The Weight of the Shortest Path Tree

Transcription

1 The Weight of the Shortest Pth Tree Reco v der Hofstd Gerrd Hooghiestr Piet V Mieghe Deceber, 005 Abstrct The iil weight of the shortest pth tree i coplete grph with idepedet d expoetil (e ) rdo li weights, is show to coverge to Gussi distributio. We prove coditiol cetrl liit theore d show tht the coditio holds with probbility covergig to. Itroductio Cosider the coplete grph K +,with +odes d ( +) lis. To ech li (or edge) we idepedetly ssig expoetilly distributed weight with e. The shortest pth betwee two odes is tht pth whose su of its lis weights is iil. (Ech of these shortest pths is.s. uique.) The shortest pth tree (SPT) is the uio of the shortest pths fro root (e.g. ode ) to ll other odes i the grph. I this pper we cosider the totl weight W of the SPT rooted t ode to ll other odes i the coplete grph. I [7, 9], we hve rephrsed the shortest pth proble betwee two rbitrry odes i the coplete grph with expoetil li weights to Mrov discovery process which strts the pth serchig process t the source d which is cotiuous tie Mrov chi with +sttes. Ech stte represets the lredy discovered odes (icludig thesourceode). IftsoestgeitheMrovdiscoveryprocess odes re discovered, the the ext ode is reched with rte λ = ( + ), which is the trsitio rte i the cotiuous-tie Mrov chi. Sice the discovery of odes t ech stge oly icreses, the Mrov discovery process is pure birth process with birth rte ( + ). We cll τ the iter-ttchet tie betwee the iclusio of the th d ( +) st ode to the SPT for =,...,. The iter-ttchet tie τ is expoetilly distributed with preter λ s follows fro the theory of Mrov processes. By the eoryless property of the expoetil distributio, the ew ode is dded uiforly to lredy discovered ode. Hece, the resultig SPT to ll odes is exctly uifor recursive tree (URT). A URT of size +isrdotreerootedtsoesourceodedwheretechstge ew ode is ttched uiforly to oe of the existig odes util the totl uber of odes is equl to +. rhofstd@wi.tue.l, Deprtet of Mthetics d Coputer Sciece, Eidhove Uiversity of Techology, P. O. Box 53, 5600 MB Eidhove, The etherlds. G.Hooghiestr@ewi.tudelft.l d P.VMieghe@ewi.tudelft.l, Electricl Egieerig, Mthetics d Coputer Sciece, Delft Uiversity of Techology, P.O. Box 503, 600 GA Delft, The etherlds.

2 The verge of the weight W of the SPT equls d the vrice is E [W ]= =, () vr [W ]= 4 ( +) = = = 5 4. () The result for the e () hs bee foud first i [8], but it is rederived i Sectio. becuse the ethod is cosiderbly sipler. The derivtio for the vrice () is i Sectio. while y pperig sus re coputed i the Appedix. The syptotic for of the verge weight is iedite fro () s µ E [W ]=ζ() + O, (3) while the correspodig result for the vrice, derived i Sectio., is µ 4ζ (3) vr [W ]= + o. (4) The third d i result i this pper is tht we show tht the scled weight of the SPT teds to Gussi. I prticulr, (W ζ ()) d 0,σ, = where σ = σ SPT =4ζ (3) ' A relted result for the iiu spig tree (MST) is worth etioig. The verge weight of the iiu spig tree W MST i the coplete grph with expoetil with e (or uifor o [0, ]) li weights hs bee coputed erlier by Frieze [3]. For lrge, Frieze showed tht E [W MST ] ζ (3). Jso [5] exteded Frieze s result by provig tht the scled weight of the MST teds to Gussi, (WMST ζ (3)) d 0,σ MST, where σ MST =ζ (4) i=0 = = (i + )! (i + ) i i!!(i + + ) i++ ' The triple su ws exctly coputed by Wästlud [0] resultig i σ MST =6ζ (4) 4ζ (3). The weight of the shortest pth tree Fro the Mrov discovery process briefly explied i Sectio, the discovery tie to the th discovered ode fro the root equls v = τ, (5) =

3 where the iter-ttchet ties τ,τ,,τ re idepedet, expoetilly distributed rdo vribles with preter λ = ( + ),. A rbitrry uifor recursive tree cosistig of +odes d with the root lbeled by zero c be represeted s (0 ) ( )...( ) (6) where ( ) es tht the th discovered ode is ttched to ode {0,..., }. Hece, is the predecessor of d this reltio is idicted by.theweightw of rbitrry SPT fro the root 0 to ll other odes is with (5) d v 0 =0d =0, W = (v v )= = = = + I the URT, the iteger,, re idepedet d uiforly distributed over the itervl {0,..., }. It is ore coveiet to use discrete uifor rdo vrible o {,...,} which we defie s A = +.Werewrite W = τ = {A }τ = τ {A }. = =A = = = = τ. The set {A } re idepedet rdo vribles with P [A = ] = dditio, we defie for {,...,} the rdo vribles for {,,...,}. I to obti B = W = {A }, (7) = B τ. (8) = The rdo vribles B,B,...,B re depedet. The e of the rdo vrible B follows fro (7) s h i E [B ]= E {A } = P [A ] =. (9) = The vrice vr [B ] d covrices cov [B,B ] re give i Le below.. The verge weight of the SPT It is iedite fro (8) d the idepedece of the A,A,...,A fro the iter-ttchet ties τ,τ,...,τ tht E [W ]= E [B ] E [τ ]= = = = which is, by the equlity (3) below, equl to (). = ( + ) = = = ( + ) = = = =+, 3

4 . The vrice of the weight of the SPT To copute the vrice of W,weusetheforul vr [W ]=vr [E[W B,...,B ]] + E[vr [W B,...,B ]]. (0) Sice for expoetil rdo vrible τ with preter λ = ( + ), the expecttio equls /λ d the vrice /λ,wehve " # E[W B,...,B ]=E B τ B,...,B = λ B, () Cobiig (0), () d (), = " # vr [W B,...,B ]=vr B τ B,...,B = vr [W ]=vr " = = λ B # + = = = λ B. () λ E[B ]. (3) To proceed, we eed expressios for the covrice of B d B, which re coputed i the followig le: Le For every,, (i) (ii) vr [B ]= cov [B,B ]= = = µ, (4) µ,. (5) Proof. Theproofof(i)followsfrothtof(ii)with =. (ii) The bilierity of the covrice yields, for, cov [B,B ]=cov {Ai }, = i= = i= = {A } Sice A i d A re idepedet for i 6=, wehvethtcov tht h i cov {Ai }, {A } = h i With cov {A }, {A } {A i(,)} = {A } for, weobti = = i= = h i cov {Ai }, {A }. h {Ai }, {A } h i cov {A }, {A }. i =0, for i 6=, such h i = E {A } {A } P [A ] P [A ] d {A } {A } = h i cov {A }, {A } = = 4 µ.

5 Applyig Le to the right side of (3) gives " # vr [W ]=vr λ B + = λ = λ = where the su T () is defied s T () = = = µ λ E[B]= + ( + ) = λ = λ λ = = = cov [B,B ]+ 3 = = T ()+T (). = λ (E[B ]), (6) = d the lst equlity is proved i the Appedix (see (46) below), while T () = = = = 4 + = ( + ) ( + ) 3 5 = 4 + = 3 = = + µ =, (7) where the lst equlity is proved i the Appedix (see (44) below). Suig T () d T () gives the explicit for () of the vrice for W. We ext ivestigte the sytotics of the vrice of W for lrge. We write the su of the lst two ters i () by Q() = (8) The, for lrge, = Q() Q( ) = 4 3 = = = 5 4 = O µ log ³ d, by sutio, Q() = Q + O log, where the liit Q =li Q () exists, by (8). It follows fro [, Corollry 4, i theore] tht so tht Q =4 Hece, syptoticlly, we rrive t (4). = 3 = 3, 5 =0, (9) 4 = µ log Q() =O. 3 Cetrl liit theore for W d I this sectio we prove cetrl liit theore for W.Weusethesybol to deote covergece i distributio d the sybol P for covergece i probbility. We deote by σ(w ), the stdrd 5

6 devitio of W,soσ (W )=vr [W ]. We deote by (0, ) rdo vrible with stdrd orl distributio. The i result proved i this sectio is the followig cetrl liit theore for W : Theore As, W E[W ] σ(w ) d (0, ). Westrtwithoutlieoftheproof.WewishtoprovethtW is syptoticlly orl, i the sese tht (W E[W ]) hs syptotic orl distributio. We first defie s = = B ( + ). (0) We ote tht s is rdo vrible, d we soeties e this explicit by writig s (ω), where ω is eleet of the probbility spce. We split W = + Y, where = = Our strtegy is to prove the followig steps: ³ τ B, d Y = ( + ) = B ( + ). (). Defie evet A such tht: () A is esurble with respect to the σ-lgebr geerted by {A } =,(b)p(ac ) δ, d (c) Uiforly for ω A,wehves (ω) σ, = o(), where σ (E[B ]), = ( + ) = T (). = Cosecutively, we show tht σ =li σ, exists.. Prove the cetrl liit theore for with vrice s, coditiolly o {A } =,whe {A } = is such tht A holds. More precisely, we will show tht uiforly o A, E A [e it ]=e t s / + o(), where E A is the coditiol expecttio give {A } =. 3. Prove tht (Y E[Y ]) coverges i distributio to orl rdo vrible with vrice σ =li T (). Together, these steps prove Theore. Ideed, we copute, usig tht A is esurble with respect to the sig-lgebr geerted by {A } =, i φ(t) =E[e it W ]=E[e it W A ]+O(P(A c )) = E he A [e it W ] A + O( δ ). We split W = + Y, d use tht Y is esurble with respect to {A } = i φ(t) =E he it Y E A [e it ] A + O( δ ). to rrive t 6

7 Accordig to Step, uiforly o A, E A [e it ]=e t s / + o(), d, ccordig to Step, uiforly o A, E A [e it ]=e t σ, / + o(). Therefore, usig tht E[W ]=E[Y ], E[e it (W E[W ]) ]=E[e it (W E[W ]) A ]+o() = e t σ, / E[e it (Y E[Y ]) A ]+o(), gi by Step. ow, by Step 3, we hve tht E[e it (Y E[Y ]) A ]=E[e it (Y E[Y ]) ]+o() = e t σ / + o(), so tht E[e it (W E[W ]) ]=e t σ / + o(), where σ = σ + σ,dσ =li σ,. We ow tur to the detils of the proof. We will prove Steps -3 i Sectios , respectively. 3. Step : The good evet d covergece i probbility of s Fix (0, ) d iteger 0, d defie A = B C, where d with B = \ = { B E[B ] E[B ]}, () \ C = {B x( 0,)log}, (3) = = /3. Lter we will see tht i fct we eed 0 lrge d > 3 4. O the evet B, with ll rdo vribles B, with, re close to their respective e E[B ]; o the evet C,wehve logrithic boud o the rdo vribles B, with. We will show two les. The first shows tht A occurs with high probbility, while the secod proves tht s is close to costt o A. Together the les iply the clis i step. Le 3 Fix (, ) d 0 sufficietly lrge. The, for sufficietly lrge, P(A c ) ( ), so tht we c te δ = > i Step. 7

8 Proof. We use Boole s iequlity to obti, P(A c ) P(B > x( 0,)log)+ P( B E[B ] E[B ]). = = ote tht B is the su of idepedet idictors, d, therefore, by the estite of Jso [6] d with 0 < <, P( B E[B ] E[B ]) e 3 8 E[B ], where E[B ] is give i (9) which we boud s µ Z log = x dx E[B ] Therefore, we hve tht Z P( B E[B ] E[B ]) e 3 8 log, µ dx = log. x which is o( ) for ll 0 d 0 sufficietly lrge. O the other hd, for 0, gi for 0 sufficietly lrge. Hece, P(B 0 log ) P( B E[B ] 0 E[B ]), P(B > x( 0,)log) = +, (4) = for 0 sufficietly lrge. We coplete the rguet s follows. For,wehvetht P( B E[B ] E[B ]) e 3 8 log e 3 6, (5) sice, uiforly for ll such tht,wehve log. Ideed, this follows sice f () = log is first icresig d the decresig. Therefore, uiforly for, We ote tht, for sufficietly lrge d (, ) fixed, d, usig tht log ( x) x, 0 <x<, f () i f ( ),f ( ). (6) f ( )=( ) log, f ( ) ( )log( ) ( ). For = /3, we c use (5), to obti, gi for sufficietly lrge, P( B E[B ] E[B ]) e 6 3 = o( + ). (7) = Cobiig the bouds (4) d (7) we obti the stteet i the le. 3 Recll tht s = P B =. We ext ivestigte s (+ ) o the evet A : 8

9 Le 4 For, d uiforly o the evet A, s (ω) = (E[B ]) ( + ) = o(). Proof. Fro (7), it follows tht B +. For sufficietly lrge, = B ( + ) for y <. Therefore, we hve tht s (ω) = O the evet C,for sufficietly lrge, = = B ( + ) + o(). = O( ), = B ( + ) = (x( 0,)) (log ) O + (log ) = o(), so tht, o C, s (ω) = = B ( + ) + o(). O B,dfor, we c sdwich ( ) (E[B ]) B ( + ) (E[B ]),so with probbility t lest O( ( ) ),wefid, s (ω) =(+O( )) Siilr estites s bove yield tht = E[B ] ( + ) = o(), (E[B ]) ( + ) + o(). = E[B ] ( + ) = o(). = This copletes the proof of the le. The rguet of covergece i probbility of s (ω) is coplete whe we prove tht For this, we ote tht = d fro (47), we fid tht σ =ζ (). = (E[B ]) ( + ) σ. (E[B ]) ( + ) = T (), 9

10 3. Step : Coditiol cetrl liit theore for I this sectio, we copute E A [e it ],where{a } = is such tht A tht, for y rdo vrible with fiite third oet, we hve tht holds. For this, we ote φ (t) =E[e it ]=e itμ t σ /+O( t 3 3 ), (8) where μ = E[], σ = vr() d 3 = E[ 3 ]. The idepedece of the τ, coditiolly o {A } =,givestht Y E A [e it ]= E A [e it τ B (+ ) ]. = By (8), d sice B,...,B re esurble with respect to the σ lgebr sped by the rdo vribles A,A,...,A, we obti tht Therefore, where E A [e it τ (+ ) h B ]=exp t B ³ ( + ) + O t 3 3/ B 3 i 3 ( + ) 3. E A [e it ]=e t s / e O( t 3 v ), v = 3/ = B 3 3 ( + ) 3. We filly show tht v = o() o A.First,weotetht,siceB + d <, 3/ = B 3 3 ( + ) 3 3/ = Whe, we c use the bouds provided by A. We strt with the cotributio due to, for which we c boud o C,forsufficietly lrge, d soe costt C depedig o 0 d, 3/ = B 3 3 ( + ) 3 3/ = (x( 0,)) 3 (log ) 3 3 ( + ) 3 C(log ) Filly, for,weobtiob,usige[b ] log, 3/ = B 3 3 ( + ) 3 ( + ) 3 3/ (E[B ])3 3 ( + ) 3 = ( + ) 3 3/ (log )3 ( + ) 3 = ( + ) 3 (log ) 3 3/ η, for y η< 3, d we ote tht η>0whe > 3 4. This copletes the proof tht v whe > 3 4,dtht,for{A } = such tht A holds, η E A [e it ]=e t s / e O( η) = e t s / + o(). 0

11 3.3 Step 3: The cetrl liit theore for Y We gi use covergece of chrcteristic fuctios to tht of orl rdo vrible with e 0. We rewrite Y E[Y ] = {A } = (Y, E[Y, ]), ( + ) where = Y, = = = = {A } ( + ). (9) The suds Y,,...,Y, re idepedet. We wish to show tht (Y E[Y ]) is syptoticllyorlwithsyptoticvricevr(y ). Fro the idepedece of the suds, E[e it (Y E[Y ]) ]= Y E[e it(y, E[Y, ]) ]. The, we ote tht, for sufficietly lrge d usig tht (+ ) = + ( + + ), Y, = = ( + ) = Therefore, we hve tht, for sufficietly lrge d t>0, ( + ) 3log. (30) E[e it(y, E[Y, ]) ]=exp (t /)vr(y, )+O( t 3, ) ª, where, = E[ Y, EY, 3 ] deotes the bsolute third cetrl oet. By (30), we hve tht Y, EY, 3log,sotht, 3log vr(y, ). Hece Y Y E[e it (Y E[Y ]) ]= E[e it(y, E[Y, ]) ]= e ( t vr(y, )/+O( t 3, ) = = = e t σ, / e O( t 3 σ, / log ). This copletes the proof becuse σ, = vr(y, )=vr(y )=vr = s show by (45) i the ppedix. = λ B = T () σ =4ζ(3) ζ(), Appedix I Sectio A we prove couple of idetities forulted s les. Le 5 to Le 0 re ll prove i ideticl wy by tig differeces. We therefore leve out soe of the detils. We will deote the prtil sus i these idetities by C(),D(),..., isted of C,D,...,iorder to distiguish the fro the stdrd ottio for rdo vribles. I Sectio B, we pply these idetities to obti syptotic expressios for the vrice of d Y.

12 A Idetities Le 5 For ll, C() = = =+ =. (3) = The idetity (3) ws proved i [8] by iductio. Erlier Coppersith d Sori [] hve proved (3) lso by iductio. We give ew d sipler proof. Proof: Clerly, C() = d C() C( ) = + = = =+ = = + µ = ( ) =. = = Suig both sides fro =to = M, usigc() = d relbelig M the leds to the right hd side i (3). A relted su which we will eed is D() = = = Reltio (3) is strightforwrd by syetry. Le 6 For ll, F () = = =+ = = = + = = =+ = = = =. (3) 3 = =. (33) Proof: The first equlity follows fter reversio of the sutios. Prllel to the proof of Le 5, the secod equlity is derived s F () F ( ) = + = = =+ = = ( ) = ( ). = = = Writig P = = P = +, d usig 3 ( ) = ( + ) o the secod sud, we fid tht F () F ( ) = 3. = As i the proof of Le 5 this leds to the quoted result by itertio fro F () =. The ext le sttes soewht ore ivolved idetity: =

13 Le 7 For ll, G() = = =+ = = =. (34) Proof: G() G( ) = = = = = = + = + = = =+ µ ( ) + = =+ =+ + = =+ + ( ) = +! ( ). ow, usig idetity (3), = ( ) =+ d the prtil frctios result: we rrive t = = =+ = D( ) = = = =, ( ) = + ( ) + ( ), (35) G() G( ) = = = = = =. As before we obti the result by itertio. Le 8 For ll, L() = = = =+ Proof: After tedious, but strightforwrd, coputtio we get = 3. (36) = L() L( ) = = =+ = =. With idetity (3), L() L( ) = = 3. = = This yields the proof. 3

14 Le 9 For ll, R() = = = =+ Proof: Oce ore we copute the differece R () R ( ) = = = = =+ =+ =5 = + Usig the prtil frctio result (35) o the lst su R () R ( ) = = = = = = = = =+ + = = = 4 = = + = = =+ = = 3 =. (37) = µ ( ) = = = + = + 3 = = = = = ( ). = = µ ( ) ( ), where we hve used (3). Usig (35) to replce the lst su o the right side, we obti, R () R ( ) = 4 3 = Suig both sides, d usig R () =, R () =6 = Filly by rguet prllel to (3), = = 4 3 = = = = = = + = = = =. = =. (38) 4. (39) Together, this yields the proof. Le 0 For ll, T () = = =+ = = 3 =. (40) = 4

15 Proof: As before T () T ( ) = = = = = = + = + = = ow fro tig prtil frctios, (3) d (3), = ( ) =+ =+ µ ( ) + = =+ =+ + = C( ) = D( ) + = We c siplify this usig the expressios for C() i (3) d D() i (3), = ( ) =+ Usig prtil frctio expsio gi yields = Cobiig these results the gives T () T ( ) = = 4 3 = = = + = ( ) = = = = + = = = = = = =+ + ( ) + = = = = 3 = = = =.. +! ( ). (4) = + 4 fro which by suig both sides fro =to = M (d the M gi) T () = = 3 = + = 4 + = Usig (39) the yields the proof. = =. =, 5

16 B The syptotic results for the vrices I this sectio we use the idetities of Sectio A, i order to copute siplified expressios for T () d T (). Cosequetly we use these results for the syptotic vrice of W. The su T () (copre (7)) equls where d R () = R () = T () =R () R (), = = = = ( + ) ( + ) ( + )( + ) We strt with the su R (), d iterchge the sus to obti R () = = = = = Splittig (( + )) ito two prts, R () = + = = = = = =,. ( + ) ( + ) ( + ) =+ = = =+ The first su equls L()/( +),wherel() ws siplified i Le 8. By the se ethod tht we used i (3) to obti D(), wefid This iplies usig (33), = =+ =+ = = = = = =+ =+ + + = (G()+F ()) = = =+ 3. = =.. (4) =+ Cobiig ll, R () = 4 + = 3 = 4ζ(3) ( + O( )). (43) 6

17 We ow tur to the su R () = = = = = =+ ( + )( + ) d R () ws siplified i Le 8. Together we fid where ( + ) = = = T () =R () R () =R () R () = 4 + = 3 5 = Fro (38) we fid tht R() R( ) = ζ() R =5 = R() =R = = 4 + ζ () = = = 3 = =+ = R(), +. (44) = + O( log ), so tht, by sutio, 3 µ log, + O = 5 ζ(4) ζ() =0, = The first equlity follows by (9), while the secod follows by [4, (9.54)]. Thus, we obti the syptotics µ 4ζ (3) ζ () log T () =R () R () = + O. (45) We filly tur to the secod su T () (see (6)), which su is equl to the su T () displyed i Le 0. Therefore, T () = = ( + ) Fro the proof of Le 0, the differece T () T ( ) = 3 = = + 4 = = = 3 = =. (46) = = + O µ log which shows, by sutio tht for lrge, T () behves syptoticlly s µ ζ () log T () =T + + O, (47) wherewewrite T = = 3 = 3, = 5 ζ(4) ζ (), (48) = d, gi, the secod equlity follows by (9), while Equtio [4, (9.54)] iplies tht T =0. 7

18 Rer. We ote tht T =0d Q =0c lso be proved directly fro the first equlity i (46) without resortig to [, Corollry 4, i theore]. We split T () s / T () = ( + ) + ( + ) = = For sufficietly lrge, the first su is bouded s / ( + ) =+/ = = =+/ < 3log, while the secod su is, ssuig tht / is iteger, / ( + ) = (/+ ) = = = / = For /, we boud the su betwee brcets s Hece, we obti =+/ =+ ( + ) Z < dx ³ x = log = < / = =+ = =+/ = µ + O ³ + O 3 / = Cobiig both estites shows tht T () 0 s, hece T =0. Fro (7), we fid tht T = li T () is Equtio (48) shows tht T = 5 T = = = = = The defiitio (8) of Q () together with () gives fro which Q =li Q () follows s = = O µ. +. (49) = + ζ () 5ζ(4) = T =0. Q () =T ()+T ()+O Q = T + T =0 This result proves (9) idepedetly fro Borwei s pper [, Corollry 4, i theore]. 8

19 Acowledgeets The wor of RvdH ws supported i prt by the etherlds Orgistio for Scietific Reserch (WO). The uthors re grteful to referee for y rers tht iprove the presettio i the pper, d, i prticulr, for outliig the rer i Appedix B. Refereces [] D. Borwei d J.M. Borwei. O itriguig itegrl d soe series relted to ζ(4). Proc. Aer. Mth. Soc., 3(4): 9-98, 995. [] D. Coppersith d G. B. Sori. costructive bouds d exct expecttios for the rdo ssiget proble. Rdo Structures d Algoriths, Vol 5, o., 3-44, 999. [3] A. M. Frieze. O the vlue of rdo iiu spig tree proble. Discrete Applied Mthetics, 0:47 56, 985. [4]I.S.GrdsteydI.M.Ryzhi. Acdeic Press, Lodo, 994. Tbles of Itegrls, Series d Products (Fifth Editio). [5] S. Jso. The iil spig tree i coplete grph d fuctiol liit theore for trees i rdo grph. Rdo Structures d Algoriths, 7(4): , 995. [6] S. Jso. O cocetrtio of probbility. Coteporry Cobitorics, ed. B. Bollobás, Bolyi Soc. Mth. Stud. 0, Jáos Bolyi Mtheticl Society, Budpest, 89-30, 00. [7] R. v der Hofstd, G. Hooghiestr, d P. V Mieghe. First pssge percoltio o the rdo grph. Probbility i the Egieerig d Ifortiol Scieces (PEIS), 5:5 37, 00. [8] R. v der Hofstd, G. Hooghiestr, d P. V Mieghe. Size d weight of shortest pth trees with expoetil li weights. Cobitorics, Probbility d Coputig, to pper 006. [9] P. V Mieghe, G. Hooghiestr, d R. v der Hofstd. A sclig lw for the hopcout i iteret. Delft Uiversity of Techology, report 0005, 000. [0] J. Wästlud. Evlutio of Jso s costt for the vrice i the rdo iiu spig tree proble. Liöpig studies i Mthetics, o. 7, Series Editor: Begt Ove Turesso,