Analysis of the Expected Number of Bit Comparisons Required by Quickselect

Transcription

1 Aalyi of the Expected Number of Bit Compario Required by Quickelect Jame Alle Fill Departmet of Applied Mathematic ad Statitic The Joh Hopki Uiverity ad ad Takéhiko Nakama Departmet of Applied Mathematic ad Statitic The Joh Hopki Uiverity ad ABSTRACT Whe algorithm for ortig ad earchig are applied to key that are repreeted a bit trig, we ca quatify the performace of the algorithm ot oly i term of the umber of key compario required by the algorithm but alo i term of the umber of bit compario. Some of the tadard ortig ad earchig algorithm have bee aalyzed with repect to key compario but ot with repect to bit compario. I thi paper, we ivetigate the expected umber of bit compario required by Quickelect alo kow a Fid. We develop exact ad aymptotic formulae for the expected umber of bit compario required to fid the mallet or larget key by Quickelect ad how that the expectatio i aymptotically liear with repect to the umber of key. Similar reult are obtaied for the average cae. For fidig key of arbitrary rak, we derive a exact formula for the expected umber of bit compario that uig ratioal arithmetic require oly fiite ummatio rather tha uch operatio a umerical itegratio ad ue it to compute the expectatio for each target rak. AMS ubect claificatio. Primary 68W4; ecodary 68P, 6C5. Key word ad phrae. Quickelect, Fid, earchig algorithm, aymptotic, average-cae aalyi, key compario, bit compario. Date. March, 9. Reearch for both author upported by NSF grat DMS 464, ad by The Joh Hopki Uiverity Acheo J. Duca Fud for the Advacemet of Reearch i Statitic.

2 INTRODUCTION AND SUMMARY Itroductio ad Summary Whe a algorithm for ortig or earchig i aalyzed, the algorithm i uually regarded either a comparig key pairwie irrepective of the key iteral tructure or a operatig o repreetatio uch a bit trig of key. I the former cae, aalye ofte quatify the performace of the algorithm i term of the umber of key compario required to accomplih the tak; Quickelect alo kow a Fid i a example of thoe algorithm that have bee tudied from thi poit of view. I the latter cae, if key are repreeted a bit trig, the aalye quatify the performace of the algorithm i term of the umber of bit compared util it complete it tak. Digital earch tree, for example, have bee examied from thi perpective. I order to fully quatify the performace of a ortig or earchig algorithm ad eable compario betwee key-baed ad digital algorithm, it i ideal to aalyze the algorithm from both poit of view. However, to date, oly Quickort ha bee aalyzed with both approache; ee Fill ad Jao [3]. Before their tudy, Quickort had bee exteively examied with regard to the umber of key compario performed by the algorithm e.g., Kuth [3], Régier [9], Röler [], Kel ad Szpakowki [], Fill ad Jao [], Neiiger ad Rüchedorf [7], but it had ot bee examied with regard to the umber of bit compario i ortig key repreeted a bit trig. I their tudy, Fill ad Jao aumed that key are idepedetly ad uiformly ditributed over, ad that the key are repreeted a bit trig. [They alo coducted the aalyi for a geeral abolutely cotiuou ditributio over,.] They howed that the expected umber of bit compario required to ort key i aymptotically equivalet to l lg a compared to the lead-order term of the expected umber of key compario, which i aymptotically l. We ue l ad lg to deote atural ad biary logarithm, repectively, ad ue log whe the bae doe ot matter for example, i remaider etimate. I thi paper, we ivetigate the expected umber of bit compario required by Quickelect. Hoare [8] itroduced thi earch algorithm, which i treated i mot textbook o algorithm ad data tructure. Quickelect elect the m-th mallet key we call it the rak-m key from a et of ditict key. The key are typically aumed to be ditict, but the algorithm till work with a mior adutmet eve if they are ot ditict. The algorithm fid the target key i a recurive ad radom fahio. Firt, it elect a pivot uiformly at radom from key. Let k deote the rak of the pivot. If k m, the the algorithm retur the pivot. If k > m, the the algorithm recurively operate o the et of key maller tha the pivot ad retur the rak-m key. Similarly, if k < m, the the algorithm recurively operate o the et of key larger tha the pivot ad retur the k m-th mallet key from the ubet. Although previou tudie e.g., Kuth [], Mahmoud et al. [5], Prodier [8], Grübel ad U. Röler [7], Let ad Mahmoud [4], Mahmoud ad Smythe [6], Devroye [], Hwag ad Tai [9] examied Quickelect with regard to key compario, thi tudy i the firt to aalyze the bit complexity of the algorithm. We uppoe that the algorithm i applied to ditict key that are repreeted a bit trig ad that the algorithm operate o idividual bit i order to fid a target key. We alo aume that the key are uiformly ad idepedetly ditributed i,. For itace, coider applyig Quickelect to fid the mallet key amog three key k, k, ad k 3 whoe biary repreetatio are....,...., ad...., re-

3 INTRODUCTION AND SUMMARY pectively. If the algorithm elect k 3 a a pivot, the it compare each of k ad k to k 3 i order to determie the rak of k 3. Whe k ad k 3 are compared, the algorithm require bit compario to determie that k 3 i maller tha k becaue the two key have the ame firt digit ad differ at the ecod digit. Similarly, whe k ad k 3 are compared, the algorithm require 4 bit compario to determie that k 3 i maller tha k. After thee compario, key k 3 ha bee idetified a mallet. Hece the earch for the mallet key require a total of 6 bit compario reultig from the two key compario. We let µm, deote the expected umber of bit compario required to fid the rak-m key i a file of key by Quickelect. By ymmetry, µm, µ + m,. Firt, we develop exact ad aymptotic formulae for µ, µ,, the expected umber of bit compario required to fid the mallet key by Quickelect, a ummarized i the followig theorem. Theorem.. The expected umber µ, of bit compario required by Quickelect to fid the mallet key i a file of key that are idepedetly ad uiformly ditributed i, ha the followig exact ad aymptotic expreio: µ, H + c l l B + l +. l + O,. H ad B deote harmoic ad Beroulli umber, repectively, ad k c : + k l k k With χ k : πik l expreed a ad γ : Euler cotat..577, the cotat c ca alteratively be c γ 9 l 4 l k Z\{} ζ χ k Γ χ k Γ4 χ k χ k..4 The aymptotic formula how that the expected umber of bit compario i aymptotically liear i with lead-order coefficiet approximately equal to Hece the expected umber of bit compario i aymptotically differet from that of key compario required to fid the mallet key oly by a cotat factor the expectatio for key compario i aymptotically. Detail of the derivatio of the formulae are decribed i Sectio 3. Complex-aalytical method are utilized to obtai the aymptotic formula. with c i the form.4 ad eem to be idipeable for obtaiig aymptotic beyod the lead term. [We remark that, although it ivolve the imagiary umber χ k, the expreio.4 i real becaue the term with idice k ad k are complex cougate.] I Sectio 3. we agai ue complex-aalytical method to reexpre.4 i the form.3. Havig doe all thi we upected that there mut be a purely real-aalytical way to obtai directly the lead-order aymptotic µ, c with c i the form.3. Ideed, there i: See Remark 3.3.

4 PRELIMINARIES 3 I Sectio 4 ad 5 we move o to derive exact ad aymptotic expreio for the expected umber of bit compario for the average cae. We deote thi expectatio by µ m,. I the average cae, the parameter m i µm, i coidered a dicrete uiform radom variable; hece µ m, m µm,. The derived aymptotic formula how that µ m, i alo aymptotically liear i ; ee More detailed reult for µ m, are decribed i Sectio 4. Latly, i Sectio 5, we derive a exact expreio of µm, for each fixed m that i uited for computatio. Our prelimiary exact formula for µm, [how i.8] etail ifiite ummatio ad itegratio. A a reult, it i ot a deirable form for umerically computig the expected umber of bit compario. Hece we etablih aother exact formula that oly require fiite ummatio ad ue it to compute µm, for m,...,,,..., 5. The computatio lead to the followig coecture: i for fixed, µm, icreae i m for m + ad i ymmetric about + ; ad ii for fixed m, µm, icreae i aymptotically liearly. Prelimiarie To ivetigate the bit complexity of Quickelect, we follow the geeral approach developed by Fill ad Jao [3]. Let U,..., U deote the key uiformly ad idepedetly ditributed o,, ad let U i deote the rak-i key. The, for i < aume, P {U i ad U are compared} m + i + m i + if m i if i < m < if m.. To determie the firt probability i., ote that U m,..., U remai i the ame ubet util the firt time that oe of them i choe a a pivot. Therefore, U i ad U are compared if ad oly if the firt pivot choe from U m,..., U i either U i or U. Aalogou argumet etablih the other two cae. For < < t <, it i well kow that the oit deity fuctio of U i ad U i give by f Ui,U, t : i t i t.. i,, i,,

5 3 ANALYSIS OF µ, 4 Clearly, the evet that U i ad U are compared i idepedet of the radom variable U i ad U. Hece, defiig P, t, m, P, t, m, P 3, t, m, m i< i<m< i< m m + f U i,u, t,.3 i + f U i,u, t,.4 m i + f U i,u, t,.5 P, t, m, P, t, m, + P, t, m, + P 3, t, m,.6 [the um i.3.5 are double um over i ad ], ad lettig β, t deote the idex of the firt bit at which the key ad t differ, we ca write the expectatio µm, of the umber of bit compario required to fid the rak-m key i a file of key a µm, β, tp, t, m, dt d.7 k l k l k k l l k l k k + P, t, m, dt d;.8 i thi expreio, ote that k repreet the lat bit at which ad t agree. 3 Aalyi of µ, I Sectio 3., we derive the exact expreio for µ, how i Theorem.. I Sectio 3., we prove the aymptotic reult tated i Theorem.. 3. Exact Computatio of µ, Sice the cotributio of P, t, m, or P 3, t, m, to P, t, m, i zero for m, we have P, t,, P, t,, [ee.4 through.6]. Let x :, y : t, z : t. The P, t,, z x i y i z i,, i,, i< z z η i< i,, i,, z η x + y + η dη z z z x i y i η dη η 3 t η + dη. 3.

6 3 ANALYSIS OF µ, 5 Makig the chage of variable v t η + ad itegratig, ad recallig z t, we fid, after ome calculatio, From.8 ad 3., µ, k + k k + k k + k k + k P, t,, k l k l k t. 3. l l k k l k l k l k l k l l k l k P, t,, dt d l k t dt d l k t [l l k l k ] dt k k {l k [l k ] } k + k k k [l l ]. 3.3 l To further traform 3.3, defie a,r B r r r if r if r if r, 3.4 B r deote the r-th Beroulli umber. Let S, : l l. The S, ee Kuth [3], ad k l [l l ] S k, k l S k, S k+, S k, S k, S k+, r From 3.3 ad 3.5, a,r k r µ, r l a,r k+ r r r a,r r a,r k r r. 3.5 k + k a,r k r r. k r

7 3 ANALYSIS OF µ, 6 Here Hece µ, k + k a,r k r r k r r k + a,r kr r k r a,r r k + kr a,r r. The um r r r r r k r a,r r r + r B r r r + r r B r r r r+ r r+ r r a,r r r. r 3.6 ca be implified a follow: r r r r r r r r r + r r r r r r + r. 3.7 r Pluggig 3.7 ito 3.6 ad recallig B k+ for k, we fially obtai [ µ, + r B r r r + r ] r r r r r + + B H + t, 3.8 H deote the -th harmoic umber ad [ B ] t :. 3.9 The lat equality i 3.8 follow from the eay idetity k k H. k k

8 3 ANALYSIS OF µ, 7 3. Aymptotic Aalyi of µ, I order to obtai a aymptotic expreio for µ,, we aalyze t i The followig lemma provide a exact expreio for t that eaily lead to a aymptotic expreio for µ, : Lemma 3.. For, let u : t + t with t ad v : u + u. Let γ deote Euler cotat..577, ad defie χ k : πik l. The i v H+ + + l γ l Σ, + + Σ : k Z\{} ζ χ k Γ + Γ χ k ; l Γ + 3 χ k ii u H + a H + γ l + + l + + Σ, a : 4 9 Σ : + 7 6γ 8 l l k Z\{} k Z\{} ζ χ k Γ χ k l χ k ζ χ k Γ χ k Γ4 χ k χ k, Γ + Γ + χ k ; iii t H + a [ H + H 7 ] l γ + l H 3 + b Σ, b : Σ : k Z\{} k Z\{} ζ χ k Γ χ k l χ k Γ3 χ k, ζ χ k Γ χ k Γ + l χ k Γ + χ k, ad H deote the -th Harmoic umber of order, i.e., H : i i.

9 3 ANALYSIS OF µ, 8 I thi lemma, u ad v are derived i order to obtai the exact expreio for t i iii. From 3.8, the exact expreio for t alo provide a alterative exact expreio for µ,. Before provig Lemma 3., we complete the proof of Theorem. uig part iii. We kow H l + γ + + O 4, 3. H π O Combiig with 3.8 ad Lemma 3.iii, we obtai a aymptotic expreio for µ, : µ, a l l l + l + O. 3. The term O i 3. ha fluctuatio of mall magitude due to Σ, which i periodic i log with amplitude maller tha.. Thu, a how i Theorem., the aymptotic lope i 3. i c a 8 9 Let S deote the um i c: S : k Z\{} + 7 6γ 9 l 4 l k Z\{} ζ χ k Γ χ k Γ4 χ k χ k k Z\{} ζ χ k Γ χ k Γ4 χ k χ k. 3.3 ζ χ k 3 χ k χ k χ k, 3.4 the formula Γ+x xγx i ued to derive the ecod expreio. Both expreio ivolve the imagiary umber χ k, but S i a real umber. We ivetigate S ad expre it uig oly real fuctio. We have the followig reult: Theorem 3.. Let S be the um defied at 3.4. The ad S : S l S ρ, 3.5 k h k, hm : m l m l m! m m, 3.6 k ρ : 7 6γ 36 l.

10 3 ANALYSIS OF µ, 9 Proof of Theorem 3.. Chooe ad fix < θ <. We how that the itegral J : θ+i θ i ζ 3 d equal πi S o the oe had ad equal πi[ρ + S/ l ] o the other had. Equatig thee two expreio give the deired reult. To get the firt expreio for J, we calculate J k θ+i θ i ζ k 3 d θ+i k ζt kt t + t + t dt. But, for ay poitive iteger m ad ay α >, θ+i θ i ζtm t t + t + t dt α+i α i k θ i ζtm t t + t + t dt πire t [ ζtm t ] t, + t + t which follow from reidue calculu, takig ito accout the cotributio of the imple pole of the itegrad at. Here [ ζtm t ] Re t t + t + t 6 m ad Further, ice α+i α i we have by Melli iverio that ζtm t t + t + t dt α+i α i /m t t + t + t dt. t t + t + 3/4 + / t t + t + /4 t +, equal πi α+i α i x t t + t + t dt fx : 3 4 l x + x 4 x for x ad equal for x. Note that thi require oly α >. So α+i ζtm t α i t + t + t dt πi [ f πi m m l m l m! 3 m ] 4 m πi [hm + 6 ] m, the um i over m or m, ad therefore θ+i θ i ζtm t t dt πihm. + t + t

11 3 ANALYSIS OF µ, Thu we obtai our firt expreio for J. We remark that the erie S coverge geometrically rapidly. To obtai the ecod expreio for J we move the horizotal i.e., real coordiate of the vertical lie of itegratio over from θ to C C i large poitive umber C. By reidue calculu, we fid a deired. [ J πi {Re ζ 3 ] + S } πi ρ + S, l l Uig Theorem 3. it i traightforward to derive the alterative expreio k c + k l 3.7 k for the liear coefficiet c i 3.3. Graber ad Prodiger [6] obtaied a earlier draft of thi maucript ad idepedetly coducted a imilar aalyi of S leadig to 3.7. They alo howed how to compute c efficietly to high preciio ad i particular computed c to 5 decimal place. Remark 3.3. The lead-order aymptotic µ, c with c i the form 3.7 ca alo be obtaied imply uig real-aalytical argumet. Start with.7 with m ad recall that P, t,, P, t,, i give by 3. to ee that µ, t k β, tt [ t + t] d dt. A eay domiated-covergece argumet the how that µ, c with c give i the itegral form Writig β, t c t β, tt d dt. ad t agree i their firt k bit k ad breakig up the double itegral accordig to the firt k bit of t lead to the ummatio form 3.7 of c. We omit the detail. We do ot kow how to obtai aymptotic for µ, beyod the lead term by thi ort of approach. Now we prove Lemma 3.: Proof of Lemma 3.. i Sice [ B + + ] u t + t [ ] B, B [ ]

12 3 ANALYSIS OF µ, it follow that + v u + u B [ ] [ ] B B k+ k k + k + [ k+ ] k k ζ k k k + [ k+ 3.8 ] k ζ! πi + [ + d, 3.9 ] C C i a poitively orieted cloed curve that ecircle the iteger,..., ad doe ot iclude or ecircle ay of the followig poit: + χ k χ k : πik l, k Z; ; ad. Equality 3.8 follow from the fact that the Beroulli umber are extrapolated by the Riema zeta fuctio take at oegative iteger: B k kζ k. [The coefficiet k do ot cocer u ice the Beroulli umber of odd idex greater tha vaih.] Equality 3.9 follow from a direct applicatio of reidue calculu, takig ito accout cotributio of the imple pole at the iteger,...,. Let φ deote the itegrad i 3.9: ζ! φ + [ + ]. We coider a poitively orieted rectagular cotour C l with horizotal ide Im λ l ad Im λ l, λ l : l+π l, l Z +, ad vertical ide Re θ ad Re λ l, < θ <. By elemetary boud o φ alog C l ad the fact that θ+i θ i φ d 3. thi i implicit o page 3 of Flaolet ad Sedgewick [5] ad explicitly proved i the Appedix, oe ca how that lim φ d. l C l Accoutig for reidue due to the pole ecircled by C l, we obtai v + Re [φ] + Re [φ] + Re +χk [φ] k Z\{} H+ + + l γ l Σ, Σ : k Z\{} ζ χ k Γ + Γ χ k. 3. l Γ + 3 χ k

13 3 ANALYSIS OF µ, ii We have u t 3 t t 3 9. Hece, from i, u u + v 9 + Here 9 v + + l 9 H H + l H + γ + + l H + γ + + l γ γ 3 l H + l + + l H Σ Σ Σ. 3.3 H H + H H H + H H H + +, 3.5 we aume 3 for 3.4, but 3.5 hold alo for. I regard to Σ, ote that Σ k Z\{} ζ χ k Γ χ k l χ k [ Γ + Γ + Γ + 3 χ k Γ + χ k ], o that Defie Σ k Z\{} Σ : ζ χ k Γ χ k l χ k k Z\{} ζ χ k Γ χ k l χ k [ Γ + Γ + χ k Γ3 ]. 3.6 Γ4 χ k Γ + Γ + χ k. 3.7 The, combiig 3.3, 3.5, ad 3.6, we obtai u H + a H + l + + γ l + + Σ,

14 4 ANALYSIS OF THE AVERAGE CASE: µ m, 3 a : γ 8 l l k Z\{} ζ χ k Γ χ k Γ4 χ k χ k. 3.8 iii Cloely followig the derivatio of u decribed above, we obtai for t t + u u H + a l 3 H γ + l [ H + H 7 H 3 + H + a l γ + l H 3 + b Σ, 3.9 ] Σ b : Σ : k Z\{} k Z\{} ζ χ k Γ χ k l χ k Γ3 χ k, 3.3 ζ χ k Γ χ k Γ + l χ k Γ + χ k Aalyi of the Average Cae: µ m, 4. Exact Computatio of µ m, Here we coider the parameter m i µm, a a dicrete radom variable with probability ma fuctio P {m i}, i,,...,, ad average over m while the parameter i fixed. Thu, uig the otatio defied i.3 through.7, µ m, m, for l,, 3, µm, β, tp, t, m, dt d m β, t P, t, m, dt d µ m, + µ m, + µ 3 m,, m µ l m, β, t P l, t, m, dt d. 4. m

15 4 ANALYSIS OF THE AVERAGE CASE: µ m, 4 Here µ m, µ 3 m,, ice P 3 t,, m +, P, t, m, by a eay ymmetry argumet we omit, ad o Therefore µ 3 m, β, t P 3, t, m, dt d m β t, P 3 t,, m +, dt d m β, t P, t, m, dt d µ m,. m µ m, µ m, + µ m,, 4. ad we will compute µ m, ad µ m, exactly i Sectio Exact Computatio of µ m, We ue the followig lemma i order to compute µ m, exactly: Lemma 4.. β, t P, t, m, dt d m B + Before provig the lemma, we complete the computatio of µ m,. Note that µ m, β, t µ, + P, t, m, dt d m β, t P, t,, dt d + β, t β, t P, t, m, dt d. m P, t, m, dt d m

16 4 ANALYSIS OF THE AVERAGE CASE: µ m, 5 Therefore, by 3.8 ad Lemma 4., we obtai µ m, B + 3 the ecod equality hold ice B B 3 B , 4.3 +!!!!!! 3 [! +!! ] I Sectio 4.. we combie the expreio for µ m, i 4.3 with a imilar expreio for µ m, to obtai a exact expreio for µ m,. The remaider of thi ectio i devoted to provig Lemma 4.. For thi, the followig expreio for P, t, m, will prove ueful: Lemma 4.. Let m ad let x :, y : t, z : t. The the quatity P, t, m, defied at.3 atifie P, t, m, x ξ + y [Υ m,, ξ, x, y, z Υ m,, ξ, x, y, z + Υ 3 m,, ξ, x, y, z] dξ, 4.4

17 4 ANALYSIS OF THE AVERAGE CASE: µ m, 6 Υ m,, ξ, x, y, z : Υ m,, ξ, x, y, z : Υ 3 m,, ξ, x, y, z : x ξ m mξ + y + z m+, m x ξ m m + zξ + y + z m, m x ξ m z m+. m Proof of Lemma 4.. By..3, P, t, m,! m + i! i!! xi y i z m i<! m i m! x i y i z m + m! i m, i, i! m i<! m i m! x i y i z. m! m + i m, i, i! m i< 4.5 I order to compactly decribe the derivatio of 4.4, we defie the followig idefiite itegratio operator T : T fx : x fξ dξ. We really hould write T fx rather tha T fx, but we would like to ue horthad uch a T x x+ + whe >. The operator T treat it argumet f a a fuctio of x; the other variable ivolved i f amely, y ad z are treated a cotat. The otatio T l will deote the l-th iterate of T. I thi otatio, for m < i, i m! i! x i T m x i m, ad the um i 4.5 equal T m m i< m + m i m, i, x i m y i z. Here m + z z m+ z η m+ dη,

18 4 ANALYSIS OF THE AVERAGE CASE: µ m, 7 o m x i m y i z m + i m, i, m i< z m+ T m m x i m y i η +m dη z i m, i, m i< z m+ T m η +m x + y + η m dη z t m z m+ T m η 3 η + dη 4.6 T m z ote that x + y t. Makig the chage of variable v t η + ad itegratig, we obtai, after ome computatio, z η 3 t η + m dη t m + m From 4.5 ad , [ m + t m+ m + + t ] m +. z z 4.7 P, t, m,! m +! T m t [ mz + t m+ m + zz + t m + z m+ ]. Here 4.8 t [ mz + t m+ m + zz + t m + z m+ ] m+ r t r Υm,, r, z, 4.9 m + m Υm,, r, z : m z m+ r m + z m+ r. 4. r r The, ice t x + y, m+ r t r Υm,, r, z m+ r r r Υm,, r, z x y r. 4.

19 4 ANALYSIS OF THE AVERAGE CASE: µ m, 8 From , P, t, m,! m +! T m r m+ r r r Υm,, r, z x y r m+! r r Υm,, r, z y r T m x m +! m+! r r Υm,, r, z y r x +m m +! + + m. r 4. Becaue of the partial fractio expaio it follow that m + + m m! l r r y r x +m + + m r r y From , r x+m m! m l l m l + l + m m x x l m l m! l l m m x x l m l m! l m! l x! P, t, m, m +!m! x m r x m l m l + l +, r y r ξ dξ ξ l r ξ l ξ + y r dξ x ξ m ξ + y r dξ. 4.3 m+ m+ r x ξ m ξ + y Υm,, r, z x x ξ m ξ + y r dξ Υm,, r, zx ξ m ξ + y r dξ m+ r Υm,, r, zξ + y r dξ. 4.4

20 4 ANALYSIS OF THE AVERAGE CASE: µ m, 9 Here, by 4., m+ r Υm,, r, zξ + y r m+ r m [ m m+ r m + r m + r z m+ r m + m r ξ + y r z m+ r m + m[ξ + y + z m+ z m+ m + ξ + yz m ] m + z[ξ + y + z m z m mξ + yz m ] z m+ r ] ξ + y r m+ r m r ξ + y r z m+ r mξ + y + z m+ m + zξ + y + z m + z m Subtitutio of 4.5 ito 4.4 give the deired 4.4. Proof of Lemma 4.. From Lemma 4., we have P, t, m, m x ξ + y [Υ m,, ξ, x, y, z Υ m,, ξ, x, y, z + Υ 3 m,, ξ, x, y, z] dξ. m Here Υ m,, ξ, x, y, z ξ + y + z m ξ + y + z d dw [ d dw m ] x ξ m w m m wξ+y+z { w [x ξ + w x ξ ] } wξ+y+z 4.6 ξ + y + z { w [x ξ + w x ξ ] + w x ξ + w } wξ+y+z x ξ + ξ + y + z 4.7 ote that x + y + z. Similarly, [ Υ m,, ξ, x, y, z z d dw m ad m m ] x ξ m w m+ m wξ+y+z z d [ x ξ + w x ξ ] z [ x ξ + w dw wξ+y+z ] wξ+y+z z, 4.8 Υ 3 m,, ξ, x, y, z m x ξ m z m+ x ξ + z x ξ. m

21 4 ANALYSIS OF THE AVERAGE CASE: µ m, Hece [Υ m,, ξ, x, y, z Υ m,, ξ, x, y, z + Υ 3 m,, ξ, x, y, z] m ξ + y + x ξ + z. 4.9 Therefore, from 4.6 ad 4.9, we obtai m x P, t, m, ξ + y [ ξ + y + x ξ + z ] dξ x x ξ + y { ξ + y + [ ξ + y] } dξ ξ + y ξ + y dξ x + y y x ξ + y dξ t t. 4. We complete the proof by uig 4. to compute β, t m P, t, m, d dt. We have β, t P, t, m, d dt m t β, t t dt d β, t β, t t dt d t Cloely followig the derivatio how i , oe ca how that β, t t dt d Thu, i order to complete the proof, it remai to how that t β, t dt d B dt d

22 4 ANALYSIS OF THE AVERAGE CASE: µ m, Ideed, we have β, t t k+ k + k k k+ k + k k k k k+ k k+ dt d k k + k v t dt d v dv d k v V k+ [ k+ v] W d dv. 4.4 Here k v V k+ [ k+ v] W { d v if v k+ k v if k+ < v k. 4.5 Thu k v k v V k+ [ k+ v] W From 4.4 ad 4.6, we obtai ad 4.3 i proved. β, t d dv [ k+ ] k v dv + v k v dv k+ k t dt d k + k k+ + k + k + k + +, 4.7 k

23 4 ANALYSIS OF THE AVERAGE CASE: µ m, 4.. Exact Computatio of µ m, ad µ m, The derivatio for obtaiig a computatioally preferable exact expreio for µ m, are etirely aalogou to thoe for µ m, decribed i the previou ectio Sectio 4... Thu we omit detail. A decribed i Sectio 3., P, t, m, i zero for m ad for m, o, from 4., µ m, β, t m P, t, m, dt d. 4.8 Therefore we firt derive a computatioally deirable expreio for Agai, let x :, y : t, z : t. The P, t, m, m m i m< m i + S m,, x, y, z i,, i,, m S m,, x, y, z m P, t, m,. x i y i z m S 3 m,, x, y, z, 4.9 S m,, x, y, z : S m,, x, y, z : S 3 m,, x, y, z : i< m i< i< m x i y i z, i + i,, i,, x i y i z, i + i,, i,, x i y i z. i + i,, i,, Fill ad Jao [3] howed that S m,, x, y, z t. Hece S m,, x, y, z m t. 4.3 Followig the derivatio how i 4.5 through 4., oe ca how that S m,, x, y, z m y x[x + z + y ] 4.3 t {[ t ] + t } t. 4.3

24 4 ANALYSIS OF THE AVERAGE CASE: µ m, 3 To obtai a imilar expreio for m S 3m,, x, y, z, we ote that, lettig m : + m, i : +, : + i, S 3 m,, x, y, z i +,, i,, i x y i z i Thu m i < S + m,, z, y, x. S 3 m,, x, y, z m Ipectig , we fid m m S + m,, z, y, x S m,, z, y, x m S 3 m,, x, y, z t From 4.9, 4.3, 4.3, ad 4.34, P, t, m, m t t t t t + Hece, from 4.8 ad 4.35, µ m, + t t t β, t β, t β, t t dt d t dt d t dt d. 4.36

25 4 ANALYSIS OF THE AVERAGE CASE: µ m, 4 Fill ad Jao [3] howed that β, t t dt d [ ] A careful term-by-term ipectio of the derivatio how i reveal that β, t β, t Combiig , we obtai µ m, 4 t dt d t dt d [ + ] [ ] [ ], [ + ] Fially, we complete the exact computatio of µ m,. From 4., 4.3, ad 4.4, we have µ m, µ m, + µ m, B B [ ] We rewrite or combie ome of the term i 4.4 for the aymptotic aalyi of µ m,

26 4 ANALYSIS OF THE AVERAGE CASE: µ m, 5 decribed i the ext ectio. We defie F :, F : F 3 : F 4 : F 5 : 3 3 B, B [ [ ]. 3 ] [ The ecod, third, fourth, ad fifth term i 4.4 ca be writte a 8 F 4, F, 4 9 F 3, ad 4F 4, repectively. The lat three term i 4.4 ca be combied a follow: [ + ] 4 [ ] 4 [ + ] 3 8 [ ] 8 F 5. Therefore 3 µ m, 8 F + 4 F F 3 4F F Aymptotic Aalyi of µ m, We derive a aymptotic expreio for µ m, how i 4.4. The computatio decribed i thi ectio are aalogou to thoe i Sectio 3.. Hece we merely ketch detail to derive the aymptotic expreio. Firt, we aalyze F. A routie complex-aalytical argumet imilar to but much eaier tha the oe decribed i Sectio 3. how that [ ] F +! Re k k + [, 3 + H + H 5 ] H H 5 l + 4 γ l + γ + + O ],

27 4 ANALYSIS OF THE AVERAGE CASE: µ m, 6 Sice F i equal to t, which i defied at 3.9 ad aalyzed i Sectio 3., we already have a aymptotic expreio for F. Next we derive a aymptotic expreio for F 3 : { } F 3! Re k [ ] k H H + l + γ l + O To obtai a aymptotic expreio for F 4, we cloely follow the approach of Sectio 3.. Let ũ : F 4 + F 4. The [ ] B ũ. 3 Let ṽ : ũ + ũ. The, by computatio imilar to thoe performed for v i Sectio 3., ṽ k ζ k k + k + [ k+3 ] k k 3 { } + ζ! Re k + + [ +3 ] [ ] k { + + ζ! Re 3+χk + + [ +3 ] [ ] Hece k Z\{} ũ ũ + ṽ ξ : k Z\{} 9 H + ã + ξ l [ γ l H + + ζ χ k Γ χ k Γ. l Γ + 3 χ k ] ξ, H H l γ + 4 l +, } ã : ξ : 7 36 l 4 7 γ l k Z\{} k Z\{} ζ χ k Γ χ k Γ l χ k Γ + χ k. ζ χ k Γ χ k l χ k Γ4 χ k,

28 4 ANALYSIS OF THE AVERAGE CASE: µ m, 7 Thu F 4 F 4 + ũ 9 H H + ã l 3 + l γ 8 l ã + b ξ + H 3 + l γ + l 4 l, 4.45 b : ξ : k Z\{} k Z\{} ζ χ k Γ χ k l χ k χ k Γ3 χ k, ζ χ k Γ χ k Γ l χ k χ k Γ + χ k. Therefore F 4 9 l + ã + 9 γ + 8 l + O Fially, we aalyze F 5. By computatio that are etirely aalogou to thoe performed for F, F, ad F 4, { } F 5 +! Re k [ ] 3 k { } + +! Re +χk [ ] 3 k Z\{} k Z\{} 4 H l H l 3 [ l H + H + l l H + l + l ] + Γ χ k Γ + l χ k χ k Γ χ k l l γ l l + l l + O Therefore, from ad , we obtai the followig aymptotic formula for µ m, : µ m, 4 + l ã 4 l l + 4 l l + O The aymptotic lope 4 + l ã i approximately 8.73.

29 5 DERIVATION OF A CLOSED FORMULA FOR µm, 8 5 Derivatio of a Cloed Formula for µm, The exact expreio for µm, obtaied i Sectio [ee.8] ivolve ifiite ummatio ad itegratio. Hece it i ot a preferable form for umerically computig the expectatio. I thi ectio, we etablih aother exact expreio for µm, that oly ivolve fiite ummatio. We alo ue the formula to compute µm, for m,...,,,...,. A decribed i Sectio, it follow from equatio.6.8 that, for q,, 3, µ q m, : µm, µ m, + µ m, + µ 3 m,, 5. k k l l k l k l k l k k + P q, t, m, dt d. 5. The ame techique ca be applied to elimiate the ifiite ummatio ad itegratio from each µ q m,. We decribe the techique for obtaiig a cloed expreio of µ m, i detail. Firt, we traform P, t, m, how i.3 o that we ca elimiate the itegratio i µ m,. Defie C i, : m i < m + i,, i,,, 5.3 m i < i a idicator fuctio that equal if the evet i brace hold ad otherwie. Sice i t i t i i t i u i u i u v t v, u v u v it follow that P, t, m, m i< m i< fm C i, C i, f h i u i u v v f fi h f i f t h f i + u t v+u i u v h + f + h i + f t h C f, h, 5.4 f+ C f, h : f+h+ im f+ i C i, f i + h + f + h i +.

30 5 DERIVATION OF A CLOSED FORMULA FOR µm, 9 Thu, from 5. ad 5.4, we ca elimiate the itegratio i µ m, ad expre it uig polyomial i l: µ m, fm Note that f h C 3 f, h k + k k l C 3 f, h : l h+ l h+ l f+ l f+ kf+h+ [l h+ l h+ ][l f+ l f+ ], + f + C f, h. h h + l h+, f [ f + l f+ ] f Hece [ l h+ l ] [ h+ l ] f+ l f+ f h f + h + which ca be rearraged to C 4 f, h, : f+h + f+h f+h+ h + V f W h Therefore, from , we obtai [ ] f+ h+ l +, C 4 f, h, l, 5.6 f + h + [ ] f+. µ m, fm fm f h f h C 3 f, h f+h+ k + k C 5 f, h, k l k + k f+h+ kf+h+ kf+h+ k C 4 f, h, l l, l

31 6 DISCUSSION 3 C 5 f, h, : C 3 f, h C 4 f, h,. Here, a decribed i Sectio 3., k l a,r k r, l r a,r i defied by 3.4. Now defie C 6 f, h,, r : a,r C 5 f, h,. The µ m, C 7 a : fm fm f h f h f+h+ f+h+ a,r C 5 f, h, k + kf+h++r r k C 6 f, h,, r[ f+h++r ] r C 7 a a, 5.7 a fm f hα f+h+ β C 6 f, h,, a + f + h +, i which α : a f ad β : f + h + a. The procedure decribed above ca be applied to derive aalogou exact formulae for µ m, ad µ 3 m,. I order to derive the aalogou exact formula for µ m,, oe eed oly tart the derivatio by chagig the idicator fuctio i C i, [ee 5.3] to i < m < ad follow each tep of the procedure; for µ 3 m,, tart the derivatio by chagig the idicator fuctio to i < m. Uig the cloed exact formulae of µ m,, µ m,, ad µ 3 m,, we computed µm, for, 3,..., ad m,,...,. Figure how the reult, which ugget the followig: i for fixed, µm, icreae i m for m + ad i ymmetric about + ; ii for fixed m, µm, icreae i ; iii max m µm, i aymptotically liear i. 6 Dicuio Our ivetigatio of the bit complexity of Quickelect revealed that the expected umber of bit compario required by Quickelect to fid the mallet or larget key from a et of key i aymptotically liear i with the aymptotic lope approximately equal to Hece aymptotically it differ from the expected umber of key compario to achieve the ame tak oly by a cotat factor. The expectatio for key compario i aymptotically ; ee Kuth [] ad Mahmoud et al. [5]. Thi reult i rather cotrative to the

32 6 DISCUSSION 3 Expectatio of bit compario 8 μm, m Figure : Expected umber of bit compario for Quickelect. The cloed formulae for µ m,, µ m,, ad µ 3 m, were ued to compute µm, for,,..., repreet the umber of key ad m,,..., m repreet the rak of the target key.

33 7 APPENDIX 3 Quickort cae i which ee Fill ad Jao [3] the expected umber of bit compario i aymptotically l lg a the expected umber of key compario i aymptotically l. Our aalyi alo howed that the expected umber of bit compario for the average cae remai aymptotically liear i with the lead-order coefficiet approximately equal to Agai, the expected umber i aymptotically differet from that of key compario for the average cae oly by a cotat factor. The expected umber of key compario for the average cae i aymptotically 3; ee Mahmoud et al. [5]. Although we have yet to etablih a formula aalogou to 3.8 ad 4.4 for the expected umber of bit compario to fid the m-th key for fixed m, we etablihed a exact expreio that oly require fiite ummatio ad ued it to obtai the reult how i Figure. However, the formula remai complex. Writte a a igle expreio, µm, i a eve-fold um of rather elemetary term with each um havig order term i the wort cae; i thi ee, the ruig time of the algorithm for computig µm, i of order 7. The expreio for µm, doe ot allow u to derive a aymptotic formula for it or to prove the three coecture decribed at the ed of Sectio 5. The ituatio i ubtatially better for the expected umber of key compario to fid the m- th key from a et of key; Kuth [] howed that the expectatio ca be writte a [ H m + H m + 3 mh + m ]. I thi paper, we coidered idepedet ad uiformly ditributed key i,. I thi cae, each bit i bit trig i with probability.5. Buildig o the preet work ad that of Fill ad Jao [3], much more geeral key-ditributio are treated by Vallée et al. []. Their geeralizatio further elucidate the complexity of Quickelect ad other algorithm. Ackowledgmet. We thak Philippe Flaolet, Svate Jao, ad Helmut Prodiger for helpful dicuio. 7 Appedix I order to prove 3., it uffice to how that, for ay poitive iteger m, θ+i θ i ζ m d + ote that ad < θ <. Lettig t :, it i thu ufficiet to how that J : ++θ+i ++θ i Uig the reidue theorem, we obtai [ ] J πi k ζ k m k k! + k! + m + +! k ζt m t dt tt + [t + + ]. +i i ζt m t dt tt + [t + + ] ; 7.

34 REFERENCES 33 The i the ecod term here could ut a well be ay real umber exceedig. Here k ζ k m k k! + k! +! + k k B k+ m k + k +! + k! k B k m k k! + k!. Therefore k ζ k m k k! + k! + k m + +! [ + k m + m +! k m +! B k +! k! + k! m+ k + m+ + k + +! m k ] k m + ; 7. for the ecod equality, ee Kuth [] Exercie O the other had, Flaolet et al. [4] howed that +i i ζt m t dt tt + [t + + ] Thu it follow from that J. Referece πi +! m k k m [] L. Devroye. O the probablitic wort-cae time of Fid. Algorithmica, 3:9 33,. [] J. A. Fill ad S. Jao. Quickort aymptotic. Joural of Algorithm, 44:4 8,. [3] J. A. Fill ad S. Jao. The umber of bit compario ued by Quickort: A averagecae aalyi. Proceedig of the ACM-SIAM Sympoium o Dicrete Algorithm, page 93 3, 4. [4] P. Flaolet, P. Graber, P. Kirchehofer, H. Prodiger, ad R. F. Tichy. Melli traform ad aymptotic: digital um. Theoretical Computer Sciece, 3:9 34, 994. [5] P. Flaolet ad R. Sedgewick. Melli traform ad aymptotic: Fiite differece ad Rice itegral. Theoretical Computer Sciece, 44: 4, 995. [6] P. J. Graber ad H. Prodiger. O a cotat ariig i the aalyi of bit compario i Quickelect. Quaetioe Mathematicae, 3:33 36, 8. [7] R. Grübel ad U. Röler. Aymptotic ditributio theory for Hoare electio algorithm. Advace i Applied Probability, 8:5 69, 996. [8] C. R. Hoare. Fid algorithm 65. Commuicatio of the ACM, 4:3 3, 96.

35 REFERENCES 34 [9] H. Hwag ad T. Tai. Quickelect ad the Dickma fuctio. Combiatoric, Probability ad Computig, :353 37,. [] C. Kel ad W. Szpakowki. Quickort algorithm agai reviited. Dicrete Mathematic ad Theoretical Computer Sciece, 3:43 64, 999. [] D. E. Kuth. Mathematical aalyi of algorithm. I Iformatio Proceig 7 Proceedig of IFIP Cogre, Lublaa, 97, page 9 7. North-Hollad, Amterdam, 97. [] D. E. Kuth. The Art of Computer Programmig. Volume : Fudametal Algorithm. Addio-Weley, Readig, Maachuett, 998. [3] D. E. Kuth. The Art of Computer Programmig. Volume 3: Sortig ad Searchig. Addio-Weley, Readig, Maachuett, 998. [4] J. Let ad H. M. Mahmoud. Average-cae aalyi of multiple Quickelect: A algorithm for fidig order tatitic. Statitic ad Probability Letter, 8:99 3, 996. [5] H. M. Mahmoud, R. Modarre, ad R. T. Smythe. Aalyi of Quickelect: A algorithm for order tatitic. RAIRO Iformatique Théorique et Applicatio, 9:55 76, 995. [6] H. M. Mahmoud ad R. T. Smythe. Probabilitic aalyi of multiple Quickelect. Algorithmica, : , 998. [7] R. Neiiger ad L. Rüchedorf. Rate of covergece for Quickelect. Joural of Algorithm, 44:5 6,. [8] H. Prodiger. Multiple Quickelect Hoare Fid algorithm for everal elemet. Iformatio Proceig Letter, 56:3 9, 995. [9] M. Régier. A limitig ditributio of Quickort. RAIRO Iformatique Théorique et Applicatio, 3: , 989. [] U. Röler. A limit theorem for Quickort. RAIRO Iformatique Théorique et Applicatio, 5:85, 99. [] B. Vallée, J. Clémet, J. A. Fill, ad P. Flaolet. The umber of ymbol compario i Quickort ad Quickelect. Preprit, 9.