x. Itrducti The k-d tree, r k-dimesial biary search tree, was prpsed by Betley i 75. I this paper, we prpse a mdicati, the squarish k-d tree, ad aalyz

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1 Squarish k-d trees Luc Devrye, Jea Jabbur ad Carls Zamra-Cura Schl f Cmputer Sciece McGill Uiversity Mtreal, Caada h3a 2k6 fluc, jabbur, czamrag@cs.mcgill.ca bstract. We mdify the k-d tree [; ] d by always cuttig the lgest edge istead f rtatig thrugh the crdiates. This mdicati makes the expected time behavir f lwer-dimesial partial match queries behave as fr perfectly balaced cmplete k-d trees des. This is i ctrast t a result f Flajlet ad Puech, wh prved that fr (stadard radm k-d trees with cuts that rtate amg the crdiate axes, the expected time behavir was much wrse tha fr balaced cmplete k-d trees. We als prvide results fr rage searchig ad earest eighbr search fr ur trees. Keywrds ad phrases. k-d trees, partial match query, rage search, expected time, prbabilistic aalysis f algrithms, data structures. Cmputig Reviews categries 3.74, 5.25, 5.5. Research f the authrs was spsred by NSRC Grat 3456 ad by FCR Grat - R-2. The secd authr was leave frm the Departmet f Mathematics, Uiversite de Versailles, Frace. ddress Schl f Cmputer Sciece, McGill Uiversity, 34 Uiversity Street, Mtreal, Caada H3 2K6. The third authr received a DGP-UNM Schlarship.

2 x. Itrducti The k-d tree, r k-dimesial biary search tree, was prpsed by Betley i 75. I this paper, we prpse a mdicati, the squarish k-d tree, ad aalyze its expected time perfrmace fr partial match queries, rthgal rage searchig, ad earest eighbr search uder the stadard radm mdel fr the iput ( pits idepedetly ad uifrmly distributed the uit hypercube. We pit ut its superirity ver the stadard k-d tree fr this mdel. Betley's k-d tree is a biary search tree that geeralizes the -d tree r rdiary biary search tree t IR k. partiti f space it hyper-rectagles is btaied by splittig alteratig crdiate axes by hyper-plaes thrugh data pits. Figure shws the partiti ad the crrespdig k-d tree. Iserti ad search are implemeted as fr the stadard biary search tree algrithms. These trees are used fr a variety f ther peratis, icludig rthgal rage searchig (reprt all pits withi a give rectagle, partial match queries (reprt all pits whse values match a give k-dimesial vectr with pssibly a umber f wildcards, e.g., we may search fr all pits with values (a ; ; ; a 4 ; a 5 ;, where detes a wild card. dditially, earest eighbr searchig is greatly facilitated by k-d trees. Fr rthgal rage searchig, a hst f particular data structures have bee develped, such as the rage tree ad variatis r imprvemets f it (fr surveys, see Betley ad Friedma (7, Ya (, Samet (a, b, ad garwal (7. Hwever, the k-d tree ers several advatages{it takes O(k space fr data pits, it is easily updated ad maitaied, it is simple t implemet ad cmprehed, ad it is useful fr ther peratis besides rthgal rage search Figure. The k-d tree ad its partiti f the plae. The query rectagle is shaded. Betley's rthgal rage search algrithm simply visits recursively all subtrees f the rt that have a empty itersecti with the query rectagle. I gure fr example, the 2

3 left ad right subtrees f the rt are visited. Nte that each de i the tree represets bth a pit f the data ad a rectagle i the partiti, amely the rectagle split by that pit. Leaf regis thus have pits strictly i their iterir. The query time fr rthgal search depeds up may factrs, such as the lcati f the query rectagle, ad the distributi f the pits. Oe may cstruct a media k-d tree -lie by splittig each time abut the media, thus btaiig a perfectly balaced biary tree, i which rdiary pit search takes (lg wrst-case time, ad a partial match query with s crdiates specied takes wrst-case time O(?s=k + N, where N is the umber f pits retured (see, fr example, Lee ad Wg, 77. Fr -lie iserti, balacig is triusly dicult. If we assume that the data are idepedet ad have a cmm distributi, the the expected query time is clearly f iterest. Fr stadard radm biary search trees, it is kw (Kuth, 7; Pittel, 4; Devrye, 6, 7; Mahmud, 2 that mst prperties f balaced search trees are iherited the expected depth f a radmly selected de is abut 2 lg ad the expected height is O(lg. Oe wuld hpe that the radm k-d tree, cstructed by csecutive iserti f data pits, wuld als have a perfrmace clse t that f the media (lie k-d tree. ssumig that the data pits are draw frm the uifrm distributi the uit k-cube, Flajlet ad Puech (6 shwed that a radm partial match query (carried ut with s values als draw uifrmly ad idepedetly [; ], s, there are k? s wildcards has expected time perfrmace (?s=k+(s=k, where (u is a strictly psitive fucti f u 2 (;, with maximum t exceedig 7. Thus, radm k-d trees behave a bit wrse tha their balaced cuterparts, the media k-d trees. Surveys f related kw prbabilistic results are prvided by Vitter ad Flajlet (, ad Get ad Baeza-Yates (. Figure 2. Tw radm k-d tree partitis clearly shw the elgated rectagles. We prpse a mir mdicati f the iserti prcedure, amely, each time a rectagle is split by a ewly iserted leaf pit, the lgest side f its rectagle is cut, that is, the cut is a (k? -dimesial hyper-plae thrugh the ew pit perpedicular t the lgest edge f the rectagle. It was shw by Chazy, Devrye, ad Zamra-Cura ( that 3

4 elgated rectagles explai the pr perfrmace f radm k-d trees. I this paper, we shw t what extet the prpsed k-d trees have mre squarish-lkig rectagles, ad will therefre call these trees squarish k-d trees. Fr the prbabilistic mdel f Flajlet ad Puech, it will be shw that the expected time fr a partial match query is (?s=k, just as fr radm media k-d trees. Furthermre, the expected cmplexity f ay rthgal rage search i media k-d trees is asympttically equivalet t that fr the simple squarish k-d trees prpsed here. I the last part f the paper, we deal with rthgal rage search i geeral whe the query rectagles may have dimesis that deped up i a arbitrary fashi. The prfs are prbabilistic, rather tha aalytical, ad d t er explicit cstats fr expected times but ly ( results. Hwever, they are shrt ad explai may f the phemea at wrk. Iterestigly, very little prbability beyd Hlder's iequality is eeded. We cclude the paper by shwig that a atural earest eighbr search (with a radmly selected prbe pit takes O(lg lg lg expected time i ay dimesi. We shuld te that there are ideed mre sphisticated data structures fr sme f the subprblems dealt with here. Fr example, if e is just iterested i partial match queries, the e culd just make j-d trees fr each f the 2 k? empty subsets f size j f the k crdiates separately, s that search i the prper tree is just a pit search, takig expected wrst-case time O(lg, while the space used is still O(2 k. Hwever, these wuld t be helpful fr geeral rthgal, simplex, r cvex rage searches. Fr a aalysis f rage search based multiattribute trees see Gardy, Flajlet, ad Puech (. x2. The radm prcesses I this secti, we will try t explai the diereces betwee alteratig cuts ad lgestedge cuts i sequeces f radmly cut rectagles. T explai the prcesses at wrk, we csider the fllwig simplicati f ur prblem i IR 2, start with a rectagle with e vertex permaetly pegged at the rigi ad the ppsite e at (;, ad let (U ; V dete the crdiates f the tp right vertex after iteratis, with (U ; V = (;. The rectagle will be reduced i size, rst by alteratig uifrm cuts, that is, if Z ; Z 2 ; are i.i.d. uifrm [; ] radm variables, the we set Clearly, at time 2, we have (Z U (U ; V =? ; V? U 2 L = V 2 L = if is dd; (U? ; Z V? if is eve. Y i= Z i L = e? P i= i L = e?g where the i are idepedet expetial radm variables, ad G detes a gamma radm variable with parameter. s U k ad V j are idepedet f each ther fr all k; j, we see that the rati U 2 V 2 L = e G?G 4

5 where G ; G are i.i.d. gamma radm variables. By the cetral limit therem, it is easy t see the that p lg U2 L! N? N ; V 2 a dierece f tw idepedet stadard rmal radm variables. Thus, the raw rati behaves asympttically like exp( p (N? N, ad thus exhibits wide swigs. I fact, if we stp at a large value fr, the rectagle will lk very skiy ideed (see Figure 2. Sice we wuld like t preserve squarish rectagles, we may pt istead t always cut the lgest side f the rectagle. Mre frmally, with tati as abve, (U ; V = (;, we have (Z U (U ; V =? ; V? if U? > V? ; (U? ; Z V? if U? < V?. I case f equality U? = V?, which ly ccurs at =, we ip a perfect ci ad pick a edge t cut at radm. Lemma. With the lgest-edge cuttig methd, the sequece U =V,, is idetically distributed. The cmm distributi is that f Z =Z 2, the rati f tw idepedet uifrm [; ] radm variables. Prf. Clearly, U =V is distributed as Z with prbability =2 ad as =Z therwise. It is easy t verify that this has the required desity =(2 max(z; 2, z >. By iducti, we eed t shw that if Z ; Z 2 ; Z are i.i.d. uifrm [; ] radm variables, the the radm variable ZZ =Z 2 I Z>Z 2 + Z =(ZZ 2 I Z<Z 2 is i tur distributed as Z =Z 2. This ca be de by stadard calculatis, r eve the methd f characteristic fuctis. Hwever, by far the quickest way t see this is by embeddig. We te that Z =Z 2 is distributed as the radm variable Z S 4 where S = ad S =? with equal prbability, ad Z 4 is ather uifrm [; ] radm variable. The case Z > Z 2 crrespds t S =?, ad thus, we see that ZZ =Z 2 I Z>Z 2 + Z =(ZZ 2 I Z<Z 2 is distributed as (Z 4 =Z S, which was t be shw, as S is idepedet f Z ad Z 4. Lemma shws that cuttig the lgest edge is extremely stabilizig. Nevertheless, as U =V has Cauchy-like tails, its mea des t exist, ad we will fte see skiy rectagles, althugh by ad large, the rectagles will be rather squarish. The abve bservatis explai why the squarish k-d trees are useful. Our aalysis is f curse mre ivlved, as rectagles participate i a evlvig cllecti f rectagles, with very itricate depedecies. s s as a rectagle becmes t small, it is ulikely t be picked agai s, ad thus, the rati f the sides f the rectagles must be csidered i cjucti with the sizes. Fr this, we itrduce a few ew aalysis methds. 5

6 3 lg (U /V Figure 3. Fr the tw prcesses abve, lg(u =V is pltted versus. The alteratig cuts prcess waders just as a radm walk. The largest edge cut strategy iduces a sequece U =V that hvers ear e ad remais statiary. Figure 4. Fr the tw prcesses abve, let L ad S p be the lg ad shrt dimesis f a leaf rectagle i the 2-d partiti. The values L =S are pltted versus p L S (rmalized s that the largest value is e fr the tw k-d trees. Fr squarish k-d trees ( the right, there are may mre rectagles i which L ad S are clse. d early all big rectagles are squarish. Fr the rdiary radm 2-d tree, the left, mst rectagles have very small edge ratis. 6

7 x3. Ntati ad prelimiaries I k-d trees, des represet rectagular regis. Betley's algrithm fr rthgal rage search ad partial match queries starts at the rt f k-d tree ad recursively visits all subtrees that have a empty verlap with the rectagular regis f the childre, ad reprts all pits that fall i the search regi. Let u ; u 2 ; ; u, dete the des i the k-d tree, ad let U ; ; U dete the data pits, which are i.i.d. ad uifrmly distributed [; ] k. Thus, U i is the data pit crrespdig t u i. The rectagle split by u i is R i. Thus, R = [; ] k. Let jr i j dete the vlume f rectagle i. The + leaf rectagles (the daglig edges i Figure are als deted R i, with the idex i w ruig frm + t 2 +. The cllecti f rectagles is deted by R. The cllecti f the idices f the + al rectagles is F. We will dete by T the k-d tree cstructed by isertig successively u ; u 2 ; ; u it a iitially empty k-d tree. Give a de u i T, we will dete by T u the subtree f T rted at u. With rtatig crdiate cuts, such a tree is called a radm k-d tree. With ur methd f cuttig the largest edges, it will be called a radm squarish k-d tree. The dimesis f rectagle R i are ij ; j k. Fr 2-d trees, we will use the lighter tati i ; Y i fr the x ad y dimesis f R i. The query rectagle Q is Z + [?m ; m ] [?m k ; m k ], m i fr all i, where the m i 's are xed (that is, they may deped up ly ad Z is uifrmly distributed [; ] k ad idepedet f (U ; ; U. Betley's rage search applied t Q is called a radm rthgal rage search. Nte that a de u i is visited by the rage search algrithm if ad ly if the query rectagle Q itersects R i. y rectagle R i is visited if ad ly if it itersects Q. Let N be the time cmplexity f Betley's rthgal rage search. The, N = 2+ i= [Ri\Q6=;] This quatity will be aalyzed further fr radm squarish k-d trees. I a radm partial match query, we specify a subset f s dimesis, j ; ; j s, ad perfrm a rthgal rage query with the i-th iterval i the rectagle either fz i g (a uifrm radm umber [; ] if i 2 fj ; ; j s g, r (?; therwise. It is assumed that the Z i 's are idepedet, ad idepedet f (U ; ; U. I this paper, we rst study radm partial match queries fr radm squarish k-d trees ad btai results that shuld be cmpared agaist the fllwig result fr radm k-d trees Therem (Flajlet ad Puech, 6. Fr a radm k-d tree ad a radm partial match query, i which s f the k elds are specied with k > s, let N (s be the umber f cmpariss that Betley's rthgal rage search perfrms. Dee (?u u? t t (u = max t + 2? 2 ; < u < ; t? u u ad te i particular that is decreasig (;, ( =, ad that? u < (u < 7? u, < u <. The = (c + ( (s=k ; N (s 7

8 where c is a cstat depedig the idices f the s xed crdiates. The fllwig prpsiti is useful i relatig partial radm partial match queries t the rage search prblem. Prpsiti. Give is a radm k-d tree based i.i.d. radm variables U ; ; U, distributed uifrmly [; ] k. Csider a radm partial match query, i which s f the k elds are specied. Let N (s be the umber f cmpariss that Betley's rthgal rage search perfrms. Let S be the set f specied crdiates. The N (s = < 2+ Y = ij ; ; where ij ; j k are the legths f the sides f rectagle R i i R. i= j2s Q Prf. Let Q be the query rectagle. Nte that P fq \ R i 6= ; j U ; ; U g = j2s ij. Thus we have, N (s = ( 2+ i= [Q\Ri6=;] = 2+ i= P fq \ R i 6= ;g = < 2+ Y = ij ; i= j2s The fllwig bservati is imprtat. It fllws immediately by csiderig the radm grwth f ur k-d trees. d f curse, it implies that that the jit distributi f the rdered vlumes f the + leaf rectagles is idetical fr bth radm k-d trees csidered here! Lemma 3. Csider a radm k-d tree r a radm squarish k-d tree. The, the vlumes f the rectagles i F are distributed as the set V f the csecutive spacigs betwee the rder statistics f i.i.d. radm variables, uifrmly distributed [; ]. x4. Radm partial match queries with squarish 2-d trees I a radm partial match query a 2-d tree, we take a uifrmly distributed value Z, ad visit all des i the tree whse rectagle cuts the vertical lie at Z. The prbability f hittig a rectagle with dimesis i Y i is f curse i, s that the expected umber f des visited, ad hece, the expected time fr a partial match query, is simply P 2+ i= i, where the sum is take ver all 2 + rectagles i the partiti. similar frmula hlds f curse fr hriztal partial match queries. I this secti, we prve that a radm partial match query i a radm squarish 2-d tree takes expected time ( p as ppsed t ( 566 fr radm 2-d trees (see Therem.

9 Therem 2. Fr a radm squarish 2-d tree, p 3 ( 2+ i= Y i p The same result hlds fr P2+ match query is ( p. i= i. Hece, the expected time fr a radm partial Of curse, attempt was made t ptimize the cstats. few techical results will be eeded i the sequel. Lemma 4. Fr p ;, bpc+ (?(p + jr i j p 4?(p + ; + p p? p? i2f fr all. Prf. Let V ; ; V + be the spacigs iduced by idepedet uifrmly distributed radm variables [; ]. It is kw that V L i = Beta(;. Thus, by Lemma 3, with B(s; t =?(s?(t?(s+t ( i2f jr i j p = ( + i= Vi p = +Z i= B(p + ; = ( + B(; p (? v? v dv B(;?( + 2 =?(p +?(p + + Nw, as?(x + = x?(x fr ay x >, ad fr ay atural umber ad ay s 2 [; ],?s?( + =?( + s ( +?s (see Mitrivic, 7, the ( i2f jr i j p?( + =?(p + ( + ( + p ( + p? bpc?( + p? bpc?(p + ( + 2?p+bpc bpc+?(p + + = p? 4?(p + ; p? 2+bpc?p

10 as 2 + bpc? p 2. Nw, fr the lwer bud, te that ( i2f ( i Y i p?( + =?(p + ( + ( + p ( + p? bpc?( + p? bpc?(p + p??(p + p??(p + p? + p bpc+ ( + p ( + p?cpb bpc+ + p bpc+ Lemma 5. I a radm squarish 2-d tree, fr every q, ( i2f >< Yi q >?q=2?q=2 ; fr q 2 [; 2; e lg ; fr 2? 2 5?(q=2+ q=2?? q 2? q=2? ; fr q > 2, lg q 2; ad fr q 2 [; 2, ( i2f Yi q bq=2c+?(q=2 +?q=2 q=2 + The same result hlds fr P i2f q i. P Prf. Let r >, ad dee S r (q = Y q i. Nte that, give U ; ; U r, S (q? r+ S(q r is distributed as Y q whe > Y ad as Y q (U q + (? U q? whe Y, where U is a uifrm [; ] radm variable, ad (; Y are the dimesis f the rectagle split whe U r+ is added. Thus, ( S (q? r+ S(q r = i Y i? [i>y i] Y q i + [i<y i] Y q i ( q + (? q? Ntice that U q + (? U q? fr q, ad as mifa; bg p ab, fr a; b, the by Lemmas 3 ad 4, ( S (q? r+ S(q r (? i Y i [i>y i] Y q i ( i Y i q=2+ 4?(q=2 + 2 r q=2

11 By summig the diereces we get, S (q =? r= (? r= S (q r+ S(q r + S (q 4?(q= r q=2 Z? 2 + 4?(q= dx xq=2 ( + 4?(q=2+2 (?q=2? ; (q 2 [; 2?q=2 5?(q= ?(q=2+2 (??q=2 (q > 2 q=2? (?q=2?q=2 ; (q 2 [; 2? q??q=2 (q > ?(q=2+2 q=2? Because?q=2?q=2, as a fucti f q, reaches its miimum at q = 2(? = lg, ad S (q is a decreasig fucti f q, we have that S (q e lg, fr q q 2. P The result fr i2f q i ca be btaied similarly just by replacig the y-legths fr the x-legths i the apprpriate places. Nw, fr the lwer bud, te that as the i 's ad the Y i 's are idetically distributed ( ( Y q i = 2 (Y q i + q i i2f i2f ( (Y i i q=2 by Lemma 4, fr q 2 [; 2. i2f bq=2c+?(q=2 + ; q=2 + q=2? Prf f Therem 2. Nte that the lwer bud fllws frm Lemma 5, as P i2f Y i is less tha P2+ i= Y i. Fr the upper bud we will use the same techique as i the prf f Lemma 5. Let S = P 2+ i= Y i. Nte that as the sum is ver all the rectagles geerated by U ; ; U, we have w that fr r, as i ad Y i are idetically distributed, fs r+? S r g = = 3 (? i Y i [i>y i] 2Y i + [i<yi] (Y i U + Y i (? U ( i Y 2 i where U L = Uifrm[; ], ad idepedet f all U ; ; U. Let q 2 (; 2 ad p > such

12 that p + q =, the by Hlder's iequality used twice, ( i Y 2 i ( 4?(p + r p? ( i Y i p =p =p ( =q Y q i =q? q=2 r q=2? by Lemmas 4 ad 5. Take p = 3, q = 3=2, ad verify that the upper bud is t mre tha 24 =3 32 2=3 = p r < 3= p r. By summig the diereces we ally btai ( 2+ Y i 5? 2 + p 5 r 2 + (p?? p i= r= P2+ Fr the lwer bud, set q = i the lwer bud f Lemma 5. The result fr i= i ca be btaied similarly just by replacig the y-legths fr the x-legths i the apprpriate places. x5. The k-dimesial case. I this secti, we btai the k dimesial geeralizati f the results i the previus secti by iducti. Give U ; ; U, we dee fr each R i 2 R, i = max j=;;k ij ad ji as the idex j 2 f; ; g fr which ij = i. Nte that j i is uique w.p.. Our mai result geeralizes Therem 2 ad establishes the expected time ptimality f radm squarish k-d trees. Therem 3. Csider a radm squarish k-d tree. C; C > such that C? `k < 2+ Y = ij ; `k C? ; i= j2i Fr ` 2 f; ; k? g, there exist fr ay I f; ; kg f cardiality ` ad all 2 IN. I particular, by Prpsiti the expected time f a radm partial match query with s specied crdiates is (?s=k. The ext lemma cmplemets Therem 3 whe ` = k. Lemma 7. Let U ; ; U be idepedet uifrmly distributed radm variables ver [; ] k. Let R = fr ; R 2 ; ; R 2+ g be the rectagles i the partiti deed by the radm squarish k-d tree based U ; ; U. Let ij be the legth the j th crdiate f the i th rectagle. The, ( 2+ i= i ik = 2H +? ; 2

13 where H is the th harmic umber. We prve the fllwig lemma that will allw us t prve the lwer bud i the previus therem. Lemma. Let ` 2 f; ; kg, the fr every x ; ; x k >, k k x j max j= I If;;kg Y j2i ` x j Prf. Let I be the subset f f; ; kg f cardiality ` fr which the maximum abve is reached. It suces t bserve that, k ` ky x j s+`? Y k? Y x Y x j Y k x j ; j= s= j=s s= j2i j2i where the subidice j must be uderstd as (j md k, if j > k. Prpsiti 2. Let I f; ; kg f cardiality ` 2 f; ; kg ad p 2 [; k`, the there are psitive cstats C ad C such that fr all 2 IN. C?p `k < Y = ij p ; C?p k ` ; j2i Prf. Fr I f; ; kg with jij = `, we dee S I;p r Y j2i p ij We rst lk at the upper bud. We dee recursively the cstats C k (`; p fr ay iteger k >, ` 2 f; ; kg ad real umber p 2 [; k` as fllws, ( 4?(p + ; if ` = k; C k (`; p = (k? `? p` k C k (k; ~q =~q C k (` + ; p~p`=(` + =~p + 2 if ` < k, where ~p; ~q > deped p; k ad `, they are such that ~p + ~q Fr the sake f clarity we will chse ~p later. =, ad p~p ` `+ < k `+. Fr ` 2 f2; ; kg, we dee the hypthesis H` statig that the upper bud hlds fr all 2 IN, all I f; ; kg such that jij = `, ad all p 2 [; k`, with cstat C k(`; p. We 3

14 will prve H` with a iductive argumet. First, te that H k hlds by Lemma 4. ssumig that H` is true, we will prve H`?. Let I f; ; kg such that `? = jij, ad k p 2 [;. The fr ay iteger r we have, `? Y S I;p? r+ SI;p r ju ; ; U r k < ij [j Y p ij i j= j2i p Z = + [ji 2I] ij (x p + (? x p? dx ; Y j2i as we are usig the lgest edge cut methd. Sice R (xp + (? x p? dx fr ay p, we ca drp the secd term abve ad take expected values s that, S I;p? r+ SI;p r t62i < k j= ij [ji Y = ij p ; j2i Let us dete by (t the expected value f the t th term abve. Observe that [j i =t] ij `? ` ij ` it. Thus we ca bud each (t as fllws, >< Y k ij > Y j2i[ftg p `? ` >= ij >; Nw, fr ay ~p; ~q > such that ~p + ~q =, we have by applyig Hlder's iequality twice that, (t >< k > j= ij ~q >= ~q >; >< Y j2i[ftg `? ij ` p~p >= >; We ca apply hypthesis H` t bud the secd term abve, if we ca chse ~p > such that p~p `? k k ` 2 [; k=`. Nte that >, as p 2 [;. Let us dee ~p = p(`? `? maxp k=p(`? ; ` (`?p, s that ~p >, yet p~p `? ` < k`. This cmpletely dees the cstat C k (`; p. We ca therefre use hypthesis H` ad btai, Ck (k; ~q (t r ~q? =~q Ck (`; p~p(`? =` r `? k p~p? =~p = C k(k; ~q =~q C k (`; p~p(`? =` ~p r `? k p We ca thus bud the diereces as fllws, S I;p? r+ SI;p r t62i (t (k? ` + C k(k; ~q =~q C k (`; p~p(`? =` =~p 4 r `? k p ~p

15 Sice p < we get, as is prved. S I;p P k, we have that `? r= r p `? k?p `? k. S, by summig the diereces p `? k S I;p `? p(`??p [C k (`? ; p? 2] k? + 2 C k (`? ; p k 2, fr every p, ad ay empty I f; ; kg. Thus, hypthesis H`? We w prve the lwer bud. s we ip a perfect ci at the begiig f the prcess t chse the side f R that we cut, all the crdiates i ; ; ik f a rectagle R i are exchageable. S, detig by S the set f all I f; ; kg f cardiality `, all the radm variables P i2f Q j2i p ij The, by Lemmas 4 ad 7, i2f are equally distributed s that < Y p ij i2f j2i = ; = Y = ij p ; j2i < Y jsj jsj p ij I 2S i2f j2i >< k > i2f j= ij p` k >= = ; >; C p` k We must te that by Lemma 4, if ` = k, the fr ay p, there are psitive cstats C ad C, depedig p such that the previus result hlds. We are w ready t prve Therem 3. Prf f Therem 3. The lwer bud fllws immediately frm the previus prpsiti. Fr ay subset I f; ; kg f cardiality ` 2 f; ; k? g, we dee S I = 2 Y i= j2i ij s we are usig the lgest edge cut methd we have that, S I r+? S I r ju ; ; U = 3 ky < ij [j 62I]2Y i j= j2i ky Y ij ij j= j2i ij + [j i 2I] Y j2i ij = ; p We chse w p = k=`, q = =(? p`=k, s that p + q =, ad apply Hlder iequality with these values t get, S I r+? S I r 3 < k j= p = ij ;=p 5 Y j2i q = ij ;=q

16 The by Lemma 4 ad Prpsiti 2, there exists a psitive cstat C depedig up ` ad k such that S I r+? S I r C r `k We add the diereces t get S I C r `k r=! + 2 C? `k + 2 `k Prf f Lemma 7. First, te that fr ay i, i ik is the vlume jr i j f the rectagle R i. Nte that if U ; ; U i have already bee iserted i [; ] k, ad U i+ is a ew pit, the the size f the tw rectagles geerated by U i+ is equal t the size f the rectagle i the al partiti f [; ] k i which U i+ falls. Let us dete by R(U i+ this rectagle. Thus, ( 2+ i=? i ik = + i= f fjr(u i+ j j U ; U i gg ; where the accuts fr the rt rectagle. We claim that fjr(u i+ jg = 2 i+2. Nte that the claim is bviusly true fr i =. Nw, suppse that U ; ; U i have already bee iserted i the squarish k-d tree, s that there are i + exteral des. These exteral des represet the i + rectagles partitiig [; ] k. Let these rectagles be S ; ; S i+, ad let the umberig be s that the leaves are take frm left t right, i rder f appearace as leaves i the squarish k-d tree f U ; ; U i. The, fjr(u i+ jg = = = ( ( i+ `= ( i+ `= ( i+ [Ui+2S`] js`j U ; U i `= js`j P U i+ 2 S` js`j 2 U ; ; U i It is well kw that (js j; ; js i+ j are jitly distributed as uifrm spacigs, that is the legths f the itervals [; ] deed by a i.i.d. uifrm [; ] sample f size i. ll these spacigs are idetically distributed fllwig a Beta(; i distributi. If B is a Beta(i radm variable, the we have fbg = =(i+ ad B 2 = 2=((i+(i+2. Therefre, fjr(u i+ jg = (i + B 2 = 2 i + 2 ad thus,? + fjr(u i+ jg = + 2(H +? i= 6

17 x6. Orthgal rage search I this secti, we btai tight upper buds fr the expected cmplexity fr Betley's rage search algrithm. Fr radm rthgal rage search, the fllwig therem establishes the stadard fr cmpariss. Therem 5 belw the states that radm squarish k-d trees are superir t radm k-d trees fr ay kid f radm rthgal rage search. Therem 4 (Chazy, Devrye ad Zamra-Cura,. Give is a radm k-d tree f size. Let Q be a radm query rectagle f dimesis k (which are determiistic fuctis f takig values i [; ], with ceter at Z which is uifrmly distributed [; ] k, ad idepedet f the k-d tree. Let N be the umber f cmpariss that Betley's rthgal rage search algrithm perfrms. The, there exist cstats > > depedig up k ly such that lg + P If;;kg jij<k fn g Q j =2I j (jij=k! ; where ( is the fucti deed i Therem. Therem 5. Let Q be a radm query rectagle f dimesis k (which are determiistic fuctis f takig values i [; ], with ceter at Z which is uifrmly distributed [; ] k, ad idepedet f the k-d tree. Let N be the umber f cmpariss that Betley's rthgal rage search algrithm perfrms. The, there exist cstats > > depedig up k ly such that fn g lg + P If;;kg jij<k Qj =2I jij? j k! We ca rewrite the previus result as fllws, fn g ky i= k? j + `=? `k If;;kg jij=` Y j =2I C j + lg ; ad therefre by allwig ay r f the j 's t be zer, the term that will dmiate the previus bud is, Y? k r j I;jIj=r j =2I Fr example, whe k = 2, = (=, ad = (=, the fn g?? + 2? + 2? + lg 7

18 By lkig at the dieret regis i the - plae we btai, fn g < (lg ; fr =2 ad =2; (maxf =2? =2? g; fr =2; =2, r =2; =2; (?? ; fr =2; =2. Nte that if = ad =2, r = ad =2, we recver the expected cmplexity time f the radm partial match query prblem. p =, Partial Match Query =2? lg Pit Search =2?? =2? =2 Figure 5. The cmplexity regis fr = (= ad = (=. Lemma. Let U ; ; U be idepedet ad uifrmly distributed ver [; ] k radm variables, let i be the largest side f the i th rectagle geerated by U ; ; U. The, fr all, ( i2f [ i > 2] 2 4k?3 P P Prf. Nte that i2f [i > 2 ] 2 k i2f Qj2I i ij, where I i = fj ij > 2 g. Dee S = P i2f Q j ij > 2 s that fr, ( i2f [ i > 2] ij. We are gig t prve that fs g is decreasig 2 k?3 fs g 2 k?3 fs g = 2 4k?3 T shw fs g fs g, we lk at the diereces ce agai. Set P i = Q j2i i ij. The, S r+? S r = jr i j [i > 2] i2f + [(? i > 2]?P i + [i > 2] P i P i (? + [jiij>] i P i P i + [jiij>] i + [i 2 ; (? i 2] 2 [jiij>] P i i ;

19 where L = Uifrm[; ], ad it is idepedet f U ; ; U r. Therefre, fs r+? S r ju ; ; U r g jr i j [i > 2] P i + (x + 4 dx 2 i Z? Z 2 i + 2 i ((? x + =4dx + =2dx? 2 i = jr i j [i > 2] P i? 4i (2i 2 Prf f Therem 5. Let T be the squarish k-d tree cstructed frm U ; ; U. Nte that a de U i i T is visited if ad ly if the query rectagle Q itersects R i, where R i is the rectagle i the al partiti f [; ] k geerated by U ; ; U i?, i which U i falls. Thus, the ruig time f the rage search algrithm is exactly the umber f rectagles i R that Q itersects, N = 2 i= [Ri\Q6=;] ls, give U ; ; U, the prbability that Q itersects R i is the prbability that Z has sme crdiate that is withi distace j =2 f R i, ad this prbability is clearly buded by the vlume f R i expaded by j i the j th directi, fr all j. Therefre, fn g < 2 ky? = ij + j ; i= j= Y < 2 Y = j ij ; + j =2I i= j2i Y jij? j k + lg C = If;;kg If;;kg jij<k j =2I ; fr sme > by Therem 3 ad the Lemma 7. Fr the lwer bud we may assume that

20 j =2 ad d the fllwig fn g = ( i2f < i2f < = [Q\Ri] [j2f;;kgij =2] ky j= ky i2f j= Y If;;kg j =2I? ij + j 2 ij + j 2 j If;;kg j =2I = [j2f;;kgij =2] ; = < ky ;? < Y = 2 ij ; Y j i2f j2i < Y 2 i2f j2i i2f j= ij + j 2 ij [j2f;;kgij >=2] We ca bud the secd term abve fr ay give I f; ; kg as fllws, < Y i2f j2i ij [j2f;;kgij >=2] = ( ; i2f [j2f;;kgij >=2] = ; [ i >=2] 2 4k?3 ; by the previus lemma. Thus, fr all large eugh, we ca chse > such that, fn g If;;kg jij<k Y j =2I jij? j k C + lg = ; x7. Nearest eighbr search We csider tw atural earest eighbr search algrithms. I algrithm, start with a rthgal rage search with a square bx f size = =k cetered at the query pit. Repeat with bxes f sizes k i=2 = =k fr i = ; ; 2; 3; util i +, where i is the idex f the rst empty bx. Reprt the earest pit i the i + -st bx. ach rthgal rage search take idividually (fr xed i takes expected time O(lg by Therem 5. We shw i fact that the ttal expected time is O(lg lg lg Therem 6. Let be a pit uifrmly distributed [; ] k. Csider a squarish k-d tree based i.i.d. pits [; ] k. The the expected time f algrithm is O(lg lg lg. Prf. Let T be the ttal time it takes algrithm t ish. Let T i be the ruig time f Betley's rage search algrithm i.i.d. pits [; ] k ad a cube Q i cetered at 2

21 f legth k i=2 = =k, ad let M i be the umber f pits i Q i. Nte that, ft g O(lg + ( T + T 2 + m i=3 T i [Mi?2=] where m = b k 2 lg k(2 k c buds the maximum P umber f iteratis the algrithms ca m perfrm. Thus, it is eugh t prve that T i=3 i [Mi?2=] = O(lg lg lg. Let t = d k 2 lg k(2 k lg e, the ( m i=3 Nw, by Therem 5, T i [Mi?2=] 2 (t + ft t+ g k lg k(2 k lg = k lg k(2 k lg = 2 k lg k(2 k lg + 2 k lg k(2 k lg k lg k(2 k lg + 2 = O(lg lg lg ; (t + ft t+ g + 2 k(t+k=2 + m i=t+2 k??`=k `= k? k (t+k=2 + `= k? k (t+k=2 + k (t+k=2 k (t+k=2 k k 2 k lg P fm i?2 = g ; If;;kg jij=` Y j =2I k (t+=2 =k k k`?`=k (t+(k?`=2 + lg `= k?(t+=2 + fr all e. Fially, fr i m, P fm i?2 = g? kk(i?2=2 2 k e?kk(i?2=2 =2 k ; ad therefre P fm t+2 = g =. Thus, 2 m i=t+2 P fm i?2 = g 2m = O(lg?`=k k k?(t+`=2 + lg ` k + lg k p k(2 k lg + + lg =k! + lg!! C Therem 6 is i ctrast with the situati fr stadard radm k-d trees, where algrithm is shw t take expected time (, where 2 (6; 64 depeds up k ly (Chazy, Devrye ad Zamra-Cura,. I algrithm B, isert i the squarish k-d tree, ad let Q be the rectagle assciated with. Let be the paret f i the tree (te 2 Q. Perfrm a rthgal rage search cetered at with dimesis 2k? k i all directis. Reprt the earest eighbr amg all pits retured by this rthgal rage search. We will aalyze this algrithm fr k = 2 ly. 2

22 Therem 7. Let be a pit uifrmly distributed [; ] k. Csider a squarish k-d tree based i.i.d. pits [; ] k. The the expected time f algrithm B is O(lg 2. The bud algrithm B is a bit wrse tha that fr algrithm, because while mst rectagles are squarish, a suciet umber f them are elgated. I fact, fr give M >, abut =M f the al (leaf rectagles r mre shuld have a edge rati exceedig M. Fr edge rati M, ad csiderig that all rectagle areas are abut =, we see that the rthgal rage search shuld take abut M pits (the lgest edge is abut p M=. The expected umber f retured elemets is at least (lg. d the expected umber f leaf rectagles visited is f the same rder. But each visited leaf rectagle als iduces a visit t all f its acestrs, ad there are abut lg f thse, hece the claim. The remaider f this secti ctais the prf f Therem 6. Let us prve the fllwig prpsiti that will allw us t simplify the remaiig prfs. Prpsiti 3. Fr all x ad, x (?(x + =x p 2e =2 + 2 x Prf. Stegu, 7, the Let x >, the as p 2x? x e x?(x + p 2x? xe x e =2x (bramwitz ad (?(x + =x =2x (2x =2x x e =2x2 p 2e =2 + 2 x ad (?(x + =x (2x =2x =2x x x ; as Lemma. Let Z; U ; ; U be idepedet ad uifrmly distributed radm variables [; ] 2. Let (Z ad Y (Z be the x-legth ad y-legth f the rectagle i the al partiti (f the squarish 2-d tree iduced by U ; ; U i which Z falls. The, bth 2 (Z ad Y 2 (Z are O(lg 2. Prf. By Lemmas 4 ad 5, fr ay p; q > such that p + q =, we have that 2 (Z = ( i2f ( 3 i Y i i2f 4?(p + ( i Y i p =p ( i2f p? =p 5?(q + q? =q 2q i q? q? =q = 4=p 5 =q (?(p + =p (?(q + =q (q q?? =q (q? =q Let us chse q = + lg, p = lg +, ad assume > e. By Prpsiti 3, there is c >, such that (?(p+ =p cp = c(lg +, ad there is c >, such that (?(q+ =q c q 4c. Furthermre, (q??=q = (lg lg lg + lg, ad (q q?? =q 2e?. Therefre 2 (Z = O(lg 2. The result fr Y 2 (Z fllws i the same maer. 22

23 Lemma (Devrye, 6. Let H be the height f a radm biary search tree f size, the fr ay iteger k maxf; lg g we have k 2e lg P fh kg k Lemma 2. Let Z; U ; ; U be idepedet ad uifrmly distributed radm variables ver [; ] 2. Let (Z ad Y (Z be the x-legth ad y-legth f the rectagle i the al partiti iduced by U ; ; U i which Z falls. The, Y (Z P 2 i= Y i, (Z P 2 i= Y i, ad Y (Z P 2 i= i (Z P 2 i= i are O(lg 2. Prf. Let F dete the cllecti f al rectagles i the squarish 2-d tree T cstructed frm U ; ; U. Fr a al rectagle R i, dete by D(R i its depth. The P P 2 i= i D(R i2f i i +. Thus if H is the height f T, ( 2 i= i (Z = < = D(R i i j 2 Y j ; + i2f j2f < = i j 2 Y j ; + i2f j2f < = i2f j2f < [H t lg ] H i H t lg + fr ay t >. Usig Lemma 2, we see that, < [H t lg ] H i2f i j 2 Y j; + i2f j2f = j 2 Y j; + ; i j 2 Y j; 3 P fh t lg g 2 t lg( 2e t j2f? 2e We chse t such that t lg t <?2 s that < [H t lg ] H We cmplete the prf by shwig that let S r = P i P i2f = i j 2 Y j; = O( j2f P i2f i = P j2f 2 j Y j j2fr 2 j Y j, fr r = ; ;?. Nte that m Y m 2 4 [m<y m] m = O(lg. Fr this, S r+? S r = j 2 Y j m2fr j2fr # + [m>ym] (( m 2 Y m + ((? m 2 Y m? m 2 Y m i ; 23

24 where = L Uifrm[; ], ad is idepedet f all U ; ; U. Nw, as ( m 2 Y m + ((? m 2 Y m? m 2 Y m, we have that S r+? S r ( i Y i 3=2 j 2 Y j Nte that fr ay p; q >, such that p + q =, fs r+? S r g j2fr (! p =p < ( i Y i j2fr 2 j Y j q = ;=q ad agai by Hlder's iequality, ad Lemma 4, by chsig q = p 4, ad p = p 4 p 4?, (! p =p ( ( i Y i 3=2 r p=q By applyig Hlder's iequality iside the expected value, j2fr 2 j Y j q = ;=q < = =q rq=p (j 2 Y j q ; j2fr B < = r ( j Y j qp ; j2fr ( i Y i 3p=2 =p p r =p < j2fr q2 j =p =q! =q 66 r =p r qp? r q2 =2? = 66 p r ; = =q =q C ; Thus, fs r+? S r g 254=r, ad by summig the diereces we ally ca cclude that P P i2f i 2 j2f j Y j is ideed O(lg. The ther expected values ca be buded i the same way. Prf f Therem 7. Give U ; ; U, we dee L (Z = 2( (Z+Y (Z. Nte that as the expected height f T is O(lg, the expected time cmplexity f the earest eighbr algrithm is buded by O(lg plus the expected time f radm rthgal rage search with query rectagle Q havig all sides f legth L (Z, ad cetered at Z. Let N be the time cmplexity f rage search. By the same argumets fllwed i Therem 3 we have, fn g ( 2+ By Lemma 5, P 2+ i Y i + 2 i= i= i Y i ( 2+ i= L (Z( i + Y i + L 2 (Z + = O(lg. Fr P 2+ i= L (Z( i + Y i, Lemma 2 abve shws that it is O(lg 2. s f (ZY (Zg = P i2f ( iy i 2, Lemma 4 shws that it is O(. Fially, by Lemma we have that L 2 (Z the expected ruig time f algrithm B is O(lg 2. = O(lg 2. Thus 24

25 x. Further wrk ad pe prblems quadtrees. Fr quadtree splittig i k dimesis, it is easy t see that the aalysis ad thus Therem are t valid. I fact, fr radm quadtrees, the expected perfrmace fr partial match queries was shw t be f the rder f that fr stadard radm k-d trees (Flajlet, Get, Puech ad Rbs (, 2. Fr rthgal rage search with query rectagles depedig up, see Chazy, Devrye ad Zamra-Cura (. expected wrst-case cmplexity. We cjecture that the expected wrst case cmplexity ver all rage search rectagles f dimesis i (but with wrst-case lcati f the ceter is als buded frm abve by the bud give i Therem 2. d the expected wrst-case time fr a s-dimesial partial match query is cjectured t be O(?s=k fr s < k (fr s = k, the cmplexity is clearly buded by the expected height f the tree, O(lg. -uifrm distributis. Fially, we als ited t study the behavir f squarish k-d trees fr uifrm distributis, althugh it appears ce agai that the upper bud f Therem 2 remais valid fr all distributis with a jit desity [; ] k. 25

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