D. DOR AND J. HνASTAD AND S. ULFBERG AND U. ZWICK min invention ws the use of fctoies which mss-poduce cetin ptil odes tht cn e esily meged with ech o

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1 ON LOWER BOUNDS FOR SELECTING THE MEDIAN DORIT DOR Λ, JOHAN HSTAD y, STAFFAN ULFBERG y, AND URI ZWICK Λ Astct. We pesent efomultion of the n + o(n) lowe ound of Bent nd John fo the nume of compisons needed fo selecting the medin of n elements. Ou efomultion uses weight function. Apt fom giving moe intuitive poof fo the lowe ound, the new fomultion opens up possiilities fo impoving it. We use the new fomultion to show tht ny pi-foming medin nding lgoithm, i.e., medin nding lgoithm tht stts y comping n=c disjoint pis of elements, must pefom, in the wost cse, t lest :0n + o(n) compisons. This povides stong evidence tht selecting the medin equies t lest cn + o(n) compisons, fo some c >. Wning: Essentilly this ppe hs een pulished in SIAM jounl on Discete Mthemtics nd is suject to copyight estictions. In pticul it is fo pesonl use only. Key wods. medin selection, compison lgoithms, lowe ounds AMS suject clssictions. 68Q5, 68R05, 06A07. Intoduction. Soting nd selection polems hve eceived extensive ttention y compute scientists nd mthemticins fo long time. Compison sed lgoithms fo solving these polems wok y pefoming piwise compisons etween the elements until the eltive ode of ll elements is known, in the cse of soting, o until the i-th lgest element mong the n input elements is found, in the cse of selection. Soting in compison sed computtionl model is quite well undestood. Any deteministic lgoithm cn e modeled y decision tee in which ll intenl nodes epesent compison etween two elements; evey lef epesents esult of the computtion. Since thee must e t lest s mny leves in the decision tee s thee e possile e-odeings of n elements, ll lgoithms tht sot n elements use t lest dlog n!e n log n n log e + o(n) ß n log n :44n + o(n) compisons in the wost cse. (All logithms in this ppe e se logithms.) The est known soting method, clled mege insetion y Knuth [9], is due to Leste Fod J. nd Selme Johnson [7]. It sots n elements using t most n log n :33n + o(n) compisons. Thus, the gp etween the uppe nd lowe ounds is vey now in tht the eo in the second ode tem is ounded y 0:n. The polem of nding the medin is the specil cse of selecting the i-th lgest in n odeed set of n elements, when i = dn=e. Although much effot hs een put into nding the exct nume of equied compisons, thee is still n nnoying gp etween the est uppe nd lowe ounds cuently known. Knowing how to sot, we could select the medin y st soting, nd then selecting the middle-most element; it is quite evident tht we could do ette, ut howmuch ette? This question eceived somewht supising nswe when Blum et l. [3] showed, in 973, how to detemine the medin in line time using t most 5:43n compisons. This esult ws impoved upon in 976 when Schnhge, Pteson, nd Pippinge [3] pesented n lgoithm tht uses only 3n + o(n) compisons. Thei Λ School of Compute Science, Rymond nd Bevely Sckle Fculty of Exct Sciences, Tel Aviv Univesity, Tel Aviv 69978, Isel. E-mil: doit@checkpoint.com nd zwick@post.tu.c.il. y Deptment of Numeicl Anlysis nd Computing Science, Royl Institute of Technology, Stockholm, Sweden. E-mil: fjohnh,stffnug@nd.kth.se.

2 D. DOR AND J. HνASTAD AND S. ULFBERG AND U. ZWICK min invention ws the use of fctoies which mss-poduce cetin ptil odes tht cn e esily meged with ech othe. This emined the est lgoithm fo lmost 0 yes, until Do nd Zwick [5] pushed down the nume of compisons little it futhe to :95n + o(n) y dding geen fctoies tht ecycle deis fom the meging pocess used in the lgoithm of [3]. The st non-tivil lowe ound fo the polem ws lso pesented, in 973, y Blum et l. [3] using n dvesy gument. Thei :5n lowe ound ws susequently impoved to :75n + o(n) y Ptt nd Yo [] in 973. Then Yp [4], nd lte Muno nd Polete [0], impoved it to n + O() nd n + O(), espectively. The 43 poofs of these lst two ounds e long nd complicted. In 979, Fussenegge nd Gow [8] poved :5n + o(n) lowe ound fo the medin using new poof technique. Bent nd John [] used the sme sic ides when they gve, in 985, shot poof tht impoved the lowe ound to n + o(n), which iscuently the est ville. Thus, the uncetinty in the coefcient ofn is lge fo nding the medin thn it is fo soting, even though the line tem is the second ode tem in the cse of soting. Since ou methods e sed on the poof y Bent nd John, let us descie it in some detil. Given the decision tee of compison sed lgoithm, they invented method to pune it tht yields collection of puned tees. Then, lowe ounds fo the nume of puned tees nd fo thei nume of leves e otined. Anl gument sying tht the leves of the puned tees e lmost disjoint then gives lowe ound fo the size of the decision tee. In Section we efomulte the poof y Bent nd John y ssigning weights to ech nodein the decision tee. The weight of node v coesponds to the totl nume of leves in sutees with oot v in ll puned tees whee v occus in the poof y Bent nd John. The weight of the oot is ppoximtely n ;we show tht evey node v in the decision tee hs child whose weight is t lest hlf the weight of v, nd tht the weights of ll the leves e smll. When the poof is fomulted in this wy, it ecomes moe tnspent, nd one cn moe esily study individul compisons, to ule out some s eing d fom the lgoithm's point ofview. Fo mny polems, such s nding the mximl o the miniml element of n odeed set, nd nding the mximl nd miniml element ofnodeed 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 discussed whethe thee e optiml pi-foming lgoithms fo ll ptil odes, nd in pticul this question ws posed s n open polem y Aigne []. Some exmples wee then found y Chen [4], showing tht pi-foming lgoithms e not lwys optiml. It is inteesting to note tht the lgoithms in [5] nd [3] e oth pi-foming. It is still n open polem whethe thee e optiml pi-foming lgoithms fo nding the medin. In Section 3 we use ou new ppoch topove tht ny pi-foming lgoithm uses t lest :07n + o(n) compisons to nd the medin. Do nd Zwick [6]hve ecently een le to extend the ides descied hee nd otin (+ffl)n lowe ound, fo some tiny ffl > 0, on the nume of compisons pefomed, in the wost cse, y ny medin selection lgoithm.

3 ON LOWER BOUNDS FOR SELECTING THE MEDIAN 3. Bent nd John evisited. Bent nd John [] poved tht n + o(n) compisons e equied fo selecting the medin. Thei esult, in fct, is moe genel nd povides lowe ound fo the nume of compisons equied fo selecting the i-th lgest element, fo ny» i» n. We concentte hee on medin selection lthough ou esults, like those of Bent nd John, cn e extended to genel i. Although the poof given ybent nd John is eltively shot nd simple, we hee pesent efomultion. Thee e two esons fo this: the st is tht the poof gets moe tnspent; the second is tht this fomultion mkes it esie to study the effect of individul compisons. Theoem. (Bent nd John []). Finding the medin equies n + o(n) compisons. Poof. Any deteministic lgoithm fo nding the medin cn e epesented y de- cision tee T, in which ech intenl node v is leled y compison :. The two childen of such node, v < nd v >, epesent the outcomes <nd >, espectively. We ssume tht decision tees do not contin edundnt compisions etween elements whose eltive ode hs ledy een estlished. We conside univese U contining n elements. Fo evey node v in T nd suset C of U we mke the following denitions: mx v (C) = min v (C) = ρ ρ C C evey compison : ove v with C hd outcome > evey compison : ove v with C hd outcome < Befoe we poceed with the poof tht selecting the medin equies n + o(n) compisons, we pesent poof of somewht weke esult. We ssume tht U contins n = m elements nd show tht selecting the two middlemost elements equies n + o(n) compisons. The poof in this cse is slightly simple, yet it demonsttes the min ides used in the poof of the theoem. We deneweight function on the nodes of T. This weight function stises the following thee popeties: (i) the weight of the oot is n+o(n). (ii) ech intenl node v hs child whose weight is t lest hlf the weight ofv. (iii) theweight of ech lef is smll. Fo evey node v in the decision tee, we keep tck of susets A of size m which my contin the m lgest elements with espect to the compisons ledy mde. Let A(v) contin ll such sets which e clled uppe hlf comptile with v. The As e ssigned weights which estimte how f fom solution the lgoithm is, ssuming tht the elements in A e the m lgest. The weight ofevey A A(v) is dened s nd the weight of node v is dened s w v(a) = jminv(a)j+jmxv( μ A)j ; w(v) = AA(v) w v(a): The supescipt in w v(a) isusedswe shll shotly hve to dene second weight function w v(b). ff ff ; :

4 4 D. DOR AND J. HνASTAD AND S. ULFBERG AND U. ZWICK cse w v < (A) w v > (A) A A o o A A μ 0 A μ A 0 A μ A μ o o Tle. TheweightofsetA A(v) in the childen of node v, eltive to its weight in v. 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() hsweight m,ndwe nd, s pomised, tht w() = m m m = n+o(n) : Conside the weight w v(a) of set A A(v)tnodev leled y the compison :. Wht e the weights of A in v's childen? This depends on which of the elements nd elongs to A (nd on which of them is miniml in A o mximl in μ A). The fou possile cses e consideed in Tle.. The weights given thee e eltive to the weight w v(a) ofa t v. A zeo indictes tht A is no longe comptile with this child nd thus does not contiute to its weight. The weight w v < (A), when ; A, fo exmple, is w v(a), if min v (A), nd is w v(a), othewise. As cn e seen, v lwys hs t lest one child in which theweight ofa is t lest hlf its weight t v. Futhemoe, in ech one of the fou cses, w v < (A)+w v > (A) w v(a). Ech lef v of the decision tee coesponds to stte of the lgoithm in which the two middlemost elements wee found. Thee is theefoe only one set A left in A(v). Since we hve identied the minimum element in A nd the mximum element in μ A, we get tht w v (A) =4. So, if we follow pth fom the oot of the tee nd epetedly descend to the child with the lgest weight, we will, when we eventully ech lef, hve pefomed t lest n + o(n) compisons. We nowpove tht selecting the medin lso equies t lest n + o(n) compisons. To mke the medin well dened 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 medin, nd not the two middlemost elements, is consideed, e not necessily smll enough: it is possile to know the medin without knowing ny eltions etween elements in μ A (which now contins m elements); this is emedied s follows. In node v whee the lgoithm is close to detemining the minimum element in A, weessentilly foce it to detemine the lgest element in μ A insted. This is done y moving n element 0 out of A nd ceting set B = μ A [f0 g. This set is lowe hlf comptile with v nd the medin is the mximum element in B. By suitle choice of 0,mostofmx v ( μ A)isinmxv (B). AsetB is lowe hlf comptile with v if jbj = m nd it my contin the m smllest elements in U. We keep tck ofbs in the multiset B(v). Fo the oot of T,weletA() contin ll susets of size m of U s efoe, nd let B() e empty. We exchnge some As fo Bs s the lgoithm poceeds. The

5 ON LOWER BOUNDS FOR SELECTING THE MEDIAN 5 cse w v < (B) w v > (B) B B o o B B μ 0 B μ B 0 B μ B μ Tle. TheweightofsetB B(v) in the childen of node v, eltive to its weight in v. weight of set B is dened s w v(b) = jmxv(b)j : The weight of B estimtes how f the lgoithm is fom solution, ssuming tht the elements in B e the m smllest elements. The weight ofnodev is now dened 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 decision tee), the weight of node is still the sum of the weights of ll As, nd theefoe w() = n+o(n). We now dene A(v) ndb(v) fo the est of T moe exctly. Fo ny nodev in T, except the oot, simply copy A(v) ndb(v) fom the pent node nd emove ll sets tht e not uppe o lowe hlf comptile with v, espectively. We ensue tht the weight ofevey lef is smll y doing the following: If, fo some A A(v) we hve jmin v (A)j = d p ne, we select n element 0 min v (A) which hs een comped to the fewest nume of elements in μ A;we then emove the set A fom A(v) nd dd the set B = μ A [f0 g to B(v). Note tht t the oot, jmin (A)j = m fo ll A A(), nd tht this quntity deceses y t most one fo ech compison until lef is eched. In lef v the medin is known; thus, A(v) is empty. Lemm.. Let A(v) nd B(v) e dened y the ules descied ove. Then, evey intenl node v (leled : ) int hs child 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 childen, eltive to the weight w v(a) ofa t v. Similly, Tle. gives the weights of set B B(v) inv's childen, eltivetotheweight w v(v) ofb t v. As w v < (A)+w v > (A) w v(a) nd w v < (B)+w v > (B) w v(b), fo evey A A(v) nd B B(v), ll tht emins to e checked is tht the weight does not decese when lowe hlf comptile set B eplces n uppe hlf comptile set A. This is coveed y Lemm.3. Lemm.3. If A is emoved fom A(v) nd B is dded in its plce tob(v), nd if fewe thn 4n compisons hve een pefomed on the pth fom the oot to v, then 4pn w v(b) >w v(a). Poof. A set A A(v) is eplced y set B = μ A [f0 g B(v) only when jmin v (A)j = d p ne. The element 0, in such cse, is n element ofmin v (A) tht hs een comped to the fewest nume of elements in μ A. If 0 ws comped to t lest p n elements in μ A, we get tht ech element ofminv (A) ws comped to t

6 6 D. DOR AND J. HνASTAD AND S. ULFBERG AND U. ZWICK lest p n elements in μ A, nd t lest 4n compisons hve een pefomed on the pth fom the oot to v, contdiction. We get theefoe tht 0 ws comped to fewe thn p n elements of μ A nd thus jmxv (B)j > jmx v ( μ A)j p n. As consequence, we get tht 4 p n+jmx v (B)j > jmin v (A)j+jmx v ( μ A)j nd thus 4 pn w v(b) >w v(a), s equied. We nowknow tht the weight of the oot is lge, nd tht the weight doesnot decese too fst; wht emins to e shown is tht the weights of the leves e eltively smll. This is estlished in the following lemm. Lemm.4. Folef v (in which the medin is known), w(v)» m 4p n. Poof. Clely, the only sets comptile with lef of T e the set A contining the m lgest elements, nd the set B contining the m smllest elements. Since jmin v (A)j = jmx v (B)j =,we get tht wv(b) =nda 6 A(v). Since thee e exctly m elements tht cn e emoved fom B to otin coesponding A, μ thee cn e t most m copies of B in B(v). Let T e compison tee tht coesponds to medin nding lgoithm. If the height oft is t lest 4n, we e done. Othewise, y stting t the oot nd epetedly descending to child whose weight is t lest hlf the weight of its pent, we tce pth whose length is t lest n + o(n) nd Theoem. follows. Let us see how the cuent fomlism gives oom fo impovement thtdid not exist in the oiginl poof. The n + o(n) lowe ound is otined y showing tht ech nodev in decision tee T tht coesponds to medin nding lgoithm hs child whose weight is t lest hlf the weight ofv. Conside the nodes v 0 ;v ;:::;v` long the pth otined y stting t the oot of T nd epetedly descending to the child with the lge weight, until lef is eched. If we could show tht sufciently mny nodes on this pth hve weights stictly lge thn hlf the weights of thei pents, we would otin n impoved lowe ound fo medin selection. If w(v i ) ( + f i) w(v i ), fo evey» i» `, then the length of this pth, nd theefoe the depth of T, is t lest n + P` i= log ( + f i )+o(n). 3. An impoved lowe ound fo pi-foming lgoithms. Let v e node of compison tee. An element x is singleton t v if it ws not comped ove v with ny othe element. Two elements x nd y fom pi t v if the elements x nd y wee comped to ech othe ove v, ut neithe of them ws comped 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 ound fo pi-foming lgoithms. Theoem 3.. A pi-foming lgoithm fo nding the medin 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 delyed until just efoe one of them is to e comped fo the second time. We cn theefoe estict ou ttention to compison tees in which the ptil ode coesponding to ech node contins t most two pis. Allowing only one pi is not enough s lgoithms should e llowed to constuct two pis f; g nd f 0 ; 0 g, nd then compe n element fomf; g with n element fomf 0 ; 0 g. We focus ou ttention on nodes in the decision tee in which n element of pi is comped fo the second time nd in which the nume of non-singletons is t most fflm, fo some ffl<. If v is node in which the nume of non-singletons is t

7 ON LOWER BOUNDS FOR SELECTING THE MEDIAN 7 A μa c c c c c c Fig. 3.. The six possile wys tht,, nd c my e divided etween A nd μ A. Note tht c is not necessily singleton element; it my e pt of lge ptil ode. most fflm, fo some ffl<, then B(v) is empty nd thus w(v) = P AA(v) w v(a) nd we donothve to conside Tle. fo the est of the section. Recll tht A(v) denotes 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 nd L μ Ag: We letw H=L (v) e the contiution of the sets of A H=L (v) totheweight ofv, i.e., w H=L (v) = AA H=L (v) w v(a): Fo evity, wewitea h:::h=l :::l s (v) foa fh;:::;hg=fl ;:::;l sg(v) ndw h:::h=l :::l s (v) fo w fh;:::;hg=fl ;:::;l sg(v). Befoe poceeding, we descie the intuition tht lies ehind the est of the poof. Conside Tle. fom the lst section. If, in node v of the decision tee, the two cses A; A μ nd A; μ A e not eqully likely, o moe pecisely, if the contiutions w = (v) nd w = (v) of these two cses to the totl weight of v e not equl, thee must e t lest one child of v whose weight is gete thn hlf the weight of v. The difculty inimpoving the lowe ound of Bent nd John lies theefoe t nodes in which the the contiutions of the two csesa; A μ nd A; μ A e lmost equl. This fct is not so esily seen when looking t the oiginl poof given in []. Suppose now thtv isnodeinwhich n element of pi f; g is comped with n ity element c nd tht the nume of non-singletons in v is t most fflm. We ssume, without loss of genelity, tht >. The weights of set A A(v) in v's childen depend on which of the elements,, nd c elongs to A, nd on whethe c is miniml in A o mximl in A. μ The six possile wys of dividing the elements,, nd c etween A nd A μ e shown in Figue 3.. The weights of the set A in v's childen, eltive totheweight wv(a) ofa t v, inech 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 diffeence, howeve. If ; ; c A, s in the st owoftle 3., then the weight ofa in v >c is equl to the weightofa in v. The weight is not hlved, s my e the cse in the st ow oftle.. If the contiution w c= (v) ofthecse; ; c A to the weight ofv is not negligile, thee must gin e t lest one child of v whose weight is gete thn hlf the weight ofv. The impoved lowe ound is otined y showing tht if the contiutions of the cses A, A μ nd A, μ A e oughly equl, nd if most elements in the ptil ode e singletons, then the contiution of the cse ; ; c A is nonnegligile. The lge the nume of singletons in the ptil ode, the lge is the eltive contiution of the weight w c= (v) totheweight w(v) ofv. Thus, wheneve

8 8 D. DOR AND J. HνASTAD AND S. ULFBERG AND U. ZWICK cse w v <c (A) w v >c (A) A A c A o A A μ c A o μ A μ A c A 0 A A c μ A 0 A μ A c μ A 0 μ A μ A c μ A o Tle 3. The weight of set A A(v) in the childen of node v, eltive to its weight in v, when the element of pi >is comped with n ity element c. n element of pi is comped fo the second time, we mke smll gin. The ove intuition is mde pecise in the following lemm: Lemm 3.. If v is node in which n element of pi > is comped with n element c, nd 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 consideing the enties in Tle 3.. To otin the second 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) nd w =c (v) +w =c (v) = w =c (v), the second inequlity follows. It is woth pointing out tht in Tle 3. nd in Lemm 3., we only need to ssume tht >; we do not use the stonge condition tht >is pi. This stonge condition is cucil howeve in the sequel, especilly in Lemm 3.4. To mke use of Lemm 3. we need ounds on the eltive contiutions of the diffeent cses. The following lemm is useful tool fo detemining such ounds. Lemm 3.3. Let G =(V ;V ;E) e iptite gph. Let f nd f e the miniml degee of the vetices of V nd V,espectively. Let nd e the mximl degee of the vetices of V nd V,espectively. Assume tht positive weight function w is dened on the vetices of G such tht w(v )= w(v ), wheneve v V, v V nd (v ;v ) E. Let w(v )= P v V w(v ) nd w(v )= P v V w(v ). Then, f w(v )» w(v )» w(v ): f Poof. Let v (e) V nd v (e) V denote the two vetices connected y the edge e. We thenhve f w(v )» v V ee w(v (e)) = ee w(v (e))» w(v ): v V

9 ON LOWER BOUNDS FOR SELECTING THE MEDIAN 9 The othe inequlity follows y exchnging the oles of V nd V. Using Lemm 3.3 we otin the following sic inequlities. Lemm 3.4. If v is node in which > is pi nd the nume of nonsingletons in v is t most fflm, then ( ffl) w c=(v)» w c= (v)» ( ffl) w c= (v)» w c= (v)» ( ffl) w =c(v)» w =c (v)» ( ffl) w =c (v)» w =c (v)» ( ffl) w c=(v) ; ffl w c=(v) ; ( ffl) w =c(v) ; ffl w =c(v) : Ech one of these inequlities eltes weight, such sw c= (v), to weight, such s w c= (v), otined y moving one of the elements of the pi >fom A to A. μ In ech inequlity we `lose' fcto of ffl. When the elements nd e joined togethe fcto of is intoduced. When the elements nd e septed, fcto of is intoduced. Poof. We pesent poof of the inequlity w c= (v)» ( ffl) w c=(v). The poof of ll the othe inequlities is lmost identicl. Constuct iptite gph G =(V ;V ;E) whose vetex sets e V = A c= (v) nd V = A c= (v). Dene n edge (A ;A ) E etween A A c= (v) nd A A c= (v) if nd only if thee is singleton d A μ such tht A = A nfg [fdg. Suppose tht (A ;A )issuch n edge. As 6 min v (A )utmin v (A ), while ll othe elements e exteml with espect to A if nd only if they e exteml with espect to A (note tht min v (A ) nd mx v ( A μ )), we get tht wv(a ) = w v(a ). Fo evey set A of size m, the nume of singletons in A is t lest ( ffl)m nd t most m. We gettheefoe tht the miniml degees of the vetices of V nd V e f ;f ( ffl)m nd the mximl degees of V nd V e ;» m. The inequlity w c= (v)» ( ffl) w c=(v) theefoe follows fom Lemm 3.3. Using these sic inequlities we otin: Lemm 3.5. If v is node in which > is pi nd the nume of nonsingletons is t most fflm, fo some ffl<, then w c= (v) ( ffl) ( ffl) w c= (v) ; w =c (v) ( ffl)(3 ffl) ( ffl) w =c (v) ; w c= (v)» ( ffl) 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 3.4 we get tht nd the st inequlity follows. w c= (v) =w c= (v)+w c= (v)+w c= (v)» w c= (v)+ ffl w c=(v)+ ( ffl) w c= (v) = ( ffl) ( ffl) w c= (v)

10 0 D. DOR AND J. HνASTAD AND S. ULFBERG AND U. ZWICK We e now edy to show tht if v is node in which n element of pi is comped fo the second time, then v hs child whose weight is gete thn hlf the weight ofv. Comining Lemm 3. nd Lemm 3.5, we get tht (w(v <c)+w(v >c )) ( ffl) w(v)+ 4( ffl) w c= (v) ; w(v >c ) w(v) ffl ( ffl) w c=(v)+ ( ffl)(3 ffl) ( ffl) w =c (v) : Let ff = w c= (v)=w(v) nd ff = w =c (v)=w(v). We get tht (w(v <c)+w(v >c )) + ( ffl) 4( ffl) ff w(v) ; w(v >c ) As consequence, we get tht mxfw(v <c );w(v >c )g mx = ffl ( ffl) 3 ffl ( ffl)(3 ffl) ff + ( ( ffl) ff) ( ffl) ff + w(v) ( ffl)(3 ffl) ( ffl) : w(v) n o + ( ffl) 4( ffl) ff; 3 ffl ( ffl) ff + ( ffl)(3 ffl) ( ffl) w(v) : The coefcient ofw(v), on the ight hnd side, is minimized when the two expessions whose mximum is tken e equl. This hppens when ff = (3 4ffl+ffl ) 7 6ffl+ffl. Plugging this vlue of ff into the two expessions, we get tht whee mxfw(v <c );w(v >c )g ( + f (ffl)) w(v) ; f (ffl) = (3 ffl)( ffl) 3 ( ffl) (7 6ffl + ffl ) : It is esy to check thtf (ffl) > 0foffl<. A pi-foming compison is compison in which two singletons e comped to fom pi. A pi-touching compison is compison in which n element of pi is comped fo the second time. In pi-foming lgoithm, the nume of singletons is decesed only y pi-foming compisons. Ech pi-foming compison deceses the nume of singletons y exctly two. As explined ove, pi-foming compisons cn lwys e delyed so tht pi-foming compison : is immeditely followed y compison tht touches the pi f; g, o y pi-foming compison 0 : 0 nd then y compison tht touches oth pis f; g nd f 0 ; 0 g. Conside gin the pth tced fom the oot y epetedly descending to the child with the lge weight. As consequence of the ove discussion, we get tht when the i-th pi-touching compison long this pth is pefomed, the nume of non-singletons in the ptil ode is t most 4i. It follows theefoe fom the emk mde t the end of the pevious section tht the depth of the compison tee coesponding to ny pi-foming lgoithm is t lest m=4 n + log ( + f ( 4i )) + o(n) m =n + n 8 i= Z 0 log ( + f (t))dt + o(n) ß :0069n + o(n) :

11 ON LOWER BOUNDS FOR SELECTING THE MEDIAN A μa c d c d c d c d c d c d c d c d c d Fig. 3.. The nine possile wys tht,, c, ndd my e divided etween A nd μ A. cse w v <c (A) w v >c (A) w v <d (A) w v >d (A) A A cd= A A cd= A A cd= 0 0 A A c=d 0 A A c=d 0 A A c=d 0 A A =cd 0 0 A A =cd 0 0 A A =cd Tle 3. The weight of set A A(v) in the childen of node v, eltive to its weight in v, when the element of pi >is comped with n element of pi c>d. This completes the poof of Theoem 3.. The wost cse in the poof ove is otined when the lgoithm convets ll the elements into qutets. A qutet is ptil ode otined y comping elements contined in two disjoint pis. In the poof ove, we nlyzed cses in which n element of pi >is comped 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 3.. Theoem 3.6. A pi-foming lgoithm fo nding the medin must pefom, in the wost cse, t lest :07n + o(n) compisons. Poof. Conside compisons in which the element fom pi >is comped with n element of pi c>d. The nine possile wys of dividing the elements,, c, ndd mong A nd μ A e depicted in Figue 3.. We my ssume, without loss of genelity, tht the element is comped with eithe c o with d. Let v e node of the compison tee in which >nd c>de pis nd which one of the compions : c o : d is pefomed. Let A A(v). The weights of set A in v's childen, eltive totheweight w v(a) ofa t v, inech one of these nine cses e given in Tle 3.. The two possile compisons : c nd : d e consideed septely. The following equlities e esily veied. Lemm 3.7. If >nd c>de pis in v then w cd= (v) = w c=d (v) ; w cd= (v) = w =cd (v) ;

12 D. DOR AND J. HνASTAD AND S. ULFBERG AND U. ZWICK w c=d (v) = w =cd (v) ; w c=d (v) =4 w =cd (v) : The following inequlities e nlogous to the inequlities of Lemm 3.4. Lemm 3.8. If >nd c>de pis in v nd if the nume of non-singletons in v is t most fflm, fo some ffl<, then ( ffl)w c=d(v)» w cd= (v)» ( ffl)w =cd (v)» w c=d (v)» ( ffl)w =cd(v)» w =cd (v)» ( ffl)w =cd (v)» w =cd (v)» ( ffl) w c=d(v) ; ffl w =cd(v) ; ( ffl) w =cd(v) ; ffl w =cd(v) : Conside st the compison : c. By exmining Tle 3. nd using the equlities of Lemm 3.7, we get tht w(v <c)+w(v >c) = w cd= (v)+ 3 4 w cd=(v)+ w cd=(v)+ 3 4 w c=d(v)+ w c=d(v) + w c=d(v)+ w =cd(v)+ w =cd(v)+ w =cd(v) = w cd= (v)+ 3 w c=d(v)+3w =cd (v)+w =cd (v)+ w =cd(v): Minimizing this expession, suject to the equlities of Lemm 3.7, the inequlities of Lemm 3.8, nd 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 (ffl)) w(v) ; whee f (ffl) = (3 ffl)( ffl)3 ( ffl) 4 : It seems intuitively cle tht the compison : d is d compison fom the lgoithm's point of view. The dvesy will most likely nswe with >d. Indeed, y solving the coesponding line pogm, we get tht w(v >d ) = w cd= (v)+ w cd=(v)+w c=d (v)+w c=d (v) + w c=d(v)+w =cd (v)+w =cd (v)+w =cd (v) = w cd= (v)+ 3 w c=d(v)+5w =cd (v)+ 3 w =cd(v)+w =cd (v) 3 4 : As ( + f (ffl))» 3,foevey 0» ffl», we my disegd the compison : d fom 4 ny futhe considetion. It is esy to veify tht ( + f (ffl)) +f (ffl). As esult, we getlowe ound of n + n 8 Z This completes the poof of Theoem log ( + f (t))dt + o(n) ß :07n + o(n) :

13 ON LOWER BOUNDS FOR SELECTING THE MEDIAN 3 4. Concluding emks. We pesented efomultion of the n + o(n) lowe ound of Bent nd John fo the nume of compisons needed fo selecting the medin of n elements. Using this new fomultion we otined n impoved lowe ound fo pi-foming medin nding lgoithms. As mentioned, Do nd Zwick [6] hve ecently extended the ides descied hee nd otined (+ffl)n lowe ound fo genel medin nding lgoithms, fo some tiny ffl>0. We elieve tht the lowe ound fo pi-foming lgoithms otined hee cn e sustntilly impoved. Such n impovement seems to equie, howeve, some new ides. Otining n impoved lowe ound fo pi-foming lgoithms my e n impotnt step towds otining lowe ound fo genel lgoithms which is signicntly ette thn the lowe ound of Bent nd John []. Pteson [] conjectues tht the nume of compisons equied fo selecting the medin is out (log 4=3 ) n ß :4n. REFERENCES [] Mtin Aigne. Poducing posets. Discete Mthemtics, 35: 5, 98. [] Smuel W. Bent nd John W. John. Finding the medin equies n compisons. In Poceedings of the Seventeenth Annul ACM Symposium on Theoy of Computing, pges 3 6, 985. [3] Mnuel Blum, Roet W. Floyd, Vughn Ptt, Ronld L. Rivest, nd Roet E. Tjn. Time ounds fo selection. Jounl of Compute nd System Sciences, 7:448 46, 973. [4] Jingsen Chen. Ptil Ode Poductions. PhD thesis, Lund Univesity, Box 8, S- 00 Lund, Sweden, 993. [5] Doit Do nd Ui Zwick. Selecting the medin. In Poceedings of 6th SODA, pges 88 97, 995. Jounl vesion in SIAM Jounl on Computing, 8:7 758, 999. [6] Doit Do nd Ui Zwick. Medin selection equies (+ffl)n compisons. In Poceedings of 37th FOCS, pges 5 34, 996. Jounl vesion to ppe in SIAM Jounl on Discete Mthemtics. [7] Leste R. Fod nd Selme M. Johnson. A tounment polem. Ameicn Mthemticl Monthly, 66: , 959. [8] Fnk Fussenegge nd Hold N. Gow. A counting ppoch to lowe ounds fo selection polems. Jounl of the Assocition fo Computing Mchiney, 6():7 38, Apil 979. [9] Donld Evin Knuth. The At of Compute Pogmming, vol. 3, Seching nd Soting. Addison-Wesley Pulishing Compny, Inc., 973. [0] I. Muno nd P.V. Polete. A lowe ound fo detemining the medin. Technicl Repot Resech Repot CS-8-, Univesity of Wteloo, 98. [] Michel S. Pteson. Pogess in selection. In 5th Scndinvin Wokshop on Algoithm Theoy, Reykjvík, Icelnd, pges , 996. [] Vughn R. Ptt nd Foong Fnces Yo. On lowe ounds fo computing the i-th lgest element. In 4th Annul Symposium on Switching nd Automt Theoy, pges 70 8, 973. [3] A. Schönhge, M. Pteson, nd N. Pippenge. Finding the medin. Jounl of Compute nd System Sciences, 3:84 99, 976. [4] C.K. Yp. New lowe ounds fo medins nd elted polems. Compute Science Repot 79, Yle Univesity, 976. Astct in Symposium on Algoithms nd Complexity: New Results nd Diections, (J. F. Tu, ed.) Cnegie-Mellon Univesity, 976.