in the wost cse. (All logithms in this ppe e se logithms.) The est known soting metho, clle mege insetion y Knuth [9], is ue to Leste Fo J. n Selme Jo

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1 On lowe ouns fo selecting the mein Doit Do John H st y Stn Ulfeg y Ui Zwick June 4, 997 Astct We pesent efomultion of the n+o(n) lowe oun of Bent n John fo the nume of compisons neee fo selecting the mein of n elements. Ou efomultion uses weight function. Apt fom giving moe intuitive poof fo the lowe oun, the new fomultion opens up possiilities fo impoving it. We use the new fomultion to show tht ny pi-foming mein ning lgoithm, i.e., mein ning lgoithm tht stts y comping n=c isjoint pis of elements, must pefom, in the wost cse, t lest :0n + o(n) compisons. This povies stong evience tht selecting the mein equies t lest cn + o(n) compisons, fo some c >. Intouction Soting n selection polems hve eceive extensive ttention y compute scientists n mthemticins fo long time. Compison se lgoithms fo solving these polems wok y pefoming piwise compisons etween the elements until the eltive oe of ll elements is known, in the cse of soting, o until the i-th lgest element mong the n input elements is foun, in the cse of selection. Soting in compison se computtionl moel is quite well unestoo. Any eteministic lgoithm cn e moele y ecision tee in which ll intenl noes epesent compison etween two elements; evey lef epesents esult of the computtion. Since thee must e t lest s mny leves in the ecision tee s thee e possile e-oeings of n elements, ll lgoithms tht sot n elements use t lest log n!e n log n? n log e + o(n) n log n? :44n + o(n) compisons Deptment of Compute Science, School of Mthemticl Sciences, Rymon n Bevely Sckle Fculty of Exct Sciences, Tel Aviv Univesity, Tel Aviv 69978, Isel. E-mil: {oit,zwick}@mth.tu.c.il. y Deptment of Numeicl Anlysis n Computing Science, Royl Institute of Technology, Stockholm, Sween. E-mil: {johnh,stffnu}@n.kth.se.

2 in the wost cse. (All logithms in this ppe e se logithms.) The est known soting metho, clle mege insetion y Knuth [9], is ue to Leste Fo J. n Selme Johnson [7]. It sots n elements using t most n log n? :33n + o(n) compisons. Thus, the gp etween the uppe n lowe ouns is vey now in tht the eo in the secon oe tem is oune y 0:n. The polem of ning the mein is the specil cse of selecting the i-th lgest in n oee set of n elements, when i = n=e. Although much eot hs een put into ning the exct nume of equie compisons, thee is still n nnoying gp etween the est uppe n lowe ouns cuently known. Knowing how to sot, we coul select the mein y st soting, n then selecting the mile-most element; it is quite evient tht we coul o ette, ut how much ette? This question eceive somewht supising nswe when Blum et l. [3] showe, in 973, how to etemine the mein in line time using t most 5:43n compisons. This esult ws impove upon in 976 when Sch nhge, Pteson, n Pippinge [3] pesente n lgoithm tht uses only 3n + o(n) compisons. Thei min invention ws the use of fctoies which mss-pouce cetin ptil oes tht cn e esily mege with ech othe. This emine the est lgoithm fo lmost 0 yes, until Do n Zwick [5] pushe own the nume of compisons little it futhe to :95n+o(n) y ing geen fctoies tht ecycle eis fom the meging pocess use in the lgoithm of [3]. The st non-tivil lowe oun fo the polem ws lso pesente, in 973, y Blum et l. [3] using n vesy gument. Thei :5n lowe oun ws susequently impove to :75n + o(n) y Ptt n Yo [] in 973. Then Yp [4], n lte Muno n Polete [0], impove it to n+o() n n+o(), espectively. 43 The poofs of these lst two ouns e long n complicte. In 979, Fussenegge n Gow [8] pove :5n + o(n) lowe oun fo the mein using new poof technique. Bent n John [] use the sme sic ies when they gve, in 985, shot poof tht impove the lowe oun to n + o(n), which is cuently the est ville. Thus, the uncetinty in the coecient of n is lge fo ning the mein thn it is fo soting, even though the line tem is the secon oe tem in the cse of soting. Since ou methos e se on the poof y Bent n John, let us escie it in some etil. Given the ecision tee of compison se lgoithm, they invente metho to pune it tht yiels collection of pune tees. Then, lowe ouns fo the nume of pune tees n fo thei nume of leves e otine. A nl gument sying tht the leves of the pune tees e lmost isjoint then gives lowe oun fo the size of the ecision tee. In Section we efomulte the poof y Bent n John y ssigning weights to ech noe in the ecision tee. The weight of noe v coespons to the totl

3 nume of leves in sutees with oot v in ll pune tees whee v occus in the poof y Bent n John. The weight of the oot is ppoximtely n ; we show tht evey noe v in the ecision tee hs chil whose weight is t lest hlf the weight of v, n tht the weights of ll the leves e smll. When the poof is fomulte in this wy, it ecomes moe tnspent, n one cn moe esily stuy iniviul compisons, to ule out some s eing fom the lgoithm's point of view. Fo mny polems, such s ning the mximl o the miniml element of n oee set, n ning the mximl n miniml element of n oee set, thee e optiml lgoithms tht stt y mking n=c piwise compisons etween singleton elements. We efe to lgoithms tht stt in this wy s eing pifoming. It hs een iscusse whethe thee e optiml pi-foming lgoithms fo ll ptil oes, n in pticul this question ws pose s n open polem y Aigne []. Some exmples wee then foun y Chen [4], showing tht pi-foming lgoithms e not lwys optiml. It is inteesting to note tht the lgoithms in [5] n [3] e oth pi-foming. It is still n open polem whethe thee e optiml pi-foming lgoithms fo ning the mein. In Section 3 we use ou new ppoch to pove tht ny pi-foming lgoithm uses t lest :07n + o(n) compisons to n the mein. Do n Zwick [6] hve ecently een le to exten the ies escie hee n otin (+)n lowe oun, fo some tiny > 0, on the nume of compisons pefome, in the wost cse, y ny mein selection lgoithm. Bent n John evisite Bent n John [] pove tht n + o(n) compisons e equie fo selecting the mein. Thei esult, in fct, is moe genel n povies lowe oun fo the nume of compisons equie fo selecting the i-th lgest element, fo ny i n. We concentte hee on mein selection lthough ou esults, like those of Bent n John, cn e extene to genel i. Although the poof given y Bent n John is eltively shot n simple, we hee pesent efomultion. Thee e two esons fo this: the st is tht the poof gets moe tnspent; the secon is tht this fomultion mkes it esie to stuy the eect of iniviul compisons. Theoem (Bent n John []). Fining the mein equies n+o(n) compisons. Poof. Any eteministic lgoithm fo ning the mein cn e epesente y ecision tee T, in which ech intenl noe v is lele y compison :. The 3

4 two chilen of such noe, v < n v >, epesent the outcomes < n >, espectively. We ssume tht ecision tees o not contin eunnt compisions etween elements whose eltive oe hs ley een estlishe. We consie univese U contining n elements. Fo evey noe v in T n suset C of U we mke the following enitions: mx v (C) = min v (C) = C C evey compison : ove v with C h outcome > evey compison : ove v with C h outcome < Befoe we pocee with the poof tht selecting the mein equies n+o(n) compisons, we pesent poof of somewht weke esult. We ssume tht U contins n = m elements n show tht selecting the two milemost elements equies n + o(n) compisons. The poof in this cse is slightly simple, yet it emonsttes the min ies use in the poof of the theoem. We ene weight function on the noes of T. This weight function stises the following thee popeties: (i) the weight of the oot is n+o(n). (ii) ech intenl noe v hs chil whose weight is t lest hlf the weight of v. (iii) the weight of ech lef is smll. Fo evey noe v in the ecision tee, we keep tck of susets A which my contin the m lgest elements with espect to the compisons ley me. Let A(v) contin ll such sets which e clle uppe hlf comptile with v. The As e ssigne weights which inicte how f fom solution the lgoithm is, ssuming tht the elements in A e the m lgest. The weight of evey A A(v) is ene s n the weight of noe v is ene s w v (A) = jminv(a)j+jmxv( A)j ; w(v) = AA(v) w v (A): The supescipt in wv (A) is use s we shll shotly hve to ene secon weight function wv (B). In the oot of T, ll susets of size m of U e uppe hlf comptile with so? m tht ja()j = m. Also, ech A A() hs weight m, n we n, s pomise, tht w() = m m m = n+o(n) : ; : 4

5 cse w v < (A) w v > (A) o o A A A A 0 A A 0 A A o o Tle : The weight of set A A(v) in the chilen of noe v, eltive to its weight in v. Consie the weight wv (A) of set A A(v) t noe v lele y the compison :. Wht e the weights of A in v's chilen? This epens on which of the elements n elongs to A (n on which of them is miniml in A o mximl in A). The fou possile cses e consiee in Tle. The weights given thee e eltive to the weight wv (A) of A t v. A zeo inictes tht A is no longe comptile with this chil n thus oes not contiute to its weight. The weight wv < (A), when ; A, fo exmple, is w v (A), if min v(a), n is wv (A), othewise. As cn e seen, v lwys hs t lest one chil in which the weight of A is t lest hlf its weight t v. Futhemoe, in ech one of the fou cses, wv < (A) + wv > (A) wv (A). Ech lef v of the ecision tee coespons to stte of the lgoithm in which the two milemost elements wee foun. Thee is theefoe only one set A left in A(v). Since we hve ientie the minimum element in A n the mximum element in A, we get tht wv (A) = 4. So, if we follow pth fom the oot of the tee n epetely escen to the chil with the lgest weight, we will, when we eventully ech lef, hve pefome t lest n + o(n) compisons. We now pove tht selecting the mein lso equies t lest n + o(n) compisons. To mke the mein well ene we ssume tht n = m?. The polem tht ises in the ove gument is tht the weights of the leves in T, when the selection of the mein, n not the two milemost elements, is consiee, e not necessily smll enough: it is possile to know the mein without knowing ny eltions etween elements in A (which now contins m? elements); this is emeie s follows. In noe v whee the lgoithm is close to etemining the minimum element in A, we essentilly foce it to etemine the lgest element in A inste. This is one y moving n element 0 out of A n ceting set B = A [ f 0 g. This set is lowe hlf comptile with v n the mein is the mximum element in B. By suitle choice of 0, most of mx v ( A) is in mx v (B). A set B is lowe hlf comptile with 5

6 v if jbj = m n it my contin the m smllest elements in U. We keep tck of Bs in the multiset B(v). Fo the oot of T, we let A() contin ll susets of size m of U s efoe, n let B() e empty. We exchnge some As fo Bs s the lgoithm pocees. The weight of set B is ene s w v (B) = jmxv(b)j : The weight of B inictes how f the lgoithm is fom solution, ssuming tht the elements in B e the m smllest elements. The weight of noe v is now ene to e w(v) = AA(v) w v (A) + 4p n BB(v) w v (B): In the eginning of n lgoithm (in the uppe pt of the ecision tee), the weight of noe is still the sum of the weights of ll As, n theefoe w() = n+o(n). We now ene A(v) n B(v) fo the est of T moe exctly. Fo ny noe v in T, except the oot, simply copy A(v) n B(v) fom the pent noe n emove ll sets tht e not uppe o lowe hlf comptile with v, espectively. We ensue tht the weight of evey lef is smll y oing the following: If, fo some A A(v) we hve jmin v (A)j = p ne, we select n element 0 min v (A) which hs een compe to the fewest nume of elements in A; we then emove the set A fom A(v) n the set B = A [ f 0 g to B(v). Note tht t the oot, jmin (A)j = m fo ll A A(), n tht this quntity eceses y t most one fo ech compison until lef is eche. In lef v the mein is known; thus, ja(v)j is empty. Lemm. Let A(v) n B(v) e ene y the ules escie ove. Then, evey intenl noe v (lele : ) in T hs chil with t lest hlf the weight of v, i.e., w(v < ) w(v)= o w(v > ) w(v)=. Poof. Tle gives the weights of set A A(v) t v's chilen, eltive to the weight wv (A) of A t v. Similly, Tle gives the weights of of set B B(v) in v's chilen, eltive the weight wv (v) of B t v. As w v < (A) + wv > (A) wv (A) n wv < (B) + wv > (B) wv (B), fo evey A A(v) n B B(v), ll tht emins to e checke is tht the weight oes not ecese when lowe hlf comptile set B eplces n uppe hlf comptile set A. This is covee y Lemm 3. 6

7 B cse w v < (B) w v > (B) B o o B B 0 B B 0 B B Tle : The weight of set B B(v) in the chilen of noe v, eltive to its weight in v. Lemm 3. If A is emove fom A(v) n B is e in its plce to B(v), n if less thn 4n compisons hve een pefome on the pth fom the oot to v, then 4pn w v (B) > w v (A). Poof. A set A A(v) is eplce y set B = A [ f 0 g B(v) only when jmin v (A)j = p ne. The element 0, in such cse, is n element of min v (A) tht hs een compe to the fewest nume of elements in A. If 0 ws compe to t lest p n elements in A, we get tht ech element of min v (A) ws compe to t lest p n elements in A, n t lest 4n compsions hve een pefome on the pth fom the oot to v, contiction. We get theefoe tht 0 ws compe to less thn p n elements of A n thus jmx v (B)j > jmx v ( A)j? p n. As consequence, we get tht 4 p n + jmx v (B)j > jmin v (A)j + jmx v ( A)j n thus 4pn wv (B) > w v (A), s equie. We now know tht the weight of the oot is lge, n tht the weight oes not ecese too fst; wht emins to e shown is tht the weights of the leves e eltively smll. This is estlishe in the following lemm. Lemm 4. Fo lef v (in which the mein is known), w(v) m 4pn. Poof. Clely, the only sets comptile with lef of T e the set A contining the m lgest elements, n the set B contining the m smllest elements. Since jmin v (A)j = jmx v (B)j =, we get tht wv (B) = n A 6 A(v). Since thee e exctly m elements tht cn e emove fom B to otin coesponing A, thee cn e t most m copies of B in B(v). Let T e compison tee tht coespons to mein ning lgoithm. If the height of T is t lest 4n, we e one. Othewise, y stting t the oot n epetely escening to chil whose weight is t lest hlf the weight of its pent, we tce pth whose length is t lest n + o(n) n Theoem follows. 7

8 Let us see how the cuent fomlism gives oom fo impovement tht i not exits in the oiginl poof. The n + o(n) lowe oun is otine y showing tht ech noe v in ecision tee T tht coespons to mein ning lgoithm hs chil whose weight is t lest hlf the weight of v. Consie the noes v 0 ; v ; : : : ; v` long the pth otine y stting t the oot of T n epetely escening to the chil with the lge weight, until lef is eche. If we coul show tht the weights of suciently mny noes on this pth e stictly lge thn the weights of thei pent, we woul otin n impove lowe oun fo mein selection. If w(v i ) ( + i)w(v i? ), fo evey i `, then the length of this pth, n theefoe the epth of T, is t lest n + P` i= log ( + i) + o(n). 3 An impove lowe oun fo pi-foming lgoithms Let v e noe of compison tee. An element x is singleton t v if it ws not compe ove v with ny othe element. Two elements x n y fom pi t v if the elements x n y wee compe to ech othe ove v, ut none of them ws compe to ny othe element. A pi-foming lgoithm is n lgoithm tht stts y constucting n=c = m? pis. By concentting on compisons tht involve elements tht e pt of pis, we otin ette lowe oun fo pi-foming lgoithms. Theoem 5. A pi-foming lgoithm fo ning the mein must pefom, in the wost cse, t lest :0069n + o(n) compisons. Poof. It is esy to see tht compison involving two singletons cn e elye until just efoe one of them is to e compe fo the secon time. We cn theefoe estict ou ttention to compison tees in which the ptil oe coesponing to ech noe contins t most two pis. Allowing only one pi is not enough s lgoithms shoul e llowe to constuct two pis f; g n f 0 ; 0 g, n then compe n element fom f; g with n element fom f 0 ; 0 g. We focus ou ttention on noes in the ecision tee in which n element of pi is compe fo the secon time n in which the nume of non-singletons is t most m, fo some <. If v is noe in which the nume of non-singletons is t most m, fo some <, then B(v) is empty n thus w(v) = P AA(v) w v (A) n we o not hve to consie Tle fo the est of the section. Recll tht A(v) enotes the collection of susets of U size m tht e uppe hlf comptile with v. If H; L U e susets of U, of ity size, we let A H=L (v) = fa A(v) j H A n L Ag: 8

9 A A c c c c c c Figue : The six possile wys tht,, n c my e ivie etween A n A. Note tht c is not necessily singleton element; it my e pt of lge ptil oe. We let w H=L (v) e the contiution of the sets of A H=L (v) to the weight of v, i.e., w H=L (v) = AA H=L (v) w v (A): Fo evity, we wite A h :::h =l :::l s (v) fo A fh ;:::;h g=fl ;:::;l sg(v) n w h :::h =l :::l s (v) fo w fh ;:::;h g=fl ;:::;l sg(v). Befoe poceeing, we escie the intuition tht lies ehin the est of the poof. Consie Tle fom the lst section. If, in noe v of the ecision tee, the two cses A; A n A; A e not eqully likely, o moe pecicely, if the contiutions w = (v) n w = (v) of these two cses to the totl weight of v e not equl, thee must e t lest one chil of v whose weight is gete thn hlf the weight of v. The iculty in impoving the lowe oun of Bent n John lies theefoe t noes in which the the contiutions of the two cses A; A n A; A e lmost equl. This fct is not so esily seen when looking t the oiginl poof given in []. Suppose now tht v is noe in which n element of pi f; g is compe with n ity element c n tht the nume of non-singletons in v is t most m. We ssume, without loss of genelity, tht >. The weights of set A A(v) in v's chilen epen on which of the elements,, n c elongs to A, n on whethe c is miniml in A o mximl in A. The six possile wys of iviing the elements,, n c etween A n A e shown in Figue. The weights of the set A in v's chilen, eltive to the weight wv (A) of A t v, in ech one of these six cses e given in Tle 3. Tle 3 is simil to Tle of the pevious section, with c plying the ole of. Thee is one impotnt ieence, howeve. If ; ; c A, s in the st ow of Tle 3, then the weight of A in v >c is equl to the weight of A in v. The weight is not hlve, s my e the cse in the st ow of Tle. If the contiution w c= (v) of the cse ; ; c A to the weight of v is not negligile, thee must gin e t lest one chil of v whose weight is gete thn hlf the weight of v. The impove lowe oun is otine y showing tht if the contiutions of the cses A, A n A, A e oughly equl, n if most elements 9

10 cse w v <c (A) w v >c (A) A A c A A A c A o o A A c A 0 A A c A 0 A A c A 0 A A c A o Tle 3: The weight of set A A(v) in the chilen of noe v, eltive to its weight in v, when the element of pi > is compe with n ity element c. in the ptil oe e singletons, then the contiution of the cse ; ; c A is non-negligile. The lge the nume of singletons in the ptil oe, the lge is the eltive contiution of the weight w c= (v) to the weight w(v) of v. Thus, wheneve n element of pi is compe fo the secon time, we mke smll gin. The ove intuition is me pecise in the following lemm: Lemm 6. If v is noe in which n element of pi > is compe with n element c, n if the nume of singletons in v is t lest m + p n, then w(v <c ) w(v) + (w c=(v)? w =c (v)) ; w(v >c ) w(v) + (w =c(v)? w c= (v) + w c= (v)) : Poof. Both inequlities follow esily y consieing the enties in Tle 3. To otin the secon inequlity, fo exmple, note tht w(v >c ) (w(v) + w c=(v)? w c= (v) + w =c (v) + w =c (v)). As w c= (v) = w c= (v) n w =c (v) + w =c (v) = w =c (v), the secon inequlity follows. It is woth pointing out tht in Tle 3 n in Lemm 6, we only nee to ssume tht > ; we o not use the stonge conition tht > is pi. This stonge conition is cucil howeve in the sequel, especilly in Lemm 8. To mke use of Lemm 6 we nee ouns on the eltive contiutions of the ieent cses. The following lemm is useful tool fo etemining such ouns. 0

11 Lemm 7. Let G = (V ; V ; E) e iptite gph. Let n e the miniml egee of the vetices of V n V, espectively. Let n e the mximl egee of the vetices of V n V, espectively. Assume tht positive weight function w is ene on the vetices of G such tht w(v ) = w(v ), wheneve v V, v V n (v ; v ) E. Let w(v ) = P v V w(v ) n w(v ) = P v V w(v ). Then, w(v ) w(v ) w(v ): Poof. Let v (e) V n v (e) V enote the two vetices connecte y the ege e. We then hve v V w(v ) ee w(v (e)) = ee w(v (e)) The othe inequlity follows y exchnging the oles of V n V. Using Lemm 7 we otin the following sic inequlities. v V w(v ): Lemm 8. If v is noe in which > is pi n the nume of non-singletons in v is t most m, then (? )w c=(v) w c= (v) (? )w c= (v) w c= (v) (? )w =c(v) w =c (v) (? )w =c (v) w =c (v) (?) w c=(v) ;? w c=(v) ; (?) w =c(v) ;? w =c (v) : Ech one of these inequlities eltes weight, such s w c= (v), to weight, such s w c= (v), otine y moving one of the elements of the pi > fom A to A. In ech inequlity we `loose' fcto of?. When the elements n e joine togethe fcto of is intouce. When the elements n e septe, fcto of is intouce. Poof. We pesent poof of the inequlity w c= (v) (?) w c= (v). The poof of ll the othe inequlities is lmost ienticl. Constuct iptite gph G = (V ; V ; E) whose vetex sets e V = A c= (v) n V = A c= (v). Dene n ege (A ; A ) E etween A A c= (v) n A A c= (v) if n only if thee is singleton A such tht A = A n fg [ fg. Suppose tht (A ; A ) is such n ege. As 6 min v (A ) ut min v (A ), while ll othe elements e exteml with espect to A if n only if they e exteml with espect to A (note tht min v (A ) n mx v ( A )), we get tht wv (A ) = w v (A ).

12 Fo evey set A of size m, the nume of singletons in A is t lest (? )m n t most m. We get theefoe tht the miniml egees of the vetices of V n V e ; (? )m n the mximl egees of V n V e ; m. The inequlity w c= (v) (?) w c=(v) theefoe follows fom Lemm 7. Using these sic inequlities we otin: Lemm 9. If v is noe in which > is pi n the nume of non-singletons is t most m, fo some <, then w c= (v) (?) (?) w c= (v) ; w =c (v) (?)(3?) (?) w =c (v) ; w c= (v) (?) w c= (v) : Poof. We pesent the poof of the st inequlity. The poof of the othe two inequlities is simil. Using inequlities fom Lemm 8 we get tht w c= (v) = w c= (v) + w c= (v) + w c= (v) n the st inequlity follows. w c= (v) +? w c=(v) + (?) w c= (v) = (?) (?) w c= (v) We e now ey to show tht if v is noe in which n element of pi is compe fo the secon time, then v hs chil whose weight is gete thn hlf the weight of v. Comining Lemm 6 n Lemm 9, we get tht (w(v <c) + w(v >c )) (?) w(v) + 4(?) w c= (v) ; w(v >c ) w(v)? (?) w c=(v) + (?)(3?) (?) w =c (v) : Let = w c= (v)=w(v) n? = w =c (v)=w(v). We get tht (w(v <c) + w(v >c )) + (?) 4(?) w(v) ; w(v >c ) =? (?) + (?)(3?) (?) (? ) w(v)? 3? (?) + w(v) (?)(3?) (?) :

13 As consequence, we get tht mxfw(v <c ); w(v >c )g mx n o + (?) 4(?) ;? 3? (?) + (?)(3?) (?) w(v) : The coecient of w(v), on the ight hn sie, is minimize when the two expessions whose mximum is tken e equl. This hppens when = (3?4+ ). Plugging 7?6+ this vlue of into the two expessions, we get tht whee mxfw(v <c ); w(v >c )g ( + f ())w(v) ; f () = (3? )(? ) 3 (? ) (7? 6 + ) : It is esy to check tht f () > 0 fo <. A pi-foming compison is compison in which two singletons e compe to fom pi. A pi-touching compison is compison in which n element of pi is compe fo the secon time. In pi-foming lgoithm, the nume of singletons is ecese only y pi-foming compisons. Ech pi-foming compison eceses the nume of singletons y exctly two. As expline ove, pi-foming compisons cn lwys e elye so tht pi-foming compison : is immeitely followe y compison tht touches the pi f; g, o y pi-foming compison 0 : 0 n then y compison tht touches oth pis f; g n f 0 ; 0 g. Consie gin the pth tce fom the oot y epetely escening to the chil with the lge weight. As consequence of the ove iscussion, we get tht when the i-th pi-touching compison long this pth is pefome, the nume of non-singletons in the ptil oe is t most 4i. It follows theefoe fom the emk me t the en of the pevious section tht the epth of the compison tee coesponing to ny pi-foming lgoithm is t lest n + m=4 i= log ( + f ( 4i )) + o(n) m Z = n + n 8 log ( + f (t))t + o(n) :0069n + o(n) : 0 This completes the poof of Theoem 5. The wost cse in the poof ove is otine when the lgoithm convets ll the elements into qutets. A qutet is ptil oe otine y comping elements contine in two isjoint pis. In the poof ove, we nlyze cses in which n 3

14 A A c c c c c c c c c Figue : The nine possile wys tht,, c, n my e ivie etween A n A. element of pi > is compe with n ity element c. If the element c is lso pt of pi, tighte nlysis is possile. By pefoming this nylsis we cn impove Theoem 5. Theoem 0. A pi-foming lgoithm fo ning the mein must pefom, in the wost cse, t lest :07n + o(n) compisons. Poof. Consie compisons in which the element fom pi > is compe with n element of pi c >. The nine possile wys of iviing the elements,, c, n mong A n A e epicte in Figue. We my ssume, without loss of genelity, tht the element is compe with eithe c o with. Let v e noe of the compison tee in which > n c > e pis n which one of the compions : c o : is pefome. Let A A(v). The weights of set A in v's chilen, eltive to the weight wv (A) of A t v, in ech one of these nine cses e given in Tle 4. The two possile compisons : c n : e consiee septely. The following equlities e esily veie. Lemm. If > n c > e pis in v then w c= (v) = w c= (v) ; w c= (v) = w =c (v) ; w c= (v) = w =c (v) ; w c= (v) = 4w =c (v) : The following inequlities e nlogous to the inequlities of Lemm 8. Lemm. If > n c > e pis in v n if the nume of non-singletons in v is t most m, fo some <, then (? )w c= (v) w c= (v) (?) w c= (v) ; (? )w =c (v) w c= (v)? w =c (v) ; (? )w =c(v) w =c (v) (?) w =c(v) ; (? )w =c (v) w =c (v)? w =c(v) : 4

15 cse w v <c (A) w v >c (A) w v < (A) w v > (A) A A c= A A c= A A c= 0 0 A A c= A A c= A A c= A A =c 0 0 A A =c 0 0 A A =c Tle 4: The weight of set A A(v) in the chilen of noe v, eltive to its weight in v, when the element of pi > is compe with n element of pi c >. Consie st the compison : c. By exmining Tle 4 n using the equlities of Lemm, we get tht w(v <c)+w(v >c) = w c= (v) w c=(v) + w c=(v) w c=(v) + w c=(v) + w c=(v) + w =c(v) + w =c(v) + w =c(v) = w c= (v) + 3 w c=(v) + 3w =c (v) + w =c (v) + w =c(v): Minimizing this expession, suject to the equlities of Lemm, the inequlities of Lemm, n the fct tht the weights of the nine cses sum up to w(v), mounts to solving line pogm. By solving this line pogm we get tht w(v <c ) + w(v >c ) w(v) ( + f ()) w(v) ; whee f () = (3? )(? )3 (? ) 4 : It seems intuitively cle tht the compison : is compison fom the lgoithm's point of view. The vesy will most likely nswe with >. 5

16 Inee, y solving the coesponing line pogm, we get tht w(v > ) = w c= (v) + w c=(v) + w c= (v) + w c= (v) + w c= (v) + w =c (v) + w =c (v) + w =c (v) = w c= (v) + 3 w c= (v) + 5w =c (v) + 3 w =c (v) + w =c (v) 3 4 : As ( + f ()) 3, fo evey 0, we my iseg the compison : 4 fom ny futhe consietion. It is esy to veify tht ( + f ()) + f (). As esult, we get lowe oun of Z n + n 8 log ( + f (t))t + o(n) :07n + o(n) : 0 This completes the poof of Theoem 0. 4 Concluing emks We pesente efomultion of the n + o(n) lowe oun of Bent n John fo the nume of compisons neee fo selecting the mein of n elements. Using this new fomultion we otine n impove lowe oun fo pi-foming mein ning lgoithms. As mentione, Do n Zwick [6] hve ecently extene the ies escie hee n otine (+)n lowe oun fo genel mein ning lgoithms, fo some tiny > 0. We elieve tht the lowe oun fo pi-foming lgoithms otine hee cn e sustntilly impove. Such n impovement seems to equie, howeve, some new ies. Otining n impove lowe oun fo pi-foming lgoithms my e n impotnt step tows otining lowe oun fo genel lgoithms which is signicntly ette thn the lowe oun of Bent n John []. Pteson [] conjectues tht the nume of compisons equie fo selecting the mein is out (log 4=3 )n :4n. Refeences [] Mtin Aigne. Poucing posets. Discete Mthemtics, 35:5, 98. [] Smuel W. Bent n John W. John. Fining the mein equies n compisons. In Poceeings of the Seventeenth Annul ACM Symposium on Theoy of Computing, pges 36,

17 [3] Mnuel Blum, Roet W. Floy, Vughn Ptt, Ronl L. Rivest, n Roet E. Tjn. Time ouns fo selection. Jounl of Compute n System Sciences, 7:44846, 973. [4] Jingsen Chen. Ptil Oe Pouctions. PhD thesis, Lun Univesity, Box 8, S- 00 Lun, Sween, 993. [5] Doit Do n Ui Zwick. Selecting the mein. In Poceeings of 6th SODA, pges 8897, 995. [6] Doit Do n Ui Zwick. Mein selection equies (+)n compisons. In Poceeings of 37th FOCS, 996. [7] Leste R. Fo n Selme M. Johnson. A tounment polem. Ameicn Mthemticl Monthly, 66:387389, 959. [8] Fnk Fussenegge n Hol N. Gow. A counting ppoch to lowe ouns fo selection polems. Jounl of the Assocition fo Computing Mchiney, 6():738, Apil 979. [9] Donl Evin Knuth. The At of Compute Pogmming, vol. 3, Seching n Soting. Aison-Wesley Pulishing Compny, Inc., 973. [0] I. Muno n P.V. Polete. A lowe oun fo etemining the mein. Technicl Repot Resech Repot CS-8-, Univesity of Wteloo, 98. [] Michel S. Pteson. Pogess in selection. In 5th Scninvin Wokshop on Algoithm Theoy, Reykjv k, Iceln, pges , 996. [] Vughn R. Ptt n Foong Fnces Yo. On lowe ouns fo computing the i-th lgest element. In 4th Annul Symposium on Switching n Automt Theoy, pges 708, 973. [3] A. Sch nhge, M. Pteson, n N. Pippenge. Fining the mein. Jounl of Compute n System Sciences, 3:8499, 976. [4] C.K. Yp. New lowe ouns fo meins n elte polems. Compute Science Repot 79, Yle Univesity, 976. Astct in Symposium on Algoithms n Complexity: New Results n Diections, (J. F. Tu, e.) Cnegie-Mellon Univesity,