive bads. As a special case, that does ot affect the bouds o the opeatios of the algoithm, the coalescig opeatio is skipped whe thee is oly oe bad ad the umbe of odes is ot eough to pefom the coalescig opeatio while maitaiig the ak coditios. If, as a esult of a isetio, k dops such that the cuet value of m is ow equal to log θ k + 2 (ote that k may dop by at most 1), the a eductio opeatio is pefomed. The pupose of the eductio opeatio is to make the ak of the leftmost bad equal to m 1. A ew leftmost bad with ak m 1 is ceated. The leftmost tee of the old leftmost bad (the bad with ak m) is moved to the ew bad. We call this opeatio a demote opeatio. If this tee has height m, the it is split. If, as a esult of the demotio, the bad whose ak is m ow has o tees, the leftmost tee of the bad whose ak is m + 1 is demoted ad split if ecessay. This demote/split may be epeated fo seveal times though cosecutive bads. We call such a pocess a popagatig demotio. Note that the eductio ad the coalescig opeatios seve to peseve the ak ad mootoicity coditios as well as (2).

7 313 Aalysis Iv(X) Lemma 2 Ou algoithm pefoms at most log φ + O() compaisos to sot a iput X of legth (φ is the golde atio = ). Poof. I view of Lemma 1, it suffices to show that a give isetio, aivig i positio L, equies at most L/s k + log φ s k + O(1) (3) compaisos, whee k s k α k ad α 1 is a abitay costat. Let m be the ak of the leftmost bad, ad m be the ak of the bad that has the tee ito which the ewly iseted item falls, ad let i be the positio of this tee (umbe of the tee coutig the tees fom the left), so that the total isetio cost is at most i + m. As a esult of the ak ad mootoicity coditios, we have i m m + 1. Next, we boud L fom below as follows. Cotibutig to L, thee is at least 1 tee i each of the bads with aks fom m to m 1. Thee is aothe i d 1 (whee d = m m 0) tees of heights at least m 1. Fo ay AVL tee, the size of a tee of height h is at least φ h. It follows that: L > (i d 1)φ m 1 + m 2 i=m 1 > (i d 2)φ m 1 + φ m 1. Choosig ou paamete to be θ = φ, it follows fom (2) that φ m 1 = s k. Hece L s k > (i d 2) + φ d. This esults i the followig elatio, which implies (3). φ i, i + m < L/s k + m + 2d φ d + 2, < L/s k + log φ s k + O(1). Lemma 3 The time spet by ou algoithm, i pefomig opeatios othe tha compaisos, is O(). Poof. The pimitive opeatios, which the algoithm pefoms othe tha compaisos, ae the spit ad combie opeatios. Each of these

8 314 opeatios equies costat time. Excludig the popagatig pomotio ad demotio, the umbe of splits ad combies pe isetio, coalescig o eductio is costat. Hece, the oly opeatios that eed to be ivestigated ae the popagatig pomotios ad demotios. We use a potetial fuctio [22] to deive the liea bouds o these opeatios. Let N s be the umbe of s-bads ad let N odd be the umbe of bads that have 1 o 3 tees. Let Φ i be the potetial fuctio afte the ith isetio, such that Φ i = c 1 N odd + c 2 N s, whee c 1 ad c 2 ae costats to be detemied ad c 1 > c 2. The value of Φ 0 is 0, ad the value of Φ = O(). What we eed to show is that the diffeece i potetial whe added to the actual amout of wok duig the ith isetio is bouded by a costat. Coside the case whee duig the i + 1 isetio a popagatig pomotio that ivolves t bads takes place. Assume fist that the iitiative of this popagatig pomotio is a isetio that causes a height of a tee to become h + 1 i a bad of ak h. As a esult of a pomotio i a bad, the umbe of tees i this bad should have bee 3, ad becomes 2 afte the pomotio. This should be the case fo the t 1 bads that popagate the pomotio, accoutig fo a decease of t 1 i the value of N odd. I the last bad, the opposite may take place ad N odd may icease by 1 as a esult. Except fo the fist bad, ito which the ewly iseted item falls, the umbe of s-bads may oly decease as a esult of ay of these pomotios. Hece, the amotized cost of the popagatig pomotio i this case is bouded by O(t) c 1 (t 2) + c 2. By selectig c 1 geate tha the costat ivolved i the O() otatio i this boud, the amotized cost of this opeatio is a costat. The aalysis is simila if the iitiative fo the popagatig pomotio is a coalescig opeatio. The oly diffeece is the fist bad that gets pomoted tees, whee the umbe of tees i this bad may emai the same (eithe 2 o 3). This bad may also be coveted to a s-bad as a esult of this pomotio. This leads to a boud of O(t) c 1 (t 3) + c 2, which is a costat as well. Coside the case that duig the i+1 isetio a eductio opeatio is pefomed. Assume that this eductio iitiates a popagatig demotio that ivolves t bads. A ew bad is ceated that may get 1 tee (iceasig N odd by 1), o 2 tees oe of them may be shot (iceasig N s by 1). Fo each of the ext t 2 bads that ivolves a demotio, the umbe of tees i each of these bads should have bee 1 befoe this popagatig demotio. If a demoted tee was ot shot, it is split esultig i 2 tees. This causes N odd to decease by 1, while N s may icease by 1. O the othe had, if a demoted tee was shot, the umbe of tees i the coespodig bad emais 1 afte the demotio, while

9 315 the umbe of s-bads deceases by 1 causig N s to decease by 1. I the last bad, the opposite may take place, ad N odd may icease by 1 as a esult. Hece, the amotized cost of the popagatig demotio is bouded by O(t) (c 1 c 2 )(t k 2) c 2 k + 2c 1, fo some k t 2. By selectig c 2 ad c 1 c 2 geate tha the costat i the O() otatio i this boud, the amotized cost of this opeatio is a costat. We have thus established the followig theoem. Theoem 4 The pecedig isetio sot algoithm sots a iput X of legth i time O( log Iv(X) Iv(X) ), ad pefoms at most log φ + O() compaisos. The space equiemet fo the algoithm is O(). Tuig the paamete I the above aalysis, to pove the boud fo the umbe of compaisos, we have chose the paamete θ = φ. I pactice, this value is a too cosevative value to isue the wost-case behavio. Fo adom sequeces the pefomace of AVL tees is vey efficiet, ad empiical data shows that the aveage height of a AVL tee of odes is about 1.02 log [10, 11]. This motivates usig a lage value of θ. Kowig that the costat facto i the height of a aveage AVL tee is close to 1, the paamete θ ca be chose to be close to 2. Notes 1. lg x is the maximum of log 2 x ad 1. Refeeces [1] A. Adesso ad T. W. Lai. Fast updatig of well-balaced tees. Scadiavia Wokshop o Algoithm Theoy (1990), [2] G. Adelso-Velskii ad E. Ladis. O a ifomatio ogaizatio algoithm. Doklady Akademia Nauk SSSR, 146(1962), [3] M. Bow ad R. Taja. Desig ad aalysis of data stuctues fo epesetig soted lists. SIAM J. Comput. 9 (1980), [4] S. Calsso, C. Levcopoulos ad O. Petesso. Subliea megig ad atual Megesot. Algoithmica 9 (1993), [5] R. Cole. O the dyamic fige cojectue fo splay tees. Pat II: The poof. SIAM J. Comput. 30 (2000), [6] A. Elmasy. Pioity queues, paiig ad adaptive sotig. 29th It. Colloquium fo Automata, Laguages ad Pogammig. I LNCS 2380 (2002),

10 316 [7] A. Elmasy ad M. Fedma. Adaptive sotig ad the ifomatio theoetic lowe boud. Symp. o Theoet. Aspect. Comput. Sc. I LNCS 2607 (2003), [8] V. Estivill-Casto ad D. Wood. A ew measue of pesotedess. Ifo. ad Comput. 83 (1989), [9] L. Guibas, E. McCeight, M. Plass ad J. Robets. A ew epesetatio of liea lists. ACM Symp. o Theoy of Computig 9 (1977), [10] L. Guibas ad R. Sedgewick. A dichomatic famewok fo balaced tees. Foudatios of Compute Sciece (1978), [11] P. Kalto, S. Fulle, R. Scoggs ad E. Kaehle. Pefomace of height-balaced tees. Ifomatio Retieval ad Laguage Pocessig 19(1) (1976), [12] D. Kuth. The at of Compute pogammig. Vol III: Sotig ad Seachig. Addiso-wesley, secod editio (1998). [13] C. Levcopoulos ad O. Petesso. Splitsot - A adaptive sotig algoithm. Ifomatio Pocessig Lettes 39 (1991), [14] C. Levcopoulos ad O. Petesso. Adaptive Heapsot. J. Alg. 14 (1993), [15] C. Levcopoulos ad O. Petesso. Exploitig few ivesios whe sotig: Sequetial ad paallel algoithms. Theoet. Comput. Sciece 163 (1996), [16] H. Maila. Measues of pesotedess ad optimal sotig algoithms. IEEE Tas. Comput. C-34 (1985), [17] K. Mehlho. Data stuctues ad algoithms. Vol.1. Sotig ad Seachig. Spige-Velag, Beli/Heidelbeg. (1984) [18] K. Mehlho. Sotig pesoted files. 4th GI Cofeece o Theoy of Compute Sciece. I LNCS 67 (1979), [19] A. Moffat, G. Eddy ad O. Petesso. Splaysot: fast, vestile, pactical. Softw. Pact. ad Expe. 126(7) (1996), [20] O. Petesso ad A. Moffat. A famewok fo adaptive sotig. Discete App. Math. 59 (1995), [21] D. Sleato ad R. Taja. Self-adjustig biay seach tees. J. ACM 32(3) (1985), [22] R. Taja. Amotized computatioal complexity. SIAM J. Alg. Disc. Meth. 6 (1985), [23] A. Tsakalidis. AVL-tees fo localized seach. If. ad Cot. 67 (1985),

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