A note on random minimum length spanning trees

Transcription

1 A ote o adom miimum legth spaig tees Ala Fieze Miklós Ruszikó Lubos Thoma Depatmet of Mathematical Scieces Caegie Mello Uivesity Pittsbugh PA15213, USA ala@adom.math.cmu.edu, usziko@luta.sztaki.hu, thoma@qwes.math.cmu.edu Submitted Jue 28, 2000, Accepted August 11, 2000 Abstact Coside a coected -egula -vetex gaph G with adom idepedet edge legths, each uifomly distibuted o [0, 1]. Let mst(g be the expected legth of a miimum spaig tee. We show i this pape that if G is sufficietly highly edge coected the the expected legth of a miimum spaig tee is ζ(3. If we omit the edge coectivity coditio, the it is at most (ζ( Itoductio Give a coected simple gaph G =(V,E with edge legths x =(x e : e E, let mst(g, x deote the miimum legth of a spaig tee. Whe X =(X e : e E isa family of idepedet adom vaiables, each uifomly distibuted o the iteval [0, 1], deote the expected value E(mst(G, X by mst(g. Coside the complete gaph K.It is kow (see [2] that, as, mst(k ζ(3. Hee ζ(3 = j=1 j Beveidge, Fieze ad McDiamid [1] poved two theoems that togethe geealise the pevious esults of [2], [3], [5]. Suppoted i pat by NSF Gat CCR ala@adom.math.cmu.edu Pemaet Addess Compute ad Automatio Reseach Istitute of the Hugaia Academy of Scieces, Budapest, P.O.Box 63, Hugay Suppoted i pat by OTKA Gats T ad T FKFP 0607/ usziko@luta.sztaki.hu Suppoted i pat by NSF gat DMS thoma@qwes.math.cmu.edu 1

2 the electoic joual of combiatoics 7 (2000, #R41 2 Theoem 1 Fo ay -vetex coected gaph G, mst(g (ζ(3 ɛ 1 whee = (G deotes the maximum degee i G ad ɛ 1 = ɛ 1 ( 0 as. Fo a uppe boud we eed expasio popeties of G. Theoem 2 Let α = α( =O( 1/3 ad let ρ = ρ( ad ω = ω( ted to ifiity with. Suppose that the gaph G =(V,E is coected ad satisfies δ (1 + α, (1 whee δ = δ(g deotes the miimum degee i G. Suppose also that (S : S / S ω 2/3 log fo all S V with /2 < S mi{ρ, V /2}, (2 whee (S : S ={(x, y E : x S, y S = E \ S}. The mst(g ζ(3 ɛ2 whee the ɛ 2 = ɛ 2 ( 0 as. Fo egula gaphs we of couse take α =0. The expasio coditio i the above theoem is pobably ot the ight oe fo obtaiig mst(g ζ(3. We cojectue that high edge coectivity is sufficiet: Let λ = λ(g deotetheedge coectivity of G. Cojectue 1 Suppose that (1 holds. The, whee ɛ 3 = ɛ 3 (λ 0 as λ. mst(g ζ(3 ɛ3 Note that λ implies. Alog these lies, we pove the followig theoem. Theoem 3 Assume α = α( =O( 1/3 ad (1 is satisfied. Suppose that λ(g ω 2/3 log whee ω = ω( teds to ifiity with. The mst(g ζ(3 ɛ4 whee the ɛ 4 = ɛ 4 ( 0 as.

3 the electoic joual of combiatoics 7 (2000, #R41 3 Remak: It is woth poitig out that it is ot eough to have i ode to have the esult of Theoem 2, that is, we eed some exta coditio such as high edge coectivity. Fo coside the gaph Γ(, obtaied fom / -cliques C 1,C 2,...,C / by deletig a edge (x i,y i fomc i, 1 i / the joiig the cliques ito a cycle of cliques by addig edges (y i,x i+1 fo 1 i /. It is ot had to see that mst(γ(, ( ζ( if with = o(. We epeat the cojectue fom [1] that this is the wost-case, i.e. Cojectue 2 Assumig oly the coditios of Theoem 1, mst(g ( ζ(3 + 1 δ 2 + ɛ 5 whee ɛ 5 = ɛ 5 (δ 0 as δ. We pove istead Theoem 4 If G is a coected gaph the whee the ɛ 6 = ɛ 6 (δ 0 as δ. mst(g δ (ζ( ɛ 6 We fially ote that high coectivity is ot ecessay to obtai the esult of Theoem 2. Sice if = o( the oe ca toleate a few small cuts. Fo example, let G be a gaph which satisfies the coditios of Theoem 2 ad suppose = o(. The takig 2 disjoit copies of G ad addig a sigle edge joiig them we obtai a gaph G fo which mst(g 1 + ζ(3 ζ(3 whee 2 =2 is the umbe of vetices of G. 2 Poof of Theoem 3 Give a coected gaph G =(V,E with V = ad 0 p 1, let G p be the adom subgaph of G with the same vetex set which cotais those edges e with X e p. Let κ(g deote the umbe of compoets of G. We shall fist give a athe pecise desciptio of mst(g. Lemma 1 [1] Fo ay coected gaph G, mst(g = 1 p=0 E(κ(G p dp 1. (3

4 the electoic joual of combiatoics 7 (2000, #R41 4 We substitute p = x/ i (3 to obtai mst(g = 1 E(κ(G x/ dx 1. x=0 Now let C k,x deote the total umbe of compoets i G x/ with k vetices. Thus mst(g = 1 x=0 E(C k,x dx 1. (4 Poof of Theoem 3 I ode to use (4 we eed to coside thee sepaate ages fo x ad k, two of which ae satisfactoily dealt with i [1]. Let A =(/ω 1/3, B = (A 1/4 so that each of Bα, AB 2 / ad A/B 0as. These latte coditios ae eeded fo the aalysis of the fist two ages. Rage 1: 0 x A ad 1 k B see [1]. 1 A B E(C k,x dx (1 + o(1 x=0 ζ(3. Rage 2: 0 x A ad k>b see [1]. 1 A E(C k,x dx = o(/. x=0 k=b Rage 3: x A. We use a esult of Kage [4]. A cut (S : S ={(u, v E : u S, v / S} of G is γ-miimal if (S : S γλ. Kage poved that the umbe of γ-miimal cuts is O( 2γ. We ca associate each compoet of G p with a cut of G. Thus ( ( ( E(C k,x O 2s/λ 1 x s = O ( 2/λ e x s/ ( ( = O ( 2/λ e x s/ 2 e xλ/ ds = O x 2 log, λ ad usig Aλ ω 2/3 log we obtai 1 x=a ( 2 e xλ/ E(C k,x dx = O x=a x 2 log dx λ = O (A 1 2 e xλ/ dx x=a ( 2 = O Aλ e Aλ/ = o(/. We complete the poof by applyig Lemma

5 the electoic joual of combiatoics 7 (2000, #R Poof of Theoem 4 We keep the defiitios of A, B ad Rages 1,2, but we split Rage 3 ad let δ =. Rage 3a: x A ad k (1 ɛ, 0<ɛ<1, abitay see [1] (hee ɛ =1/2 but the agumet woks fo abitay ɛ. 1 (1 ɛ E(C k,x dx = o(/. x=a Rage 3b: x A ad k>(1 ɛ. Clealy ad hece 1 x=a k=(1 ɛ k=(1 ɛ C k,x (1 ɛ E(C k,x dx (1 ɛ. We agai complete the poof by applyig Lemma 1. 2 Refeeces [1]A.Beveidge,A.M.FiezeadC.J.H.McDiamid,Miimum legth spaig tees i egula gaphs, Combiatoica 18 ( [2] A. M. Fieze, O the value of a adom miimum spaig tee poblem, Discete Applied Mathematics 10 ( [3]A.M.FiezeadC.J.H.McDiamid,O adom miimum legth spaig tees, Combiatoica 9 ( [4] D. R. Kage, A Radomized Fully Polyomial Time Appoximatio Scheme fo the All Temial Netwok Reliability Poblem, Poceedigs of the twety-seveth aual ACM Symposium o Theoy of Computig ( [5] M. Peose, Radom miimum spaig tee ad pecolatio o the -cube, Radom Stuctues ad Algoithms 12 (