Random sampling of colourings of sparse random graphs with a constant number of colours

Random sampling of colourings of sparse random graphs with a constan t n um b er of colours Charilaos Efth ymiou ∗ P aul G. Spirakis † Abstract In this work we presen t a simple and eﬃcient algorithm which, with high pr obabilit y , provides an almost unifor m sample from the set of pro per k -colour ings on an instance of sparse random graphs G n,d/n , where k = k ( d ) is a suﬃcien tly large constant. Our algo r ithm is not based on the Mar k ov Chain Monte Carlo metho d (M.C.M.C.). Instead, w e pr o vide a no vel proof of correctness of our Algorithm that is based on interesting “spatial mixing” prop erties of colourings of G n,d/n . Our r esult improv es up on prev io us results (based on M.C.M.C.) that r equired a num b er of co lours growing unbo undedly with n . 1 In tro du ction. F or a graph G = ( V , E ), a (pr oper) k -colouring is an assignmen t σ : V → [ k ] suc h that adjacen t vertice s receiv e diﬀerent colours, where for some p ositiv e in teger k , [ k ] indicates the set { 1 , . . . , k } . It is w ell known that it is NP-hard to estimate the minimum n umber of colours in a p rop er k -colouring, i.e. estimate the c hromatic num b er of G . Ho we ve r, in m any cases there are estimates and upp er b ounds of the c hromatic num b er e.g. if ∆ is the maxim um degree of G , then one can k -colour G for k = ∆ + 1. F urth ermore, for sp ecial classes of graphs the chromatic n umb er has b een estimated very accurately , e.g. in [1], Ac hlioptas and Naor hav e found th e t w o p ossible v alues of the c hromatic num b er that an in sta nce of a sp arse random graph has with high p robabilit y , i.e. with p robabilit y that tends to 1 as the size of the graph tends to inﬁnity . All these raise the in teresting computational c hallenge of ﬁnding the num b er of p rop er k -co lourings for k greater than the chromatic n umber. In [ 14], V alian t, in tro duced the n oti on of # P -hardness and prov ed that counting k -colourings is # P -complete. The existence of a p olynomial-ti me algorithm f or exact counting is considered highly unlikely . Thus, w e fo cus on d esig ning p olynomial-time algorithms for appr oximate c ount- ing . Practically , the closer k gets to the c hromatic num b er of G , the more diﬃcult it b ecomes to estimate the num b er of its k -co lourings. By [7], [8 ] w e can red u ce the estimation of the numb e r of k -colo ur in gs of G to sampling almost uniformly from the s et of all its prop er k -colourings. By “almost” we mean with distribution close, in some sen se, to the uniform distribution. In this wo rk, w e use f ocus on sampling k -c olourings of instances of a s p arse random graph, i.e. random graphs w ith v ertices ha ving exp ected degree equal to some constan t, d , and k = k ( d ) is a suﬃcient ly large constan t which sca les as d 14 , for suﬃcien tly large d . ∗ Researc h Academic Computer T ec hnology Institute, N. K aza ntzaki str. Rio Patras, 26 500, Greece and Computer Engineering and Informatics Department of the Universit y of Pa tras, 26500, Greece. E-mail: euthimio@c eid.upatras.gr . CORRESPONDING AUTHOR † Researc h Academic Computer T echnology I nstitute N . Kazan tzaki str. Rio Patras, 26500, Greece and Com- puter Engineering and Iinformatics Department of Universit y of Patras, 26500, Greece. E- mai l : spirakis @cti.gr 1 Deﬁnition A L et n b e a p ositive inte g e r and p , 0 ≤ p ≤ 1 . The r andom gr aph G n,p is a pr ob ability sp ac e over the set of gr aphs on the vertex set { 1 , . . . , n } determine d by P r [ { i, j } is an e dge of G ] = p with these ev e nts b eing mutual ly indep endent. F or a spars e random graph the parameter p is of the form p = d/n , where d is a p ositiv e real constan t. W e take d > 1 (otherwise the p roblem of sampling is trivial). The mathematica l to ol that we us e for studying the problem of sampling k -colo urin g s of instances of a sp arse random graph is the “spin systems” and m ore sp eciﬁcally the prop er colour- ing mo del, also r efereed to as antiferr omagnetic Potts mo del at zer o temp er atur e in statistical physic s. Colouring Mo del on a ﬁnite graph . Th e colouring model on a ﬁnite graph G = ( V , E ) and set of colo urs [ S ], for some p ositiv e in teger S , is d eﬁned as follo ws. The s ystem consists of a set of sites , wh ic h corresp ond to the vertic es of G , and eac h site is assigned a spin , i.e. a mem b er of [ S ]. A c onﬁgur ation is an assignment of spins to V . Not all conﬁgurations can o ccur in the colouring mo del. A conﬁguration that ma y o ccur is called a fe asible c onﬁgur ation . The set of feasible conﬁgurations is the set of prop er S -col ourings of the under lyin g graph G . F or any vertex set V ′ ⊆ V , let ∂ V ′ = { v ∈ V \ V ′ | ∃ u ∈ V ′ s.t. { u, v } ∈ E } . C onsider the colouring C ( ∂ V ′ ) ∈ [ S ] ∂ V ′ , whic h is such that, there is a p rop er co louring in [ S ] V with the vertice s in ∂ V ′ coloured as C ( ∂ V ′ ). In a system where th e vertice s in ∂ V ′ are coloured as C ( ∂ V ′ ), the colour assignments of the vertice s in V ′ is distribu te d uniformly ov er, al l the co lour assignmen ts of the v ertices in V ′ ∪ ∂ V ′ that agree with C ( ∂ V ′ ) on the vertice s in ∂ V ′ 1 . F requen tly , one imp oses b oundary c onditions on the model, wh ich corresp onds to ﬁxing the colour assignmen t at s ome “ b oundary ” v ertex set of G ; we u s e the term “ fr e e b oundary ” when there are no b oundary conditions sp eciﬁed. In this w ork, wh en we consider a system with b oundary conditions, w e alw a ys assume that th er e exists at least one feasible conﬁguration a v ailable for this system. The p r obabilit y to ﬁnd a colouring mo del at a sp eciﬁc conﬁguration is the un iform distri- bution o v er all p rop er colourings of the un derlying graph . Generally , the probabilit y of ﬁn ding a system in a sp eciﬁc conﬁguration is giv en b y its Gibb s m easur e s p eciﬁed by this system. Deﬁnition B F or a ﬁnite gr aph G = ( V , E ) and an inte ger S , let P C S ( G, S , C ( L )) b e a c olourings mo del with underlying gr aph G , fe asible c onﬁgur ations al l the pr op er S - c olourings of G and the b oundary L ⊆ V is c olour e d as C ( L ) . L et Ω( G, S , C ( L )) b e the set of fe asible c onﬁgur ations of the system. The omission of the b oundary conditions p arameter implies free b oun d ary . If th e ﬁrst para- meter is a class of random graphs, e.g. G n,p , then w e consider that the underlying graph is an instance of this class. Clearly , Ω ( G , S , C ( L )) is the set of all prop er S -colourings of G that hav e the v ertices in the s et L ⊆ V coloured as s p eciﬁed by the assignment C ( L ). F or a system P C S ( G, S , C ( L )) w e alwa ys assume that the b oundary C ( L ) is such that Ω( G, S , C ( L )) 6 = ∅ . F or conv enience, w e use the follo wing notation r ules thr oughout this w ork. In P C S ( G = ( V , E ) , S , C (Λ)), for Λ ⊆ V the colour assignmen t of eac h v ertex v ∈ V , or the set of vertic es V ′ ⊆ V are considered to b e equal to the r andom v ariables X C Λ v ∈ [ S ] and X C Λ V ′ ∈ [ S ] V ′ , corresp ondingly . 1 A rigorous d eﬁnition of a colouring mo del invo lves the d eﬁnition of a set of functions, the c omp atibi lity functions (see [16]). Ho wev er, the deﬁnition w e giv e here is a direct consequence of that with t he compatibility functions. 2 Deﬁnition C F or the system P C S ( G = ( V , E ) , S ) , the function µ ( · ) : 2 [ S ] V → [0 , 1] indic ates the Gibbs me asur e sp e c iﬁe d by this system. In the system P C S ( G = ( V , E ) , S , C ( L )), for ∀ v ∈ V , w e denote with µ ( X v | C ( L )), the marginal Gibbs measure of the random v ariable X v . 1.1 Our w ork and related work. Previous work. The pioneering work of Dy er et.al., in [4], p roposes a very in teresting Ma rko v Chain Mon te Carlo (MCMC) based algorithm, which with high probab ility (w.h.p.), i.e. w ith probabilit y that tends to 1 as the size of graph tends to in ﬁnit y , p ro vides an almost un iform sample from the set of pr op er colourings of G n,d/n whic h uses at least Θ( log log n log log log n ) colours. Noting that, w.h .p. the maxim um degree of a s parse random graph is Θ(log n/ log log n ), to our kno wledge, this w ork w as the ﬁr st to presen t a p r ocedure for samp lin g colourings that uses few er colours than the maxim u m degree. In parallel and indep enden tly , E. Mossel a nd A. Sly , ha ve recen tly deriv ed es- sen tially the same result as w e do he re , (i.e. a random sampling k -colourings of a sparse ra ndom graph where k is a constan t) by using an MCMC approac h, [12]. Our w ork. Ou r approac h is quite diﬀerent. It is based on sh o wing the v alidit y of a sp eciﬁc sp atial mixing pr op erty for the system P C S ( G n,d/n , S ), I.e. we sho w that if S is greater than a sp eciﬁc v alue wh ic h dep ends only on th e exp ected d eg ree d , th en an asymptotic indep endenc e b et ween th e colour assignmen t of an y ve rtex v and th e colour assignment of an y su b set of v ertices whic h is at suﬃciently large (graph) distance from v holds, in the system. W e, then, p resen t an algorithm whic h exp loits this kind of asymptotic indep endence and pro duces, eﬃcientl y , an al most uniform sample of the S -colourings of the und erlying graph G n,d/n , wh ic h uses a constan t n umb er of colours (where the constan t is an increasing fu nctio n of d ). F or an earlier v ersion of our r esult, see [5]. Ou r algorithm exploits ideas whic h are similar to those presen ted in [11] and [15], for count ing satisﬁable tru th assignments in a r andom k -SA T formula and sampling indep endent s ets of general graphs, corresp ondin gly . Ho w ev er, our pr oof tec hn iqu es are no ve l and some tec hn ica l results are of ind epen d en t in terest. A p ossible schema of our algorithm is the follo win g one. The inpu t is the ﬁ n ite graph G = ( V , E ), an in sta nce of G n,d/n , and an in teger S . W e consider the system P C S ( G, S ) an d the algorithm pr o vides a sample wh ic h is distributed “clo se” to the Gibbs measure sp eciﬁed b y this system. The algorithm assumes an arb it rary p ermutatio n of the vertice s of the input graph, e.g. ( v 1 , v 2 , . . . , v n ), and in turn it assigns them a col our as follo ws: F or 1 ≤ i ≤ n , let A i ⊆ V b e the set of i − 1 ﬁrst coloured v ertices, by the algorithm, and let C ( A i ) b e their colour assignmen t. Assume that the colour assignment C ( A i ), of the v ertices in A i , is done according to a probabilit y measure wh ic h is suﬃ ci entl y close to µ ( X A i = C ( A i ))), w ith µ ( · ) b eeing the Gibbs measur e that is sp eciﬁed b y P C S ( G, S ). The algorithm computes eﬃcientl y a “go od ” estimation of µ ( X v i = s | C ( A i )), ∀ s ∈ [ S ], and assign v i a colouring according to this probabilit y measure. The notion of “go od” estimation of µ ( X v i = s | C ( A i )), ∀ s ∈ [ S ], implies that this estimation should b e so accurate, that it will b e p ossible for the algorithm to colour the remaining v ertices with distribu tio n as clo se to Gibbs measure of P C S ( G, S ) as we initially w an ted. The spatial mixin g prop ert y of P C S ( G n,d/n , S ), for su ﬃcien tly large S , is exploited by our algorithm in conjunction w ith the structural prop ert y of G n,d/n stated in Lemma A. Lemma A L et G = ( V , E ) b e an instanc e of G n,d/n , wher e d ≥ 1 is a ﬁxe d p ositive r e al. With high pr ob ability (w.h.p.) the gr aph has no v ertex v with the fol lowing pr op erty: The induc e d 3 sub gr aph of G tha t c ontains v and al l vertic es within gr aph distan c e ǫ log n fr om v , c ontains mor e than one cycle, for any r e al ǫ > 0 such that ǫ ≤ (4 log( e 2 d/ 2)) − 1 . The pro of of Lemma A is giv en in section 2.1. Sho wing an asymptotic ind epend ence b et ween the colour assignment of an y v ertex v and the colour assignmen t of any v ertex set at distance greater than ǫ log n , for suﬃcien tly small ǫ = ǫ ( d ), implies that w h en the algorithm has to assign a colour to the i -th v ertex the follo wing holds: If the colouring C ( A i ) is done with probabilit y measure whic h is suﬃcientl y close to µ ( X A i = C ( A i )), then the alg orithm can ha v e a “go od” estima tion of µ ( X v i | C ( A i )) by j ust c hec king the colour assignment s of a v ery simple structured neigh b orho od of v i . The notion of “go od ” estimation is the same as the one stated previously . This kind of structure in the neigh b orho od of the v ertex v i is highly d esirable since th en, it allo ws us to get a colouring of v i , whic h is distribu te d as this “go od” estimation of µ ( X v i | C ( A i )), in time whic h is u pp er b ound ed b y a p olynomial of n . 1.2 F urther D eﬁnitions (Spatial Dep endency) F or the graph G = ( V , E ) and any tw o ve rtex sets V ′ , V ′′ ⊂ V we d enot e by dist ( V ′ , V ′′ ) the graph d istance of the tw o s ets, i.e. the min im um length shortest p at h b et w een all the pairs of v ertices ( v 1 , v 2 ) ∈ V ′ × V ′′ . Deﬁnition D L et G = ( V , E ) b e an i nstanc e of G n,d/n and let l b e a p ostive r e al. F or the vertex v ∈ V , let G v,d, l b e the induc e d sub g r aph of G , which c ontains the ve rtex v and al l the vertic es within gr aph distanc e ⌊ l ⌋ fr om v . F or a measure of co mparison b et wee n pr obabilit y measures w e use the total variation distanc e . Deﬁnition E F or me asur es µ and ν on the same discr ete sp ac e Ω , the total variation distanc e d T V ( µ, ν ) b etwe en µ and ν is deﬁne d as d T V ( µ, ν ) = 1 2 X x ∈ Ω | µ ( x ) − ν ( x ) | Deﬁnition F (Spat ia l Dep endency .) Consider the gr aph G = ( V , E ) , an instanc e of G n,d/n , the p ositive inte gers S , l and the p ositive r e al s . F or e ach v ∈ V c onsider the sub gr aph G v,d, l with vertex set V v,l . F or a given v ∈ V , c onsider also any two S - c olourings C 1 ( V 1 ) and C 2 ( V 1 ) with V 1 ⊂ V v,l and dist ( { v } , V 1 ) ≥ l , having Ω( G v,l , S , C 1 ( V 1 )) , Ω( G v,l , S , C 2 ( V 1 )) 6 = ∅ . If for ∀ v ∈ V it holds d T V ( ˜ µ ( X v | C 1 ( V 1 )) , ˜ µ ( X v | C 2 ( V 1 ))) ≤ s with ˜ µ ( · ) sp e ciﬁe d by the system P C S ( G v,d, l , S ) , then we say that “ ∀ v the distance l Spatial Dep endency of the S -colourings of G is s ” . This wil l b e denote d as ∀ v ∈ V S D ( v, l ) = s . Clearly , the ab o ve deﬁ nitio n extends, dir ec tly , to any system with und erlying graph any t yp e of graph. It is easy to see that if “ ∀ v ∈ V the distance l S patia l Dep endency of th e S -colourings of G = ( V , E ) is s ”, then in the system P C S ( G, S ) and for eac h v ertex v ∈ V any t wo colourings C ′ 1 ( V 1 ) and C ′ 2 ( V 1 ) with V 1 as deﬁn ed in Deﬁnition F and C ′ 1 and C ′ 2 ha ving Ω( G, S , C ′ 1 ( V 1 )) 6 = ∅ and Ω( G, S , C ′ 2 ( V 1 )) 6 = ∅ it h olds d T V ( µ ( X v | C 1 ( V 1 )) , µ ( X v | C 2 ( V 1 ))) ≤ s with µ ( · ) sp eciﬁed, now, by the system P C S ( G, S ). Th e ab o v e holds since for eac h v ∈ V , if Ω 1 = { C ( V 1 ) | Ω( G v,l , S , C ( V 1 ) 6 = ∅} and Ω 2 = { C ( V 1 ) | Ω( G, S , C ( V 1 ) 6 = ∅} , then clearly Ω 2 ⊆ Ω 1 . 4 1.3 Structure of the remaining pap er. The remainin g of our pap er has the follo wing structure. In sectio n 2, we present t wo basic prop erties of the spin systems wh ic h w e deal with. Th en, it follo ws a detail d escrip tio n of our sampling algorithm in a form of ps eu do co de accompanied by a discus sion and a statemen t, without pro of, of t wo theorems whic h deal with the acc ur acy of the r esu lt that is returned and the execution time of the algo rithm, coresp ondingly . The pro ofs and an analytic d iscu ssion ab o ut th e prop erties of the spin-system that our algorithm co nsid ers, are giv en in section 3. This w ork ends b y presen ting the pro ofs of the theorems, that d eal with the accuracy and the time eﬃciency of the our samplin g algorithm, section 4. F or easiness of v eriﬁcation of our pro ofs, we provide here the dep endence of eac h lemma and theorem on lemmas logically p rece ding it: Lemma A, Lemma B and Lemma C, h a ve no lemmas preceding them. Lemma D has pr edec essors Lemma A, Lemma B and Lemma C. Lemma E has predecessors L emm a A, Lemma B, Lemma C and Lemma E. Lemma F has no pr edec essors. Lemma G has predecessors Lemma C and Lemma B . Lemma H has pr ed ec essors, Lemma F and Lemma G. Theorem A h as precedings all the lemmas of this pap er. Theorem B has predecessors, Theorem A, Lemma A. Theorem C, has predecessors, Theorem A , T heorem B and Lemma A. Finally , Th eo rem D h as n o p receding. 2 Statemen t of results. 2.1 Prop erties of t he spin system. In this section w e state t w o cru cia l prop erties th at th e colouring mo del has with h igh probability , when the underlying graph is an in stance of a sparse random graph. Th ese prop erties are stated in the f ol lo wing lemma and theorem. The ﬁrst one, already stated in section 1, refers to the structure of the neighborho o d of eac h vertex in an instance of a sp arse rand o m graph. The second one refers to a prop ert y of the colourings (conﬁgurations) of such a spin-system. I .e. in a P C S ( G n,d/n , S ), if S is greater than a sp eciﬁc v alue wh ic h dep ends only on the exp ected degree d , then an asymptotic indep endenc e b et w een the colour assignmen t of an y v ertex v and the colour assignmen t of an y subset of v ertices whic h is at graph d ista nce, at least, j 0 . 9 4 log( e 2 d/ 2) log n k from v holds. Lemma A L et G = ( V , E ) b e an instanc e of G n,d/n , wher e d ≥ 1 is a ﬁxe d p ositive r e al. With high pr ob ability (w.h.p.) the gr aph has no v ertex v with the fol lowing pr op erty: The induc e d sub gr aph of G tha t c ontains v and al l vertic es within gr aph distan c e ǫ log n fr om v , c ontains mor e than one cycle, for any r e al ǫ > 0 such that ǫ ≤ (4 log( e 2 d/ 2)) − 1 . Pro of: T o sh o w the lemma assu me the contrary , i.e. there is some v ertex v ∈ V whose corresp onding graph G v,d, ǫ l og n (see Deﬁnition D) cont ains t w o cycles, i.e. C 1 and C 2 eac h of length at most 2 ǫ log n , the v alue of ǫ will b e determined later. Th e ab o v e assump tio n imp lie s that there are t w o p airs of paths starting f r om v , suc h th at: The paths in eac h pair do not ha v e al l their edges common and there is some v ertex in G v,d, ǫ l og n that can b e reac h ed from v b y both paths of the pair. Th e existence of such t wo pairs of paths implies that in G v,d, ǫ l og n there is a set of, at most 4 ǫ log n , v ertices whic h ha v e among eac h other a num b er of edges wh ich exceeds th e num b er of v ertices by one. 5 Th us, the pro of of the le mma reduces to sho w in g that in G n,d/n , there is no set of, at most, 4 ǫ log n , v ertices whic h con tains a num b er of edges that exceed the num b er of vertices in the set by one, for suﬃciently small ǫ . Let D b e the ev en t “suc h a s et exists”. Setting r = 4 ǫ log n w e hav e P r [ D ] ≤ r X k =1 n k !  k 2  k + 1 !  d n  ( k +1) ≤ r X k =1  ne k  k  ek ( k + 1) 2( k + 1)  ( k +1)  d n  ( k +1) ≤ de 2 n r X k =1 e 2 2 ! k k d k ≤ d 2 e 3 4 4 ǫ log n n r − 1 X k =0 de 2 2 ! k ≤ d 2 e 3 ǫ log n n ( de 2 / 2) r − 1 de 2 / 2 − 1 taking ǫ suc h that 4 ǫ · log ( d e 2 / 2) < 1 the r.h.s. of the last equation is o (1), as the nominator is o ( n ). Th us, for suﬃciently small ǫ w e hav e that P r [ D ] = o (1) ⋄ Theorem A L et G = ( V , E ) b e an instanc e of G n,d/n , wher e d > 1 . If S is a suﬃciently lar ge inte ger, which dep ends on d , and ǫ = 0 . 9 4 log( e 2 d/ 2) , then w.h.p., i.e. with pr ob ability 1 − 2 n − 0 . 25 , for every vertex v ∈ V the distanc e ⌊ ǫ log n ⌋ Sp atial De p endency of the S -c olourings of P C S ( G, S ) is n − 1 . 25 . F or suﬃciently lar ge d , we should have S ≥ d 14 . W e mentio n that, if d is relativ ely small, then for P C S ( G n,d/n , S ), the d ista nce ⌊ ǫ log n ⌋ Sp at ial Dep endency of the S -colourings of P C S ( G, S ) can b ecome n − 1 . 25 , h o we ve r, for a num b er of colours which is a constan t greater than d 14 . Section 3 is devo ted to the pro of of Theorem A. 2.2 Our Algorithm W e start by presenti ng our s amp ling algorithm in the form of pseud o cod e. The in put of the algorithm is the graph G = ( V , E ) an instance of G n,d/n with d > 1, and the integ er S . Conditioning on the pr operties of the system stated in the previous s ec tion, whic h hold w.h.p. b y taking suﬃ ciently large constan t S , the algorithm outputs a S -colouring of the input graph distributed within total v ariation distance n − 0 . 25 from the uniform d istribution o v er the set of all S -colourings of the input graph. W e hav e to men tion, here, that our algorithm is based on p rop er ties of the spin-system that already hold w.h.p.. This means that we can exp ose the en tire instance of the inpu t graph at the b eginning and exp ect these, desired, prop erties to hold, which is highly lik ely . In what follo ws, we assum e that v i is the i -th v ertex to b e coloured by the algorithm 2 and A i is the set of v ertices that ha v e already b een coloured, b efore v i . W e, also , assu m e that A i is coloured as C ( A i ). W e denote with ( V i , E i ) the verte x set and th e edge set, co rresp ondin gl y , of the graph G v i ,d,ǫ l og n , where ǫ = 0 . 9 4 log( de 2 / 2) 2 W e remind the reader that the algorithm colours one vertex th e time. 6 Sampling Algorithm Input: G = ( V , E ), instance of G n,d/n , num b er of colours S T ak e an arbitrary p erm utation of the v ertices in V , i.e. ( v 1 , . . . , v n ) A 1 = ∅ F or i = 1 , . . . , n -Create the sub g raph G v i ,d,ǫ log n = ( V i , E i ) - If G v i ,d,ǫ log n = ( V i , E i ) is not a tree or a u nicycli c graph Then Return F ailure -Colour v i according to ˜ µ i ( X v i | C ( A i ∩ V i )) using dynamic programming - A i +1 := A i ∪ { v i } Return Colouring of G The probability measur e ˜ µ i ( · ) is th e Gibb s measur e sp eciﬁed by the system P C S ( G v i ,d,ǫ log n , S ). Note that if the algorithm, at eac h iteration of the for loop, had assigned to the v ertex v i a colouring, according to µ ( X v i | C ( A i )), instead of ˜ µ i ( X v i | C ( A i ∩ V i )), then it would hav e b een exact. I.e., th e distribution ˜ µ i ( X v i | C ( A i ∩ V i )) is an estimation of µ ( X v i | C ( A i )) for our algorithm. There are tw o issues to b e clariﬁed ab out the algorithm ab o v e. T he ﬁrst one is its accuracy , i.e. how close is the d istr ibution of the colouring that is returned , to the uniform distribution o v er all prop er S -colourings of the inpu t graph . Th e second one is its eﬃcienc y , i.e. ho w m uch time is needed f or th e execution of the algorithm with resp ect to the size of the input graph. As f ar as the accuracy of the algorithm is regarded, we use Theorem A. By the discussion at the en d of section 1.2, w e see that Th eo rem A implies that, w.h.p . the Gibbs measur e sp eciﬁed by P C S ( G n,d/n , S ), wh er e S is a suﬃcien tly large co nstant, is suc h that an asymptotic indep endenc e (spatial mixing) holds, for the colour assignment of any v ertex v and the colour assignmen t of any v ertex set at distance ⌊ ǫ log n ⌋ from v , where ǫ = 0 . 9 4 log( e 2 d/ 2) . W e claim that at the i -th iteration of the for lo op of the algorithm, the follo wing h o lds: If the algorithm has assigned a colouring to the ve rtices in A i , according to the Gibbs measure, µ ( · ), that sp ec iﬁes P C S ( G, S ), then it, still, holds the same asymptoti c indep endence as th at it is implied by Theorem A, b etw een the colouring of the v ertex v i and the colourings of ve rtices at distance, at least, ⌊ ǫ log n ⌋ fr om v i , where ǫ = 0 . 9 4 log( e 2 d/ 2) . Equiv alen tly , we can think of the follo w ing situtation. Consider tw o systems, i.e. S 1 = P C S ( G, S ) and S 2 = P C S ( G, S ), with u nderlying graph G , the in put of our algorithm and with eac h sys tem b eing indep endent of the other. Assum e that, in b oth systems, we ﬁx the colour assignmen ts of the ve rtices in A i ⊂ V according to µ ( X A i ), with µ ( · ) sp eciﬁed by P C S ( G, S ). Assume that after ha ving ﬁx ed the colour assignments, of the vertice s in A i , we lo ok at the colour assignments of th e v ertices at graph d istance , at least, ⌊ ǫ log n ⌋ from the verte x v i , in b oth systems. L et V ′ b e the verte x set whose colourings ha ve b een seen and let C ( V ′ ) b e the colouring w e see in S 1 and C ′ ( V ′ ) b e the colouring we see in S 2 . T he ab o ve claim, is equiv alent to sa ying that knowing the colo ur assignment s of the vertices in V ′ in b oth sy s te ms, then the total v ariation distance b et w een the probabilit y measures of the colour assignmen ts of v i in th e t w o systems, corresp ondingly , is upp er b ounded b y the quanitit y S D ( v i , ⌊ ǫ log n ⌋ ). I.e. d T V  E [ µ ( X v i | X A i , X V ′ = C ( V ′ )) , E [ ˜ µ i ( X v i | X A i , X V ′ = C ′ ( V ′ ))  ≤ n − 1 . 25 . (1) Where b oth exp ectati ons, are tak en o v er all colourings of X A i where the pr obabilit y of eac h colouring is according to Gibb s measure µ ( · ). 7 The follo wing theorem, Theorem B, (in the pr oof of wh ic h it is shown the v alidity of the claim ab o ve ) give s a c haracterization of the distribution of the colouring th at is returned by the algorithm in terms of its total v ariation distance from th e uniform o ver all th e prop er S -colourings of the inpu t graph, if S is as large as indicated in Theorem A. Theorem B If S i s a suﬃciently lar ge inte ger c onstant, then, with pr ob ability 1 − O ( n − 0 . 1 ) , the sampling algorithm is suc c essfu l and r eturns a S -c olouring of the input gr aph G , whose distribution is within total variation dista nc e n − 0 . 25 fr om the uni f or m over al l the pr op er S - c olourings of G . The pro of of Theorem B is giv en in section 4. As far as the execution time of the algorithm is concerned , we mak e the follo wing remark. According to Lemma A, the set { G v i ,d,ǫ log n , for i = 1 , . . . , n } , for ǫ = 0 . 9 4 log( e 2 d/ 2) , w.h.p., i.e. with probabilit y 1 − n − 0 . 1 , con tains graphs whic h are eit her unicyclic or trees. If th is is not the case, then w e consid er that the algo rithm fails. As argued in [4], we can hav e a colouring of the v ertex v i according to ˜ µ i ( X v i | C ( A i ∩ V i )) by generating a random colouring of G v i ,d,ǫ log n where the v ertices in A i ∩ V i are colored as C ( A i ∩ V i ) in time upp er b ounded b y l · k 3 , where l = | V i | and k = S (for more details see the pro of of Theorem C and [4]). Theorem C The time c omplexity of the sampling algorithm is w.h.p. asymptotic al ly upp er b ounde d by O ( n 2 ) , wher e n is the numb er of vertic es of the input gr aph. The pro of of Theorem C is giv en in section 4. W e n ot e that at the i -th iteratio n of the for-lo op of the algorithm, we can app ly the Junction tree algorithm (see [16]) to assign the v ertex v i a colour according to the probabilit y measure ˜ µ i ( X u | C ( A i ∩ V i )). The execution time of the ju n ctio n tree is asymptotically b ounded b y O ( n 2+ c ), where c < 1 is a suﬃcien tly large constan t. 3 Spatial mixing. According to Lemma A, if G = ( V , E ) is an instance of G n,d/n , then the set of graphs G v,d, ǫ l og n , for v ∈ V , with ǫ = 0 . 9 log( e 2 d/ 2) , w.h.p. conta ins graph s whic h, eac h of them, is either unicyclic or trees. Instead of p roo ving Theorem A, equiv alen tly , we sh ow the tw o follo wing lemmas, whic h are prov ed in sectio ns 3.2 and 3.3, corresp ondingly . Lemma E Consider the system P C S ( G v,d, ǫ l og n , S ) , for d > 1 , ǫ = 0 . 9 4 log( e 2 d/ 2) and for G v,d, ǫ l og n we c ondition that it is a tr e e. If S is a suﬃciently lar ge c onstant, then with pr ob ability at le ast 1 − 2 n − 1 . 25 , for the ab ove system i t holds that S D ( v, ⌊ ǫ log n ⌋ ) = n − 1 . 25 . F or suﬃciently lar ge d , we should have S ≥ d 14 . Lemma H Consider the system P C S ( G v,d, ǫ log n , S ) , for d > 1 , ǫ = 0 . 9 4 log( e 2 d/ 2) and for G v,d, ǫ l og n we c ondition that it is a unicyclic gr aph. If S is a suﬃcie ntly lar ge c onstant, then with pr ob ability at le ast 1 − 2 n − 1 . 25 , for the ab ove system it holds that S D ( v , ⌊ ǫ log n ⌋ ) = n − 1 . 25 . F or suﬃciently lar ge d , we should have S ≥ d 14 . One can see that th e lemmas E and H imply Theorem A, see C orol lary A. Corollary A If L emma E and L emma H ar e true, then The or em A is true, as wel l. 8 Pro of: Assum e that Lemma E and Lemmas H are tru e. Consider the sys tem P C S ( G n,d/n , S ), where S is a suﬃ ci entl y large constan t and for su ﬃcien tly large d , S ≥ d 14 . T heorem A holds for P C S ( G n,d/n , S ) if the follo wing ev en t holds with probabilit y at least 1 − 2 n − 0 . 25 E v ent 1 = “ for ev ery graph G v,d, ǫ l og n of the set of graph s that G n,d/n sp eciﬁes, it holds that the P C S ( G v,d, ǫ log n , S ) has the p roper ty that S D ( v , ⌊ ǫ log n ⌋ ) = n − 1 . 25 ” By Lemma E and Lemma H w e hav e that for suﬃcien tly large S , which for suﬃcien tly large d b ecomes S ≥ d 14 , suc h that, for ev ery v ertex v in G n,d/n the E v ent v = “ the system P C S ( G v,d, ǫ log n , S ) has the p roper ty that S D ( v , ⌊ ǫ log n ⌋ ) = n − 1 . 25 ” holds with prob ab ility at least 1 − 2 n − 1 . 25 . C le arly P r [ E v ent 1 ] = 1 − P r h ∪ v E v ent v i By the union b ound w e get that P r [ E v ent 1 ] ≥ 1 − 2 n − 0 . 25 , whic h prov es corollary . ⋄ Note. In b oth cases the crucial p oint is to sho w that a certain quan tit y (the “disagreement probabilit y”) has a sm al l exp ected v alue. Th is quanti t y measures ho w m uch we deviate if, for colouring some v ertex v in the algorithm, w e consider only a heigh b orho od of its, instead of the whole graph. In the sequel we carefully establish the necessary u p p er b ounds for this. Remark. Both Lemma E and Lemma H are based on the fact th at we exp ect a very large prop ortion of the v ertices of an instance of G n,d/n to hav e constan t degrees. I.e. there is a constan t c 0 = c 0 ( d ) suc h that for an y c > c 0 the exp ected prop ortion of vertice s that ha v e degree less than c tend s to 1, exp onen tially fast with c . This argument is justiﬁed by the follo win g corollary that is pr o v ed in [6]. Corollary B If a r andom variable Z is distribute d as in B ( n, q ) , the binomial distribution with p ar ameters n and q , with λ = nq then P r [ Z ≥ x ] ≤ e − x x ≥ 7 λ. 3.1 The pro cess ColourRo ot and a coupling. T o w ards pr oving Lemm a E and Lemma H, we intro d uce, here, the sto c hastic pro cess ColourR o ot ( T , S , C ( L )), where T = ( V , E ) is a tree, S is a p ositiv e intege r and the v ertices in L ⊂ V are assigned a colouring C ( L ), suc h that Ω ( T , S , C ( L )) 6 = ∅ . The p rocess ColourR o ot ( T , S, C ( L )) assigns a colouring (not necessarily p roper ), to the vertic es in V \ L suc h that ∀ u ∈ V \ L its colour assignmen t is distribu te d as in µ ( X u | C ( L ∩ T u )), where T u is the subtree of T ro oted at u while the Gibb s measure µ ( X v | C ( L ∩ T u )) is sp eciﬁed by the system P C S ( T u , S , C ( L ∩ T u )). When the third parameter of the ColourRo ot is omitted, it is implied that there is no ﬁx ed colour assignmen t to any v ertex. The ColourRo ot( T , S , C ( L )) assigns a colouring to eac h verte x u in the tree T , based on the follo wing observ ation. F or the v ertex u of T consider the v ertex set C H u whic h con tains the c hildren of u in T u and the system P C S ( T u , S , C ( L ∩ T u )). F or the graph T 0 = ∪ w ∈ C H u T w consider the set of S -colorings Ω 0 = Ω( T 0 , S , C ( L ∩ T 0 )) Assume that eac h C ∈ Ω 0 sp eciﬁes a colouring of the ve rtices in C H u that us es all but W C colours fr om the set [ S ]. Note that if the system P C S ( T u , S , C ( L ∩ T u )) is in equilibr ium, the p robabilit y for the v ertices in T 0 = ∪ w ∈ C H u T w to b e coloured as sp eciﬁed by C ∈ Ω 0 is pr oportional to the qu an tit y W C , i.e. W C P C ∈ Ω 0 W C . 9 Deﬁnition G Wi th the ab ove notation, the pr o c ess ColourR o ot( T , S , C ( L ) ) assigns a c olour to the ve rtex u of T , as fol lows: 1. E ach C ∈ Ω 0 is assigne d weight, W C , e qual to the numb er of c olours in the set [ S ] that do not app e ar in the c olour assignment that C sp e ciﬁes for the vertic es in C H u . 2. Sele ct fr om Ω 0 such that the pr ob ability for e ach memb er to b e chosen is pr op ortional to the weight i t has b e en assigne d to it. L et C ′ b e the chosen memb er. 3. A ssign to the vertex u a c olour that is chosen uniformly at r andom among the c olours in the set [ S ] that do not app e ar in the c olouring of the vertic es in C H u , as this i s sp e ciﬁe d by C ′ . F or a c oupling of the p r ocesses ColourRo ot( T , S , C ( L )) and ColourRo ot ( T , S , C ′ ( L )), we in tro duce the notion of disagr e ement pr ob ability . Deﬁnition H Consider a c oupling of ColourR o ot ( T , S , C ( L )) and ColourR o ot ( T , S , C ′ ( L )) . The disagr e e ment pr ob ability for a vertex u in T , denote d b y p u , is e qual to the pr ob ability for the c oupling to assign diﬀer ent c olours to u . The coupling of ColourRo ot ( T , S , C ( L )) and C olourRo ot( T , S , C ′ ( L )) is of our main inte rest here, due to the follo wing, v ery signiﬁ cant fact. Theorem D Consider the tr e e T = ( V , E ) r o ote d at the vertex r , some set A ⊆ V and any two c olourings of the vertex set A , suc h that Ω( T , S , C ( A )) , Ω( T , S , C ′ ( A )) 6 = ∅ . Assume that ther e is a c oupling of the ColourR o ot ( T , S , C ( A )) and the ColourR o ot ( T , S , C ′ ( A )) , for some inte ger S , such that the pr ob ability of disagr e ement f or the r o ot r is p r . Then, it holds that d T V ( µ ( X r | C ( A )) , µ ( X r | C ′ ( A ))) ≤ p r . wher e µ ( X r | C ( A )) and µ ( X r | C ′ ( A )) ar e sp e ciﬁe d by the system P C S ( T , S ) . Pro of: Th e theorem follo ws directly from the Cou p ling Lemma (see [2]). ⋄ F or the sys tem P C S ( T , S ), where T is a tree ro oted at v ertex r , one can d er ive up p er b ound s for S D ( r , l ), for some p ositiv e in teger l , by u sing the ab o ve theorem an d the coupling of th e ColourRo ot, as describ ed in the follo win g deﬁnition. Deﬁnition I Consider the tr e e T = ( V , E ) r o ote d at vertex r , an i nte ger S and the set V 1 ⊂ V such that dist ( { r } , V 1 ) ≥ l for som e inte ger l . L et C ( T , S , l ) b e th e c oupling of the pr o c esses ColourR o ot( T , S , C 1 ( V 1 ) ) and ColourR o ot( T , S ). The c olour assignment C 1 ( V 1 ) , is taken so as to m axi mize the disagr e ement pr ob ability at the r o ot of T . The c oupling C ( T , S , l ) assigns c olours to the vertex u of T as fol lows: • Couple step 2 of the two pr o c esses so as to maximize the pr ob ability for the vertic es in C H u to have the same c olour assignment. • Conditional on the choic es the two pr o c esses have made at their step 2, assign c olours to u , so as to minimize the disagr e ement pr ob ability p u . In the coupling C ( T , S , l ), if the h eig ht of T is less th an l , then the set V 1 , as giv en in Deﬁnition I, is empt y . It is easy for one to see that in that case, the d isag reement p robabilit y for all v ertices in T , is zero. 10 Corollary C Consider a tr e e T , r o ote d at vertex r . If the c oupling C ( T , S , l ) , for some p ositive inte gers S , l , has disagr e ement pr ob ability p r for the r o ot of T , then for the system P C S ( T , S ) it holds that S D ( r , l ) ≤ 2 p r . Pro of: F or the tree T , ro oted at v ertex r , and the in tegers S , l , assume that the coupling C ( T , S , l ) has d isag reement pr ob ab ility on the ro ot p r . Consider the v ertex set L , wh ic h con tains v ertices at distance, at least, l from the ro ot r . Let, also, ˜ C ( L ) and ˆ C ( L ) b e the t wo colourings whic h maximize th e total v ariation distance of th e measures µ ( X r | C ( ˜ C ( L )) and µ ( X r | ˆ C ( L )), as these are sp eciﬁed by th e system P C S ( T , S ). It holds that S D ( r , l ) = d T V  µ ( X r | C ( ˜ C ( L )) , µ ( X r | ˆ C ( L ))  ≤ d T V  µ ( X r | C ( ˜ C ( L )) , µ ( X r )  + d T V  µ ( X r ) , µ ( X r | ˆ C ( L ))  ≤ 2 p r the second direv ation follo ws b y the triangle in equalit y for measures. T he co rollary follo ws. ⋄ W e note to the reader that, here, we will not need to giv e an explicit d esciptio n of the coupling C ( T , S , l ). It suﬃces to sho w that C ( T , S , l ) h as tw o sp eciﬁc prop erties, those that indicated by Lemma B and Lemma C 3 . In the rest of this section, w e state and prov e Lemm a B and Lemma C. T hese t wo lemmas pro vide means to derive upp er b ounds for the probabilit y of d isag reement , in the coupling C ( T , S , l ), for eac h v ertex u of T . More sp eciﬁcally , Lemma B and Lemma C p ro vide an inductiv e description of the couplin g C ( T , S , l ), in terms of the disagreemen t probabilities. I.e. considering the vertex u and the set C H u of its c hildr en, if the coupling C assigns colours to eac h ve rtex w ∈ C H u suc h that the probabilit y of disagreement is p w , then for the vertex u the probabilit y of disagreement p u , in C , can b e b ounded as follo ws p u ≤ a ( | C H u | , S ) ·   X w ∈ C H u p w   where a ( | C H u | , S ) is a constan t th at dep ends on th e cardinalit y of C H u and S . W e distinguish t wo classes of v ertices in T regarding the relation b et ween their n umb er of c hildren and the n umb er of av ailable colours S , i.e. the mixing ve rtices and th e nonmixing v ertices. Th e mixing vertice s ha ve a n umb er of children whic h is smaller than S and the constant a ( | C H u | , S ) is very small for them, i.e. << 1. The nonmixing v ertices h a v e high d eg rees an d for them the constant a ( | C H u | , S ) ma y b ecome v ery large. Deﬁnition J E ach vertex u of the tr e e T is “mixing” if, for a given t , the numb er of its childr en in T is less than or e qual to t , otherwise it is “nonmixing”. The v alue of t , the maxim um num b er of children of a mixing vertex, in the coupling C ( T , S , l ) is always less than the numb er of available c olours . Generally , for a giv en tree T and n u mb er of colours S , w e tak e t so large as to minimize th e prob ab ility of disagreemen t of the ro ot of the tree T in C ( T , S , l ). 3 How ever, if the reader is keen on ﬁnding one, t hen he can deduce on e from t he proofs of Lemma B an d Lemma C and the pro ofs of the claims inside th em. 11 Lemma B L et u b e a vertex of the tr e e T which is mixing . If for eve ry vertex w ∈ C H u the pr ob ability of disagr e ement, in the c oupling C ( T , S , l ) , is p w , then for the vertex u the p r ob ability of disagr e ement p u , in C , is b ounde d as p u ≤ t · S ( S − t ) 2 ·   X w ∈ C H u p w   wher e t is the maximum numb er of c hildr en of a mixing vertex. Pro of: Assume that in the coupling C ( T , S , l ), when the v ertex u is to b e coloured, the pro cesses ColourRo ot( T , S , C 1 ( V 1 )) and ColourRo ot( T , S ) at their second s te p choose from the set of colourings Ω C and Ω F , corresp ondingly . Let A b e the ev ent th at , when the vertex u is to b e coloured in C the m em b ers of Ω F and Ω C , that are chosen at step (2) of th e pro cesses ColourRo ot, sp ecify diﬀerent colour assignm ents for the ve rtices in C H u . Then P r [disagreemen t on u ] = P r [disagreemen t on u |A ] P r [ A ] + P r [disagree ment on u | A ] P r [ A ] W e mentio n that, if th e ev en t A do es not hold ( A holds), then there is a coup lin g for step (3) of the C olo urRo ot that assigns the same colour to the v er tex u in C , i.e. P r [disagreemen t on u | A ] = 0. Th us, P r [disagreemen t on u ] = P r [disagreemen t on u |A ] P r [ A ] (2) T o show the lemma w e pr ovide app ropriate up per b ounds f or the pr ob ab ilities in (2), i.e. P r [ A ] and P r [disagree ment on u |A ]. W e start with b oundin g P r [ A ], W e note that the assu m ption that for eac h w ∈ C H u , the disagreement pr o bability in C , is p w can b e seen as follo w s: There is a coupling, call it K 1 , whic h c ho oses u niformly at random (u.a.r.) from the sets Ω F and Ω C and the tw o c h ose n elemen ts sp ecify diﬀeren t colour assignmen ts for the v ertices in C H u with probabilit y whic h is upp er b ounded b y P w ∈ C H u p w . Note that | Ω F | 6 = | Ω C | . W e create the set Ω ′ F suc h that, eac h elemen t of Ω F app ears | Ω C | times in Ω ′ F . S imila rly , w e creat e the set Ω ′ C suc h that, eac h elemen t of Ω C app ears | Ω F | times in Ω ′ C . Clearly , | Ω ′ C | = | Ω ′ F | . Claim A We c an have a c oupling, c al l i t K 2 , tha t cho oses uniformly at r andom (u.a.r.) an element fr om e ach of the sets Ω ′ C and Ω ′ F , such that the pr ob ability for the two chosen elements to sp e cify diﬀer e nt c olour assignment for any vertex in C H u is u pp er b ounde d by P w ∈ C H u p w . Assume that in C , eac h of th e executions of ColourRo ot, at step (2), considers the sets Ω ′ C and Ω ′ F , corresp ondingly , in stea d of Ω C and Ω F . C learly , the fact that eac h of the pro cesses ColourRo ot considers the set Ω ′ C instead of Ω C and Ω ′ F instead of Ω F , corresp ondingly , do es not change the m argi nal p robabilit y measure of the colour assignment of the ve rtex v , in the coupling C . Claim B Assume that the numb er of childr en of the ve rtex u is k and the disagr e ement pr ob- ability for w ∈ C H u is p w . If at the c oupling of the se c ond step of the pr o c esses ColourR o ot in C ( T , S , l ) , e ach C ∈ Ω ′ C ∪ Ω ′ F is assigne d weight W C , then for the event A we have that P r [ A ] ≤ 1 q k , S max C ∈ Ω ′ F ∪ Ω ′ C { W C } min C ∈ Ω ′ F ∪ Ω ′ C { W C | W C > 0 } X w ∈ C H u p w . wher e q k , S is the pr ob ability of the e v ent that after k trials, not al l elements of [ S ] have b e en chosen, when at e ach trial we cho ose u.a.r. a memb er of [ S ] . 12 Note that the n umb er of c hildren of a mixing v ertex is less than the num b er of a v ailable colours, th us, there is no colouring of the v ertices in C H u that lea v e n o a v ailable colour for u . Sin ce w e hav e assumed that u is mixing , it is clear that in our case q k , S = 1. Also n ote th at at th e coupling of s tep (2) of the ColourRoot, for col ouring the ve rtex u , no mem b er of either Ω ′ C or Ω ′ F is assigned a weig ht wh ic h is more than S and less than S − t , wh ere t is equal to th e maxim um num b er of children that a mixing v ertex can ha ve. Th us, w e h av e P r [ A ] ≤ S S − t X w ∈ C H u p w W e p roceed to derive a b ound for P r [disagreemen t on u |A ]. F or this we use the follo wing claim. Claim C Consider the c oupling C ( T , S , l ) when it assigns c olourings to the vertex u . Assume that the two pr o c esses chose memb ers of Ω ′ F and Ω ′ C that sp e cify c olourings f o r the vertic e s in C H u such that for the vertex u , ther e ar e two lists of available c olours, l 1 and l 2 , c orr esp ondingly. Assuming that | l i | > 0 , for i = 1 , 2 , ther e is a c oupling that c an cho ose the same c olour for the vertex u with pr ob ability at le ast 1 − max {| l 1 \ l 2 | , | l 2 \ l 1 |} min {| l 1 | , | l 2 |} . By Claim C we ha ve that P r [disagreemen t on u |A ] ≤ t |S | − t since, | l 1 \ l 2 | , | l 2 \ l 1 | ≤ t and | l 1 | , | l 2 | ≥ S − t , where l 1 , l 2 are as deﬁned in the statemen t of Claim C. Com bin ing all the ab o ve w e get the lemma. ⋄ W e now pro ce ed to pr o v e the claims stated in the pr oof of Lemma B. Claim A We c an have a c oupling, c al l it K 2 , tha t cho oses uniformly at r andom an element fr om e ach of the sets Ω ′ C and Ω ′ F , suc h that the pr ob ability for the two chosen elements to sp e cify diﬀer ent c olour assignment for any vertex in C H u is upp er b ounde d by P w ∈ C H u p w . Pro of: The coup ling K 2 is d eﬁned as follo ws : C hoose u.a.r. a member of Ω C , let C b e the c hosen elemen t, b y using K 1 , take the corresp ondin g elemen t of Ω F , let C ′ b e that element. Then, c ho ose u.a.r. one among the copies of C in Ω ′ C and one of the copies of C ′ in Ω ′ F . Clearly , eac h of the elemen ts of Ω ′ C and Ω ′ F is c h osen, uniformly at r an d om. Ou r claim follo ws by noting that, the c hosen elemen ts of Ω ′ C and Ω ′ F diﬀer in the colo ur assignments of the vertic es in C H u iﬀ C and C ′ do so. ⋄ Claim B Assume for the vertex u that | C H u | = k and the disagr e ement pr ob ability for w ∈ C H u is p w . If at the c oupling of the se c ond step of the pr o c esses ColourR o ot i n C ( T , S , l ) , e ach C ∈ Ω ′ C ∪ Ω ′ F is assigne d a weight W C , then for the event A we have that P r [ A ] ≤ 1 q k , S max C ∈ Ω ′ F ∪ Ω ′ C { W C } min C ∈ Ω ′ F ∪ Ω ′ C { W C | W C > 0 } X w ∈ C H u p w . (3) wher e q k , S is the pr ob ability of the e v ent that after k trials, not al l elements of [ S ] have b e en chosen, when at e ach trial we cho ose u.a.r. a memb er of [ S ] . 13 Pro of: Consid er that we c ho ose from Ω ′ F suc h that the elemen t C is c hosen w ith pr obabilit y prop ortional to its w eigh t, W C . Consider the s ame for the set Ω ′ C . If there is a coupling of these t w o random weigh ted selections ab o v e, such that the pr obabilit y of the ev en t A to b e upp er b ounded as in (3), then we are done. The assumption that in the coupling C ( T , S , l ), for eac h w ∈ C H u , the disagreemen t prob- abilit y is p w , is equiv alen t to the follo win g : T here is a mapp ing, call it f : Ω ′ F → Ω ′ C , w h ic h is one to one (and ‘on to’, s in ce | Ω ′ F | = | Ω ′ C | ) and f or an y pair of colourings ( C, f ( C )) ∈ Ω ′ F × Ω ′ C c hosen u.a.r. the pr obabilit y to s p ecify diﬀerent colourings for the vertice s in C H u is up p er b ounded b y P w ∈ C H u p w . Clearly the mapp ing f deﬁ n es a coupling for the “nonw eigh ted” join t ran d om selectio n of the elements the sets Ω ′ F and Ω ′ C , sin ce the tw o sets are equal sized. Based on f we deﬁne a coupling for the “we igh ted” join t r an d om sele ction of the elemen ts of the sets Ω ′ F and Ω ′ C . F rom Ω ′ i w e pr o duce the set Ω W i , for i ∈ { C, F } , as follo w s: F or eac h C ∈ Ω ′ i , ins ert into Ω W i , W C copies of C , i.e. the elemen ts { C 1 , . . . , C W C } . The w eigh ted random s el ection from Ω ′ i is equiv alent to consider that we h a v e chosen C ∈ Ω ′ i if a r an d om uniform selection from Ω W i ha v e chose n one of { C 1 , . . . , C W C } . Th us, the construction of a coupling of the w eigh ted joint selection from the s ets Ω ′ F and Ω ′ C can, equiv alen tly , b e redu ce d to creating a coupling that selects uniformly at random one element from eac h of the sets Ω W F and Ω W C . Th is is what are w e doing in wh at follo ws. First, w e create a mappin g f ′ : (Ω W F ∪ ω 2 ) → (Ω W C ∪ ω 1 ), where ω 2 ⊂ Ω W F and ω 1 ⊂ Ω W C and they will b e deﬁned soon after. T he mapping f ′ will b e created based on the mapp ing f . Then we d eﬁne the coupling whic h consists of choosing u.a.r. a member of Ω W F ∪ ω 2 and then applying the c hosen element to f ′ so as to get a mem b er of Ω W C ∪ ω 1 . In this coupling, th e marginal p r obabilit y for eac h mem b er in Ω W F to b e chosen, will b e the same for all members. This should also hold for th e memb ers of Ω W C . The claim will follo w b y b ounding, appropr ia tely , the quantit y P r [ A ]. W e will deﬁ ne the sets ω 1 , ω 2 as w e construct f ′ . The mapping f ′ is deﬁned as follo w s: F or eac h C ∈ Ω ′ F , with f ( C ) = Q and W C = W f ( C ) > 0, set f ′ ( C i ) = Q i for i = 1 , . . . , W C . F or eac h C ∈ Ω ′ F , with W C > W f ( C ) and f ( C ) = Q , set f ′ ( C i ) = Q i for i = 1 , . . . , W f ( C ) and for i = W f ( C ) + 1 , . . . , W C set f ′ ( C i ) a u.a.r. c h osen mem b er of Ω W C . Let ω 1 b e the set of all th e element s of Ω W C that were randomly selected, as describ ed ab o v e. F or eac h C ∈ Ω ′ F , with W C < W f ( C ) and f ( C ) = Q , w e set f ′ ( C i ) = Q i for i = 1 , . . . , W C , and for i = W C + 1 , . . . , W f ( C ) , for the cop y Q i c ho ose u.a.r. a mem b er of Ω ′ F to corresp ond to. Let ω 2 b e the set of all the element s of Ω ′ F that w ere r andomly selec ted, as describ ed ab o v e. If we choose uniformly at random from Ω W F ∪ ω 2 , eac h elemen t of Ω W F app ears equiprobably . Similarly , if we choose u.a.r. from Ω W C ∪ ω 1 , eac h element of Ω W C app ears equip robably . F ur- thermore, if w e c ho ose u.a.r. from Ω W F ∪ ω 2 , and apply f ′ to get a memb er from Ω W C ∪ ω 1 , all the memb ers of Ω ′ C ∪ ω 1 ha v e the s ame probabilit y to b e c hosen, since the mapping f ′ is one to one and on to ( | Ω W F ∪ ω 2 | = Ω ′ C ∪ ω 1 ). Thus, in the co upling where w e choose u.a.r. Ω W F ∪ ω 2 and apply the mapping f ′ and get a member of Ω ′ C , th e marginal p robabilit y for all the members of Ω W F (and Ω W C ) to b e c hosen is the same. What remains to b e shown is that, in the coup lin g ab o v e, the even t A o ccurs with probabilit y P r [ A ] whic h is upp er b ounded as in (3). Clearly , f or the colourings in the pairs ( C, f ( C )) ∈ Ω ′ F × Ω ′ C that deﬁne the same colour assignmen ts for the v ertices in C H u , w e h av e W C = W f ( C ) . F or C ∈ Ω ′ F in su ch a pair of colourings, it holds that the cop y C i , that C has in Ω W F , is corresp onded through f ′ to the cop y Q i , that Q = f ( C ) h as in Ω W C , i.e. Q i = f ( C i ), for i = 1 , . . . , W C . Note that the ev ent A do es not hold for these p airs, ( C i , f ′ ( C i )) for i = 1 , . . . , W C . F or eac h pair ( C, f ( C )) ∈ Ω ′ F × Ω ′ C that d eﬁne a diﬀeren t colour assignmen ts for th e v ertices 14 in C H u , it d oes not necessarily hold W C = W f ( C ) . C onsider, ﬁrst, the case where W C = W f ( C ) . Then, for C ∈ Ω ′ F in suc h a pair of colourings, it holds th a t eac h copy C i , that C has in Ω W F , is corresp ond ed through f ′ to the cop y Q i , that Q = f ( C ) has in Ω W C , for i = 1 , . . . , W C . The ev en t A do es hold for these pairs, ( C i , f ′ ( C i )) for i = 1 , . . . , W C . Finally , we consider the case where the pair ( C, f ( C )) ∈ Ω ′ F × Ω ′ C deﬁnes a diﬀerent col our assignmen ts for the v ertices in C H u and W C 6 = W f ( C ) . W.l.o.g . w e assume that W C > W f ( C ) . Then, for C ∈ Ω ′ F in suc h a pair of colourings, it holds that eac h cop y C i , that C has in Ω W F , is corresp onded through f ′ to a cop y of Q i , that Q = f ( C ) has in Ω W C , for i = 1 , . . . , W f ( C ) . The ev en t A do es hold for the pairs ( C i , f ′ ( C i )), i = 1 , . . . , W f ( C ) . The remaining copies C i , that C has in Ω W F , through f ′ are mapp ed to u.a.r. chosen memb er of Ω W C , for i = W f ( C ) + 1 , . . . , W C . Note that the ev ent A d oes not necessarily hold for these p air. How ev er, we assume that it do es, whic h , clea rly , is an ov erestimate for the probabilit y P r [ A ]. Let, Ω W A ⊂ Ω W F ∪ ω 2 b e suc h that Ω W A = { C ∈ Ω W F ∪ ω 2 | for ( C, f ′ ( C )) the eve nt A holds } and Ω A ⊂ Ω ′ F b e Ω A = { C ∈ Ω ′ F | for ( C, f ( C )) the ev en t A holds } . C learly , P r [ A ] = | Ω W A | | Ω W F ∪ ω 2 | . Let Ω ( > 0) i ⊂ Ω ′ i b e suc h that Ω ( > 0) i = { C ∈ Ω ′ i | W C > 0 } and q i = | Ω ( > 0) i | | Ω ′ i | , for i ∈ { C, F } . One can see th at | Ω W A | ≤ | Ω A | m ax C ∈ Ω ′ C ∪ Ω ′ F { W C } , and | Ω W F ∪ ω 2 | ≥ | Ω ( > 0) F | · min C ∈ Ω > 0 C ∪ Ω > 0 F { W C } . F rom the fact that | Ω > 0 F | = q F | Ω F | we ge t th at P r [ A ] ≤ max C ∈ Ω ′ C ∪ Ω ′ F { W C } q F · min C ∈ Ω > 0 C ∪ Ω > 0 F { W C } | Ω A | | Ω F | . Clearly , q F = q k , S , where q k , S is as deﬁned in the statemen t of the claim. The claim follo w s b y noting that | Ω A | | Ω F | ≤ P w ∈ C H u p w . ⋄ Claim C Consider the c oupling C ( T , S , l ) when it assigns c olourings to the vertex u . Assume that the two pr o c esses chose memb ers of Ω F and Ω C that sp e ci fy c olourings for the vertic es in C H u such that for the vertex u , ther e ar e two lists of availa ble c olours, the l 1 and the l 2 , c orr esp ondingly. Assuming that | l i | > 0 , for i = 1 , 2 , ther e is a c oupling that c an cho ose the same c olour for the vertex u with pr ob ability at le ast 1 − max {| l 1 \ l 2 | , | l 2 \ l 1 |} min {| l 1 | , | l 2 |} . Pro of: The coup ling that can choose the same colour for the v ertex u with probab ility indicated in the statemen t of the claim is the maximal c oupling (see [10]). More s peciically , we assum e, w.l.o.g., that | l 1 | ≥ | l 2 | . Let U b e a random v ariable un iformly distributed in (0 , 1). W e assume that if i − 1 | l 1 | < U < i | l 1 | w e c ho ose the color i ∈ l 1 , for = 1 , . . . , | l 1 | . Also, ∀ i ∈ l 1 ∩ l 2 assume that if i − 1 | l 1 | < U < i | l 1 | w e c ho ose i in l 2 . F or U ev er y w here else in (0 , 1) mak e an arbitrary arrangemen t so as ea c h elemen t of l 2 to b e chosen with probabilit y 1 /l 2 . By the assump tion that | l 1 | ≥ | l 2 | , to all m em b ers i ∈ l 1 ∩ l 2 w e hav e assigned in terv als whic h corresp ond to probability 1 / | l 1 | ≤ 1 / | l 2 | . Clearly , th e in terv al in (0 , 1) that corr esp ond s to c ho osing diﬀerent colourings from l 1 and l 2 is of length | l 1 \ l 2 | | l 1 | . The claim follo ws by the fact that | l 1 \ l 2 | | l 1 | ≤ max {| l 1 \ l 2 | , | l 2 \ l 1 |} min {| l 1 | , | l 2 |} 15 ⋄ Lemma C L et u b e a vertex of the tr e e T which is nonmixing and has k child r en. If for every w ∈ C H u the pr ob ability of disagr e ement, in C ( T , S , l ) , i s p w , then for the vertex u , the pr ob ability of disagr e ement p u , in C , is b ounde d as p u ≤ S 1 q k , S   X w ∈ C H u p w   (4) and q k , S is the pr ob ability of the event that after k trials, not al l elements of the set [ S ] have b e en chosen, when at e ach trial we cho ose uniformly at r andom a memb er of [ S ] . Pro of: Assume that in the coupling C ( T , S , l ), when the v ertex u is to b e coloured, the pro cesses ColourRo ot( T , S , C 1 ( V 1 )) and ColourRo ot( T , S ) at their second s te p choose from the set of colourings Ω C and Ω F , corresp ondingly . Let A b e the even t that, wh en th e ve rtex u is to b e coloured in C the members of Ω F and Ω C , that are c hosen at step (2) of the t wo p rocesses ColourRo ot, corresp ondingly , sp ecify diﬀerent colour assignmen ts f o r the v ertices in C H u . Then P r [disagreemen t on u ] = P r [disagreemen t on u |A ] P r [ A ] + P r [disagree ment on u | A ] P r [ A ] W e mentio n that, if th e ev en t A do es not hold ( A holds), then there is a coup lin g for step (3) of the C olo urRo ot that assigns the same colour to the v er tex u in C , i.e. P r [disagreemen t on u | A ] = 0. Th us, P r [disagreemen t on u ] = P r [disagreemen t on u |A ] P r [ A ] (5) T o sho w the lemma w e derive appropriate up p er b ound s for the probabilities in (5), P r [ A ] and P r [disagreemen t on u |A ]. W e w ork exactly in the same manner as in the pro of of Lemma B so as to get an u pp er b ound for the term Pr[ A ], i.e. as in Lemma B w e hav e P r [ A ] ≤ 1 q k , S max C ∈ Ω ′ F ∪ Ω ′ C { W C } min C ∈ Ω ′ F ∪ Ω ′ C { W C | W C > 0 } X w ∈ C H u p w . Note that at the coupling, of the step (2) of the ColourRo ot, for colouring u , no mem b er of either Ω ′ C or Ω ′ F is assigned w eight more than S and the minimum non zero weigh t is 1. F ur thermore, for a n onmixing ve rtex u , of suﬃcientl y h igh degree, there are colourings of its c hildren that us e ev ery colour in [ S ], these colourings are assigned weig ht zero, in in th is ca se we ha v e q k , S ≤ 1. The lemma follo ws by assuming that P r [d isag reement on u |A ] = 1 w h ic h is, clearly , an o v erestimate. ⋄ 3.2 The case of a t ree - H ow to pro v e Lemma E. Consider an in sta nce of G n,d/n , where d > 1, and for eac h verte x v consider the graph G v,d, ǫ l og n , where ǫ = 0 . 9 log( e 2 d/ 2) . By L emm a A it holds that w.h.p. G v,d, ǫ l og n is either a unicyclic graph or a tree. Here, w e condition that th e graph G v,d, ǫ l og n is a tree. Deﬁnition K The gr aph G v,d, ǫ l og n when we c ondition that it is a tr e e, deﬁnes a pr ob ability sp ac e over the tr e es which we c al l T d . 16 Note th at eac h nonleaf v ertex of an instance of T d has a num b er of c hildren wh o se distribution is dominated by B ( n, d/ n ), i.e. the bin omia l d istribution with parameters n and d/n . Clearly , Lemma E will follo w by sh o wing that if S and ǫ are as large as sp eciﬁed by this lemma, then w ith probabilit y , at least, 1 − 2 n − 1 . 25 , for the tree T , an instance of T d ro oted at the v ertex r , the disagreemen t p robabilit y p r in the coupling C ( T , S , ⌊ ǫ log n ⌋ ) is b ounded as p r ≤ n − 1 . 25 / 2 (see Corollary C). It is easy to see that if, T , an instance of T d , is of heigh t less than some in teger l , then the disagreemen t p robabilit y on th e r o ot of T in C ( T , S , l ), is zero. In C ( T , S , l ), where T is an instance of T d ro oted at r , the disagreement probabilit y p r dep ends only on the structure of the ins tance of T d , for a give n S . W e remin d th e reader that in C , w e assume that the b oundary conditions are set so as to maximize th e d isag reement probabilit y p r . Clearly , p r is a r andom variable . W e use the Lemma B and L emma C to derive an upp er b ound for the exp ect ation of p r . The exp ectation of p r dep ends on l , S and t , the maxim um n umber of c hildren of a mixing vertex. Let q ( t ) b e the probability for a random v ariable, distribu ted as in B ( n − 1 , d/ n ), for ﬁxed d , to b e less than t . Lemma D F or p ositive inte gers S , l , r e al d > 1 , in the c oupling C ( T , S , l ) , wher e T is an instanc e of T d , the exp e ctation of the disagr e ement pr ob ability p r , on the r o ot of T , is b ounde d as E [ p r ] ≤  d t · S ( S − t ) 2 q ( t ) + 2 d  S (1 − q ( t )) + S S − 1  exp  d ( S − 1)  − q ( t )  l (6) Pro of: W e r emind the reader that t stands for the maximum num b er of c hildr en of a mixing v ertex. Let q ( t ) b e the probabilit y f o r a random v ariable, d istr ibuted as in B ( n − 1 , d/n ), for ﬁxed d , to b e less than t . Let a ( i ) =          t · S ( S − t ) 2 if i ≤ t S q i, S otherwise where q i, S , is as deﬁned in the state ment of Lemma C. Consider the coup lin g C ( T , S , l ), wh ere T is an instance of T d ro oted at th e vertex r . Let E [ p r ] b e the exp ecta tion of the disagreemen t probabilit y on the r oot r . Conditioning on the n umb er of c hildren of r and the d isag reement pr obabilit y p w , ∀ w ∈ C H r in C ( T , S , l ), by Lemma B and Lemma C we ha ve E [ p r | p w , ∀ w ∈ C H r ] = a ( | C H r | ) X w ∈ C H r p w By deﬁnition, ∀ w ∈ C H r , p w is upp er b oun ded b y the disagreemen t probability on the v ertex w in the coupling C ( T w , S , l − 1) where T w is the subtree of T ro ote d at v ertex w . Call this disagreemen t p robabilit y p ∗ w . W e clear out that p w refers to the couplin g C ( T , S , l ) wh ile p ∗ w to C ( T w , S , l − 1). It is direct that E [ p r ] ≤ n X i =0 ia ( i ) P r [ | C H r | = i ] E [ p ∗ w ] for w ∈ C H r Note that the random v ariables p ∗ w for w ∈ C H r are identica lly distribu ted and indep endent of th e n umber of c h ildren of r . Also, noting that the function f ( i ) = i · a ( i ) is increasing f or 17 t << S and b y the fact that the distribution of the n umber of c hildren of r is dominated by the B ( n, d/n ), (b y p roposition 9.1.2. of [13]), it holds that E [ p r ] ≤ n X i =0 i · a ( i ) n i ! p i (1 − p ) n − i E [ p ∗ w ] for w ∈ C H r (7) where p = d/n . Let S 1 = P t i =0 i · a ( i )  n i  p i (1 − p ) n − i and S 2 = P n i = t +1 i · a ( i )  n i  p i (1 − p ) n − i . S 1 ≤ t ·S ( S − t ) 2 P t i =0 i  n i  p i (1 − p ) n − i = t ·S ( S − t ) 2 np P t − 1 i =0  n − 1 i  p i (1 − p ) n − 1 − i = t ·S ( S − t ) 2 q ( t ) d Before calculating S 2 , we eliminat e the probabilit y term q i, S from the a ( i ) for i > t . F or q i, S it holds that q i, S ≥ S  1 − 1 S  i  1 − q i, ( S − 1)  i.e. the p robabilit y of the ev ent “n ot choosing some el ement of [ S ] after i trial s” is greater th an , or equal to the pr obabilit y of the ev ent “not choosing exactly one elemen t of [ S ]”, since the second ev en t is a sp ecial case of the ﬁrst one. F urthermore, since q k , ( S − 1) ≤ q k , S w e get that q i, S ≥ S  1 − 1 S  i (1 − q i, S ) Let Ω = { 1 , . . . , n } and let t 0 = sup { t ∈ Ω | q t, S ≥ 1 / 2 } . Instead of using q i, S w e mak e the follo win g simpliﬁcation. F or i > t 0 w e b ound the quan tit y 1 /q i, S as 1 q i, S ≤ 2 S  1 − 1 S  i = 2 S  S S − 1  i . Also, for i ≤ t 0 , clearly , 1 /q i, S ≤ 2. S 2 ≤ 2 S t 0 X i = t +1 i n i ! p i (1 − p ) n − i + 2 n X i = t 0 +1 i n i !  S S − 1  i p i (1 − p ) n − i ≤ 2 S n X i = t +1 i n i ! p i (1 − p ) n − i + 2 n X i = t +1 i n i !  S S − 1  i p i (1 − p ) n − i ≤ 2 S np n − 1 X i = t n − 1 i ! p i (1 − p ) n − 1 − i + 2 np S S − 1 n − 1 X i = t n − 1 i !  S S − 1  i p i (1 − p ) n − 1 − i ≤ 2 S d  1 − P t − 1 i =0  n − 1 i  p i (1 − p ) n − 1 − i  + +2 S S − 1 d   1 − p + S S − 1 p  n − 1 − P t − 1 i =0  n − 1 i   S S − 1  i p i (1 − p ) n − 1 − i  ≤ 2 S d ( 1 − q ( t )) + 2 S S − 1 d  1 + 1 S − 1 p  n − 1 − t − 1 X i =0 n − 1 i ! p i (1 − p ) n − 1 − i ! ≤ 2 S d ( 1 − q ( t )) + 2 S S − 1 d  1 + 1 S − 1 p  n − 1 − q ( t ) ! ≤ 2 d  S (1 − q ( t )) + S S − 1 (exp { d/ ( S − 1) } − q ( t ))  18 Substituting the b ounds for S 1 and S 2 in (7) we get E [ p r ] ≤  d t S ( S − t ) 2 q ( t ) + 2 d  S (1 − q ( t )) + S S − 1 (exp { d/ ( S − 1) } − q ( t ))  E [ p ∗ w ] for w ∈ C H r . W e can substitute E [ p ∗ w ] in the same manner as E [ p r ]. Using induction and assuming that for the vertic es at distance l fr om the ro ot the exp ectation of the probabilit y of disagreemen t is 1, th e lemma follo ws. ⋄ Finally , L emm a E follo ws by s etting appropr iate quan tities for S and l , in (6) and th en applying the Mark ov inequalit y . Here it is crucial to remark that if d is suﬃcien tly large, then for t ≥ 7 d it holds q ( t ) ≥ 1 − d − 28 . Lemma E Consider a system P C S ( G v,d, ǫ log n , S ) , for d > 1 , ǫ = 0 . 9 4 log( e 2 d/ 2) and for G v,d, ǫ l og n we assume tha t it is a tr e e. If the c ar dinality of S is a suﬃciently lar ge c onstant, then with pr ob ability at le ast 1 − 2 n − 1 . 25 , for the ab ove system it holds that S D ( v , ⌊ ǫ log n ⌋ ) = n − 1 . 25 . F or suﬃciently lar ge d , we should have S ≥ d 14 . Pro of: By Lemma D we ha v e that, in the coupling C ( G v,d, ǫ l og n , S , C ( L )), the exp ectation of p v , is b ounded as E [ p v ] ≤  d t · S ( S − t ) 2 q ( t ) + 2 d  S (1 − q ( t )) + S S − 1 (exp { d/ ( S − 1) } − q ( t ))  l (8) where l is the min imum distance of v and th e b oundary set L , q ( t ) is equal to the p robabilit y for a random v ariable, distributed as in B ( n − 1 , d/n ), for ﬁxed d , to b e less than t , the maxim um n umb er of children of a mixing ve rtex. Set l = ǫ log n , with ǫ = 0 . 9 log( e 2 d/ 2) in (8). T o p ro ve the lemma it suﬃces to sho w that for S as d esc rib ed in the statemen t of the lemma and appropriately large t we hav e E [ p v ] ≤ n − 2 . 5 . Clearly , for E [ p v ] ≤ n − 2 . 5 b y using the Marko v Inequalit y (see [3]) we can get that P r [ p v ≥ 2 n − 1 . 25 ] ≤ E [ p v ] 2 n − 1 . 25 ≤ n − 1 . 25 / 2 By Deﬁnition F and T heorem D w e get that if E [ p v ] ≤ n − 2 . 5 , then with pr ob ab ility at least 1 − P r [ p v ≥ 2 n − 1 . 25 ] ≥ 1 − 2 n − 1 . 25 for the system P C S ( G v,d, ǫ l og n , S ) it holds that S D ( v , ⌊ ǫ log n ⌋ ) ≤ n − 1 . 25 , which pro ves the lemma. Th us, what remains to b e sh o wn is that there are ap p ropriate v alues for t and S such th a t, E [ p v ] ≤ n − 2 . 5 . First, we sh o w th at if d is a suﬃciently large constant, th en for S ≥ d 14 and t suc h that q ( t ) ≥ 1 − d − 28 w e get E [ p v ] ≤ n − 2 . 5 . Using Corollary B (from [6]) w e see that when t ≥ max { 7 d, 28 log d + 1 } it h olds q ( t ) ≥ 1 − d − 28 . Corollary B If a r andom variable Z is distribute d as in B ( n, q ) with λ = nq then P r [ Z ≥ x ] ≤ e − x x ≥ 7 λ. (9) F or a pr oof of the ab o v e corollary , see in [6] the pr oof of Corollary 2.4. Assuming that d is a suﬃcien tly large constant, w e su b stitute the v alues of S with d 14 and t = 7 d , in (8) and w e get E [ p v ] ≤ 7 d 16 ( d 14 − 7 d ) 2 + 2 d d 14 d − 28 + d 14 d 14 − 1 1 + d d 14 − 1 + e ξ 2! d 2 ( d 14 − 1) 2 − 1 + d − 28 !!! ǫ log n 19 where 0 < ξ < d/ ( d 14 − 1). In the ab o ve inequ a lit y we used the fact that 1 − d − 28 ≤ q ( t ) ≤ 1, and substituted e d/ ( S − 1) b y using the MacLaurin series of the function f ( x ) = e x , for r ea l x . Th us, we ge t E [ p v ] ≤ 7 d − 12 (1 − 7 d − 13 ) 2 + 2 d d − 14 + 1 1 − d − 14 d − 13 1 − d − 14 + e 2 d − 26 (1 − d − 14 ) 2 + d − 28 !!! ǫ log n ≤ d − 12 7 (1 − 7 d − 13 ) 2 + 2 d − 1 + 2 (1 − d − 14 ) 2 + ed − 13 (1 − d − 14 ) 3 + 2 d − 15 1 − d − 14 !! ǫ log n T aking d at least 20, w e get that E [ p v ] ≤ n ǫ log(9 . 2 d − 12 ) Replacing ǫ , we see that it su ﬃces to hold 0 . 9 log (9 . 2 d − 12 ) ≤ − 10 log ( e 2 d/ 2), or 9 . 2 d − 12 ≤ ( e 2 d/ 2) − 11 . 11 whic h clearly holds for suﬃcien tly large constant d . F or r ela tiv ely smaller d , one can easily see that setting S = d x , f or an app ropriately large constan t exp onen t x and arran ging the quantit y t s o as q ( t ) ≥ 1 − d − 2 x similarly to the p revious case, can get E [ p v ] ≤ d − x +2 t/d (1 − d − x t ) 2 + 2 d − 1 + 2 d − x − 1 1 − d − x + 2 (1 − d − x ) 2 + e d − x +1 (1 − d − x ) 3 !! ǫ log n W e tak e x suﬃcien tly large so as to ha v e 1 − d − x ≥ 1 − 10 − 3 and xd − x ≤ 10 − 3 . If t = 7 d , then, with the ab o ve assumptions, we can easily derive that E [ p v ] ≤ ( d − x +2 16) ǫ log n . F or this case, if E [ p v ] ≤ n − 2 . 5 , then w e s h ould ha v e 16 d − x +2 ≤ ( e 2 d/ 2) 11 . 11 , wh ic h clea rly holds for suﬃcien tly large x . If 2 x log d + 1 > 7 d , then by Corollary B we should h a v e t = 2 x log d + 1. With the assumptions w e hav e made f or x we get that E [ p v ] ≤ ( d − x +2 (2 . 1 x log d d +9)) ǫ log n . If E [ p v ] ≤ n − 2 . 5 , then it sh ould hold ( d − x +2 (2 . 1 x log d d + 9)) ≤ ( e 2 d/ 2) 11 . 11 , which clearly h olds for s u ﬃcien tly large x . The lemma follo ws ⋄ 3.3 The case of a unicyclic graph - Ho w to prov e Lemma H. Consider an in stance of G n,d/n , and the set of its subgraph s G v,d, ǫ l og n , as in 3.2. By Lemm a A, we ha ve that w.h.p. G v,d, ǫ l og n is either a un icyc lic graph or a tree. Here, w e condition that that G v,d, ǫ l og n is a unicyclic graph . First, w e sho w ho w can we extend the tec hn iques for pro vin g Lemma E, i.e. pro vin g spatial mixing pr op erties of system with u nderlying graph a tree, to showing Lemma H, w hic h refers to systems with a un icyc lic underlying graph. Consider the depth ﬁrs t search in G v,d, ǫ l og n , that starts from the v ertex v and let u b e the ﬁrst v ertex of the unique cycle that is reac hed by the searc h. Clearly , there are t wo p ossible c hoices for this searc h to explore the v ertices of the cycle that u b elongs to. If w 1 and w 2 are the ve rtices on the cycle that are also adjacen t to u , then let T 1 and T 2 b e th e t wo depth-ﬁrs t searc h trees of G v,d, ǫ l og n , ro ot ed at v , with the ﬁrst tree ha ving u adjacen t only to w 1 (not adjacen t to w 2 ) and the second ha vin g u adjacen t only to w 2 . 20 Deﬁnition L With the ab ove notation, the tr e e T r,d,ǫ log n is isomorphic to the tr e e that c omes up fr om the u nion of T 1 and T 2 w 2 plus an e dge c onne cting the vertic es u in T 1 with w 2 in T 2 w 2 . The r o ot r of T r,d,ǫ log n c orr esp onds to the v ertex v i n T 1 . Note that the n um b er of children of a nonleaf ve rtex of T r,d,ǫ log n has distribution whic h is dominated b y th e B ( n, d/n ) with the condition that it is at least 2. Eac h of the trees T 1 and T 2 w 2 , in the deﬁnition of T r,d,ǫ log n , are isomorphic to some subgraph of G v,d, ǫ l og n , i.e. there is a corresp ondence b et ween the vertic es in T 1 and T 2 w 2 with the vertice s in G v,d, ǫ l og n . Based on this co rresp ond ence , w e can deﬁne a function h : V T → V G where, V T is the set of ve rtices of T r,d,ǫ log n and V G the set of ve rtices in G v,d, ǫ l og n . Let L b e the set that con tains all the v ertices in G v,d, ǫ log n that are at graph distance, at least, ⌊ ǫ log n ⌋ from v . Consider the S -colouring C 1 ( L ) whic h is such that, the set the total v ariation distance of the Gibbs measures µ ( X v | C 1 ( L )) and µ ( X v ), as these are sp eciﬁed b y the system P C S ( G v,d, ǫ log n , S ), is maximized. F or the tree T r,d,ǫ log n deriv ed by G v,d, ǫ l og n , the in tegers S and l , let C ′ ( T r,d,ǫ log n , S , l ), b e the coupling of the sto c hastic pro cesses ColourRo ot( T r,d,ǫ log n , S , C T 1 ( A )) and ColourRo ot( T r,d,ǫ log n , S ). The set A is su c h that ∀ ˆ v ∈ A ∃ ˆ u ∈ ( V 1 ∪ L ) such that h ( ˆ v ) = ˆ u and C T 1 ( ˆ v ) = C 1 ( ˆ u ). Th e diﬀerence of C ′ from C regarding the coupling of the pro cesses ColourRo ot is that, if for a non mixing v ertex u in T , whic h has i c hildr en , eac h v ertex w ∈ C H u has disagreemen t prob ab ility p w , then the d isag reement probabilit y p u , is b ounded as p u ≤ S q i,S, 2 X w ∈ C H u p w . (10) where q i,S, 2 is the pr obabilit y of the ev en t that after k trials, not all element s of the set [ S ] ha ve b een c hosen, when at eac h trial we c h oose u.a.r. a memb er of [ S ] and conditioning that the ﬁrst t w o trials c h ose diﬀeren t elemen ts of [ S ]. Comparing the b ound in (10) with that was giv en in (4) in the statemen t of Lemma C, it is easy to see that q i,S, 2 ≤ q i,S . Th is implies that th e coupling C ′ can exist as, for all the vertice s of T , it giv es the same, or w orse b ound s for the probabilities of disagreement than C , on th e same input. Lemma F Consider the gr aph G v,d, ǫ l og n and the c orr esp onding tr e e T r,d,ǫ log n , for d > 1 and ǫ = 0 . 9 4 log( e 2 d/ 2) . If p r is the b ound for the disagr e ement pr ob ability that we derive fr om L emma B and (10) for the c oupling C ′ ( T r,d,ǫ log n , S , ⌊ ǫ log n ⌋ ) , then it holds that S D ( v , ⌊ ǫ log n ⌋ ) ≤ 2 p r in the system P C S ( G v,d, ǫ l og n , S ) , for any p ositive inte ger S . Pro of: Let u b e the v ertex in G v,d, ǫ l og n whic h b elongs to the uniqu e cycle of G v,d, ǫ l og n and among al l the vertic es in th e cycle it has the smallest distance from v . Let G u b e the c onne cte d subgraph of G v,d, ǫ log n that con tains the v ertex u and the v ertices whose distance from v is greater than that of the v ertex u fr o m v . It is easy to see that that G u is a unicyclic graph and G v,d, ǫ l og n \ G u is a tree. Let L b e the set that con tains all the v ertices in G v,d, ǫ log n that are at graph distance, at least, ⌊ ǫ log n ⌋ from v . Consider the S -colouring C 1 ( L ) whic h is such that the total v ariation distance of the Gib b s measures µ ( X v | C 1 ( L )) and µ ( X v ), as these are sp eciﬁed by the system P C S ( G v,d, ǫ l og n , S ), to b e maximized. Assume that there is a coupling suc h that c ho osing uniform ly at random a mem b er from eac h of the sets of S -colourings Ω( G u , S , C 1 ( L )) an d Ω( G u , S ), co rresp ondin gly , the probabilit y for the t w o memb ers to sp ecify a d iﬀe rent colour assignment for the v ertex u is Q . Clearly , using Lemma B and (1 0) we are able to derive an up per b ound for the d isag reement prob ab ility 21 p v , of the coupling of the pro cesses ColourRo ot( T , S , C 1 ( T ∩ L )) and ColourRo ot( T , S ), wh ere T = G v,d, ǫ l og n \ ( G u \{ u } ) and with the assum p tio n that the disagreemen t probabilit y of the v ertex u is set apriori to Q . Note that th e graph G v,d, ǫ l og n \ ( G u \{ u } ) is a tree, i.e. it in cludes all the ve rtices in G v,d, ǫ l og n apart from the ve rtice in G u \{ u } . This b ound of p v is also an upp er b ound for the total v ariatio n d istance of the Gibbs m ea sur es, µ ( X v | C 1 ( L )) an d µ ( X v ), as these are sp eciﬁed b y the system P C S ( G v,d, ǫ l og n , S ). The lemma will follo w by sho wing that for an appr op r iat ely constructed tree T u , with resp ect to G u , and appropriate b ound ary condition C ′ 1 , if in coupling of the pro cesses ColourRo ot( T u , S , C ′ 1 ) and ColourRo o t( T u , S ) has disagreemen t probab ility P at the v ertex u , the ro ot of T u , then it h ol ds Q ≤ P , where the pr obabilit y Q is as previously deﬁn ed. W.l.o.g. assume that the ve rtex v b elongs to the uniqu e cycle of G v,d, ǫ l og n , i.e. the G u and G v,d, ǫ l og n are ident ical. Eac h of the trees T 1 and T 2 w 2 , in the deﬁnition of T r,d,ǫ log n (Deﬁnition L), are isomorph ic to some su bgraph of G v,d, ǫ l og n , i.e. there is a corresp ond en ce b et wee n the vertic es in T 1 and T 2 w 2 with the ve rtices in G v,d, ǫ l og n . Based on this corresp ondence, we can ha v e a function h : V T → V G where, V T is the set of vertic es of T r,d,ǫ log n and V G the set of v ertices in G v,d, ǫ l og n . Let L ′ b e the set of vertices in T r,d,ǫ log n suc h that L ′ = { u ∈ V T | h ( u ) ∈ L } . It is direct that the verte x set L ′ is at distance, at least, ⌊ ǫ log n ⌋ from r , in T r,d,ǫ log n . If N v is the vertex set that con tains all th e adjacent v ertices of v in G v,d, ǫ l og n , then for w 0 ∈ N v , let G w 0 b e th e c onne cte d comp onen t of G v,d, ǫ l og n \{ v } that con tains w 0 . It is straigh tforward that the comp onen t G w 0 is a tree which is isomorphic to the subtree T w of T r,d,ǫ log n , where h ( w ) = w 0 . Let C i T ( L ′ ) b e a colouring suc h that, for eac h v ertex w ∈ L ′ it sp eciﬁes the same colour assignmen t as C i ( L ) sp eciﬁes for w 0 ∈ L , where h ( w ) = w 0 , for i = 1 , 2. The fact, that, for an y w 0 ∈ N v and w ∈ C H r suc h that h ( w ) = w 0 , the graphs G w 0 and T w are isomorphic implies that there is a corresp ond en ce b et wee n the elemen ts of the sets Ω( G w 0 , S , C 1 ( L )) and Ω( T w , S , C 1 T ( L ′ )) such that for an y t wo corresp onding colourings C 1 and C 2 it holds that ∀ u 1 ∈ T w C 1 ( h ( u 1 )) = C 2 ( u 1 ). Clearly , there is a similar co rresp onden ce b etw een the mem b ers of the sets of S -colourings Ω ( G w 0 , S ) and Ω( T w , S ), for w 0 ∈ N v , w ∈ C H r and h ( w ) = w 0 . Using th e ab o v e corresp ondences b et we en the pairs of sets Ω ( G w 0 , S , C 1 ( L )), Ω( T w , S , C 1 T ( L ′ )) and Ω( G w 0 , S ), Ω( T w , S , C 2 T ( L ′ )), with the assum ption that ∀ w ∈ C H r , in th e coupling C ′ ( T r,d,ǫ log n , S , ⌊ ǫ log n ⌋ ) the disagreemen t probabilit y for the verte x w is u pp er b ounded by p w , w e get the follo wing: There is a coup lin g such that choosing u .a .r. from the sets of S -colourings Ω( G w 0 , S , C 1 ( L )) and Ω ( G w 0 , S ) suc h that the probabilit y for the tw o c hosen el ements to assign diﬀeren t colour to the v ertex w 0 is upp er b ounded b y p w , where h ( w ) = w 0 . F urth er m ore, there is a coupling that chooses u.a.r. from th e s ets Ω( G v,d, ǫ l og n \{ v } , S , C 1 ( L )) and Ω( G v,d, ǫ l og n \{ v } ) and there is at least one v ertex N v that the tw o c h oi ces sp ecify a d iﬀeren t colour assignment with probabilit y upp er b oun d ed as P w ∈ C H r p w . Th e b ound for this p robabilit y is der ived by using the union b ound . Remark. Note that if w 1 and w 2 are the v ertices in N v that al so b elong to the cycle, then the ev en ts e i “ ther e is disagr e ement on the ve rtex w i ”, for i = 1 , 2, are correlate d. Ho w ev er, the union b ound we used f or b oun d ing the probability for at least one vertex in N v to b e d iﬀe rently coloured, in the couplin g, still holds ev en for correlated ev ents. Claim D If, the c oupling C ′ ( T r,d,ǫ log n , S , ⌊ ǫ log n ⌋ ) , has pr ob ability disagr e ement on the ve rtex r u pp er b ounde d b y p r , then we c an have a c oupling that cho oses u.a.r. a memb er fr om e ach of the two sets Ω( G v,d, ǫ l og n , S , C 1 ( L )) and Ω( G v,d, ǫ l og n , S ) such that, the pr ob ability for v to have diﬀer ent c olour assignments to b e upp er b ounde d by p r . With the ab o v e claim and w h at follo ws, we get the pro of of the lemma. Let ˜ C ( L ) and ˆ C ( L ) b e the t w o colourings whic h maximize the total v ariation distance of the measures µ ( X v | C ( ˜ C ( L )) 22 and µ ( X v | ˆ C ( L )), as these are sp eciﬁed by the system P C S ( G v,d, ǫ log n , S ). S D ( v , ǫ log n ) = d T V  µ ( X v | C ( ˜ C ( L )) , µ ( X v | ˆ C ( L ))  ≤ d T V  µ ( X v | C ( ˜ C ( L )) , µ ( X v )  + d T V  µ ( X v ) , µ ( X v | ˆ C ( L ))  ≤ 2 p r where p r is the b ound of the disagreemen t probabilit y on the ve rtex r , at the coupling C ′ ( T r,d,ǫ log n , S , ⌊ ǫ log n ⌋ ). ⋄ W e now pro ve the claim that app ears in the pro of of Lemma F. Claim D If, the c oupling C ′ ( T r,d,ǫ log n , S , ⌊ ǫ log n ⌋ ) , has pr ob ability disagr e ement on the ve rtex r u pp er b ounde d b y p r , then we c an have a c oupling that cho oses u.a.r. a memb er fr om e ach of the two sets Ω( G v,d, ǫ l og n , S , C 1 ( L )) Ω( G v,d, ǫ l og n , S ) such that, the pr ob ability for v to have diﬀer ent c olour assignments to b e upp er b ounde d by p r . Pro of: Assume that in the coupling C ′ ( T r,d,ǫ log n , S , ⌊ ǫ log n ⌋ ) eac h v ertex of T r,d,ǫ log n , wh ose n umb er of c h ildren is at m ost t ′ , for some positive integ er t ′ , is considered mixing. The v er tices r and v ha ve the same degree in G v,d, ǫ l og n and T r,d,ǫ log n , corresp ondingly . Assuming that ∀ w ∈ C H r , in the coup lin g C ′ ( T r,d,ǫ log n , S , ⌊ ǫ log n ⌋ ) the disagreemen t proba- bilit y for the vertex w is u pp er b ounded by p w , w e can see th at there is a coupling that c h o oses u.a.r. a mem b er fr om eac h of the sets Ω( G v,d, ǫ log n \{ v } , S , C 1 ( L )) an d Ω( G v,d, ǫ l og n \{ v } , S ) and there is at least one verte x N v that the t wo c hoices sp ecify d iﬀe rent colour assignments with probabilit y upp er b ounded as P w ∈ C H r p w . Thus, th e claim follo ws if, based on the ab o v e as- sumption, we give a coupling that selects u.a.r. a mem b er f rom Ω( G v,d, ǫ l og n , S , C 1 ( L )) and Ω( G v,d, ǫ l og n , S ) s uc h that, the probabilit y for v to ha v e diﬀerent colour assignmen ts to b e upp er b ounded b y the probability of disagreemen t p r in the coupling C ′ ( T r,d,ǫ log n , S , ǫ log n ). If the degree of the ve rtex v is at most t ′ , then we use Lemma B to get an u pp er b ound for the probability for a coupling that c h ooses u.a.r. from the sets Ω( G v,d, ǫ l og n , S , C 1 ( L )) Ω( G v,d, ǫ l og n , S ) to c ho ose t w o members that sp ecify diﬀerent colour assignments for the v er- tex v . Clearly , we get the same b ound for the probabilit y of disagreemen t p r in the coupling C ′ ( T r,d,ǫ log n , S , ⌊ ǫ log n ⌋ ). W e note that despite the fact that the colour assignmen ts of tw o adjacen t v ertices of v are correlated, we can still app ly Lemma B. This is b ecause in Lemma B it is assumed that, if there is a d isag reement in the vertic es in C H v (to b e exact with the con text of th e pro of w e ha v e to wr ite N v ), then all the ve rtices ha ve diﬀeren t colour assignment s. Th is leads clearly to an o ve restimate for b ound in g the d isag reement probabilit y for p v ev en for the case w here the t w o colo urin gs are correlated. If th e degree of the v ertex v is i , greater than t ′ , then w e us e Lemma C, with a little mo diﬁcation, to ge t an upp er b ound for the pr obabilit y for a coupling that c h ooses u .a .r. from the sets Ω( G v,d, ǫ l og n , S , C 1 ( L )) and Ω( G v,d, ǫ log n , S ) to c ho ose t wo m em b ers that sp ecify d iﬀeren t colour assignmen ts f or the v ertex v . One can s ee that the term 1 /q i, S in (4) of the statemen t of Lemma C is not exact for our case. More sp eciﬁcally , in the last p arag raph of the pro of of Claim B, for our case the quantit y q F is not equal to q i, S due to the fact that t wo v ertices do not c ho ose indep endently their colour assignm ent. Ho w ev er, it is direct that q F is lo we r b ounded b y th e probabilit y of the ev en t that after i tr ials, not all the elemen ts of [ S ] hav e b een c hosen, when at eac h trial w e c ho ose u.a.r. a mem b er of [ S ] and conditioning that the ﬁrs t t wo trials c ho ose diﬀerent elements of [ S ]. Wit h this mo diﬁcation w e can see that the coupling of c h oosing u.a.r. fr om Ω( G v,d, ǫ l og n , S , C 1 ( L )) and Ω( G v,d, ǫ l og n , S ) can b e done s u c h that the v ertex v to 23 ha v e diﬀeren t colour assignmen ts w ith probabilit y p v ≤ S q i,S, 2 X w ∈ C H r p w . where q i,S, 2 is th e probabilit y of the ev en t that after k trials, not all elemen ts of [ S ] ha v e b een c hosen, when at eac h trial w e c ho ose u.a.r. a member of [ S ] and conditioning that the ﬁrst t wo trials c hose diﬀerent elemen ts of [ S ]. The claim follo w s. ⋄ T o w ards proving Lemma H, w e use Lemma F w hic h allo ws us to consider the tree T r,d,ǫ log n deriv ed by unicyclic graph G v,d, ǫ l og n , instea d of G v,d, ǫ l og n . In turn , w e can use the same tec h n iques as in s ection 3.2 for sho w ing the desired s patia l mixing prop erties for systems w ith underlying graph the tree T r,d,ǫ log n . Note that no w the coupling is C ′ . Let q ( t ) b e equal to the probabilit y for a random v ariable, distributed as in B ( n − 1 , d/n ), for ﬁxed d , to b e less than t . Lemma G F or p ositive inte gers S , l , r e al d > 1 , i n the c oupling C ′ ( T , S , l ) , wher e T is an instanc e of T r,d,ǫ log n , the exp e ctation of the disagr e ement pr ob ability p r , on the r o ot of T , is b ounde d as E [ p r ] ≤  1 1 − ( d + 1) e − d  d t · S ( S − t ) 2 q ( t ) + 2 d  S (1 − q ( t )) + exp  d S − 2  − q ( t )  l Pro of: W e remind the reader that t stands for the maxim um n umb er of a c hildr en of a mixing v ertex. Let q ( t ) b e the probabilit y f o r a random v ariable, d istr ibuted as in B ( n − 1 , d/n ), for ﬁxed d , to b e less than t . One can see than in the T r,d,ǫ log n , the n umber of children of a non leaf vertex h as distribu tio n whic h is d ominate d b y the B ( n, d/n ) w ith the condition that th ere are at least tw o c hildren. Let Z b e a random v ariable distribu ted as in B ( n, d/n ), clearly P r [ Z ≥ 2] = 1 −  1 − d n  n − n d n  1 − d n  n − 1 ≥ 1 − ( d + 1) e − d Let a ( i ) =          t · S ( S − t ) 2 if i ≤ t S q i, S , 2 otherwise where q i, S , 2 , is as deﬁ n ed in (10). Consider the coup ling C ′ ( T , S , l ), where T is an in stance of T r,d,ǫ log n ro oted at the v ertex r . Let E [ p r ] b e the exp ectat ion of the disagreemen t probability on the ro ot r . Conditioning on th e num b er of c hildren of r and the disagreemen t probab ility p w , ∀ w ∈ C H r in C ′ ( T , S , l ), b y L emm a B and Lemma C w e hav e E [ p r | p w , ∀ w ∈ C H r ] = a ( | C H r | ) X w ∈ C H r p w By deﬁnition, ∀ w ∈ C H r , p w is upp er b oun ded b y the disagreemen t probability on the v ertex w in the coupling C ′ ( T w , S , l − 1) w here T w is the su btree of T rooted at v ertex w . Call this disagreemen t probabilit y p ∗ w . W e clear out th at p w refers to the coupling C ′ ( T , S , l ) w hile p ∗ w to C ′ ( T w , S , l − 1). It is direct that E [ p r ] ≤ n X i =0 ia ( i ) P r [ | C H r | = i ] E [ p ∗ w ] for w ∈ C H r (11) 24 Also, noting that the function f ( i ) = i · a ( i ) is increasing for t << S and b y the fact th a t the distribution of the num b er of children of r is dominated b y the B ( n, d/n ), (by p roposition 9.1.2. of [13]), it holds th at E [ p r ] ≤ 1 1 − ( d + 1) e − d n X i =0 a ( i ) n i ! p i (1 − p ) n − i E [ p ∗ w ] for w ∈ C H r where p = d/n . Let S 1 = P t i =0 i · a ( i )  n i  p i (1 − p ) n − i and S 2 = P n i = t +1 i · a ( i )  n i  p i (1 − p ) n − i . Using the deriv ation of L emm a D w e ha ve that S 1 = t · S ( S − t ) 2 q ( t ) d Before calculating S 2 , we eliminate the p robabilit y term q i, S , 2 from the a ( i ) for i > t . Note that q i, S , 2 > q i − 1 , S − 1 , i.e. q i − 1 , S − 1 is th e p robabilit y for not choosing all the elemen ts of a set of cardinalit y S − 1 after i − 1 trials when at eac h trial w e choose u.a.r. a mem b er of the set. F or q i, S , 2 it holds that q i, S , 2 ≥ q i − 1 , S − 1 ≥ ( S − 1)  1 − 1 S − 1  i − 1  1 − q i − 1 , ( S − 2)  i.e. the p robabilit y of the ev en t “not choosing some of the S − 1 elemen ts after i − 1 trials” is greater than, or equal to the probabilit y of the ev ent “not c ho osing exactly one of S − 1 elements after i − 1 trials”, sin ce th e second ev en t is a sp ecial case of the ﬁrst one. F urthermore, since q i − 1 , ( S − 2) ≤ q i − 1 , S − 1 w e get that q i − 1 , S − 1 ≥ ( S − 1)  1 − 1 S − 1  i − 1 (1 − q i − 1 , S − 1 ) (12) Let Ω = { 1 , . . . , n } and let t 0 = su p { t ∈ Ω | q t − 1 , S − 1 ≥ 1 / 2 } . Instead of using q i − 1 , S − 1 w e mak e the follo wing simp liﬁca tion. F or i > t 0 w e b ound the quantit y 1 /q i − 1 , S − 1 as 1 q i − 1 , S − 1 ≤ 1 ( S − 1)  1 − 1 S − 1  i − 1 = 2 S − 1  S − 1 S − 2  i − 1 Also, for i ≤ t 0 , clearly , 1 /q i, S ≤ 2. With deriv ations similar to those in the pro of of lemma D for S 2 w e get that S 2 ≤ 2 d  S (1 − q ( t )) + S S − 1 (exp { d/ ( S − 2) } − q ( t ))  Substituting the b ounds for S 1 and S 2 in (11) we get E [ p r ] ≤ 1 1 − ( d + 1) e − d  d t · S ( S − t ) 2 q ( t ) + 2 d  S (1 − q ( t )) + S S − 1 (exp { d/ ( S − 2) } − q ( t ))  E [ p ∗ w ] for w ∈ C H r . W e can substitute E [ p ∗ w ] in the same manner as E [ p r ]. Using induction and assuming that for the vertic es at distance l fr om the ro ot the exp ectation of the probabilit y of disagreemen t is 1, th e lemma follo ws. ⋄ Lemma H, follo w s by combining the lemmas F and G. 25 Lemma H Consider a system P C S ( G v,d, ǫ log n , S ) , for d > 1 , ǫ = 0 . 9 4 log( e 2 d/ 2) and f or G v,d, ǫ l og n we assume that it is a unicyclic gr aph. If the c ar dinality of S is a suﬃcie ntly lar ge c onstant, then with pr ob ability at le ast 1 − 2 n − 1 . 25 , for the ab ove system it hold s that S D ( v, ⌊ ǫ log n ⌋ ) = n − 1 . 25 . F or suﬃciently lar ge d , we should have S ≥ d 14 . Pro of: T o pr o v e the lemma w e ﬁr s t s ho w that using th e coup lin g C ′ ( T r,d,ǫ log n , ⌊ ǫ log n ⌋ ) for the system P C S ( T r,d,ǫ log n , S ), it holds that S D ( r , ⌊ ǫ log n ⌋ ) = n − 1 . 25 with probabilit y at least 1 − 2 n − 1 . 25 , when S is a suﬃcien tly large constant and for suﬃcien tly large d , w e should h a v e S ≥ d 14 . Then, the lemma will follo w by using Lemma F. By Lemma G w e hav e that, for the coupling C ′ o v er the trees T r,d,ǫ log n when S colours are used and the b oundary set L is at distance, at least, l from the ro ot r , the exp ecta tion of p r is b ounded as E [ p r ] ≤  1 1 − ( d + 1) e − d  d t · S ( S − t ) 2 q ( t ) + 2 d  S (1 − q ( t )) + S S − 1 (exp { d/ ( S − 2) } − q ( t ))  l (13) where l is the d ista nce of v an d the b ound ary L , whic h in our case is l = ⌊ ǫ log n ⌋ , with ǫ = 0 . 9 log( e 2 d/ 2) . Also, q ( t ) is equal to the probabilit y for a random v ariable, distributed as in B ( n − 1 , d/n ), for ﬁxed d , to b e less than t , the maxim um num b er of children of a mixing v ertex. Set l = ǫ log n , with ǫ = 0 . 9 log( e 2 d/ 2) in (8). T o p ro ve the lemma it suﬃces to sho w that for S as d esc rib ed in the statemen t of the lemma and appropr iat ely large t we hav e E [ p r ] ≤ n − 2 . 5 . Clearly , for E [ p r ] ≤ n − 2 . 5 b y using the Marko v Inequalit y (see [3]) we can get that P r [ p r ≥ 2 n − 1 . 25 ] ≤ E [ p r ] 2 n − 1 . 25 = n − 1 . 25 / 2 By Deﬁnition F and Th eo rem D w e get that if E [ p r ] ≤ n − 2 . 5 , then with p robabilit y 1 − P r [ p r ≥ 2 n − 1 . 25 ] ≥ 1 − 2 n − 1 . 25 the s y s te m P C S ( T r,d,ǫ log n , S ) it holds S D ( r , ⌊ ǫ log n ⌋ ) ≤ n − 1 . 25 , which pro v es th e lemma. First we sh o w that for s uﬃcien tly large d , for S ≥ d 14 and t such that q ( t ) ≥ 1 − d − 28 w e get E [ p r ] ≤ n − 2 . 5 . By Corollary B (in the proof of lemma E) w e ha v e that if a random v ariable Z is d istributed as in B ( n, q ) with λ = nq then P r [ Z ≥ x ] ≤ e − x x ≥ 7 λ. (14) Th us, for t = max { 7 d, 28 log d + 1 } it h olds q ( t ) ≥ 1 − d − 28 . Assuming that d is a suﬃcien tly large constan t, w e subs tit ute S and t in (13) and we get E [ p r ] ≤ 1 1 − ( d + 1) e − d 7 d 16 ( d 14 − 7 d ) 2 + + 2 d 1 − d − 14 d − 14 + 1 + d d 14 − 1 + e ξ 2! d 2 ( d 14 − 1) 2 − 1 + d − 28 !!! ǫ log n where 0 < ξ < d/ ( d 14 − 1). In the ab o ve inequ a lit y we used the fact that 1 − d − 28 ≤ q ( t ) ≤ 1, and we substituted e d/ ( S − 1) b y u sing the MacLaurin series of th e f unction f ( x ) = e x , f or real x . Th us, w e get E [ p r ] ≤ d − 12 1 − ( d + 1) e − d 7 (1 − 7 d − 13 ) 2 + 2 d − 1 1 − d − 14 + 2 (1 − d − 14 ) 2 + ed − 13 (1 − d − 14 ) 3 +2 d − 15 1 − d − 14 !! ǫ log n 26 T aking d at least 20, w e get that E [ p r ] ≤ n ǫ log(9 . 2 d − 12 ) Replacing ǫ , we see that it su ﬃces to hold 0 . 9 log (9 . 2 d − 12 ) ≤ − 10 log ( e 2 d/ 2), or 9 . 2 d − 12 ≤ ( e 2 d/ 2) − 11 . 11 whic h clearly holds for suﬃcien tly large constant d . F or r ela tiv ely smaller d , one can easily see that setting S = d x , f or an app ropriately large constan t exp onen t x and arran ging the quantit y t s o as q ( t ) ≥ 1 − d − 2 x similarly to the p revious case, can get E [ p r ] ≤ d − x +2 1 − ( d + 1) e − d t/d (1 − d − x t ) 2 + 2 d − 1 1 − d − x + 2 d − x − 1 1 − d − x + 2 (1 − d − x ) 2 + e d − x +2 (1 − d − x ) 3 !! ǫ log n W e tak e x suﬃcien tly large so as to ha v e 1 − d − x ≥ 1 − 10 − 3 and xd − x ≤ 10 − 3 . If t = 7 d , then, with the ab o ve assumptions, ab out x , we can easily derive that E [ p r ] ≤ ( d − x +2 44) ǫ log n . F or this case to ha ve E [ p r ] ≤ n − 2 . 5 w e should ha ve 44 d − x +2 ≤ ( e 2 d/ 2) 11 . 11 , whic h clearly holds for suﬃcien tly large x . If 2 x log d + 1 > 7 d , then by Corollary B we should h a v e t = 2 x log d + 1. With the assumptions we ha v e made f or x w e get that E [ p v ] ≤ ( d − x +2 (8 x log d d + 16)) ǫ log n . Th us, if E [ p r ] ≤ n − 2 . 5 , then w e s h ould ha v e ( d − x +2 (8 x log d d + 16) ) ≤ ( e 2 d/ 2) 11 . 11 , whic h clea rly holds for suﬃcien tly large x . The lemma follo ws by using Lemma F. ⋄ 4 Prop erties of the algorithm W e close this wo rk by restating and pro v in g the t wo theorems that deal with the issues of accuracy and eﬃciency of the sampling algorithm. Theorem B If S i s a suﬃciently lar ge inte ger c onstant, then, with pr ob ability 1 − O ( n − 0 . 1 ) , the sampling algorithm is suc c essfu l and r eturns a S -c olouring of the input gr aph G , whose distribution is within total variation dista nc e n − 0 . 25 fr om the uni f or m over al l the pr op er S - c olourings of G . Pro of: The algorithm is considered successful if the spin-system it considers has the p rop ertie s stated in section 2.1, i.e. the follo w ing tw o hold: First , for eac h iteration i , th e induced subgraph, of the input graph, that con tains v i and all the v ertices that are within graph d istance ⌊ ǫ log n ⌋ from v i , with ǫ = 0 . 9 4 log( de 2 / 2) , is either unicyclic or a tree. According to Lemma A this h olds with pr obabilit y at least 1 − n − 0 . 1 . Se c ond , the s patia l mixing p r operty stated in Theorem A holds. F or suﬃcient ly large constan t S , the spatial mixing pr operty stated in Theorem A holds with probabilit y at least 1 − n − 0 . 25 . Consider that G is the inpu t graph of the algorithm, which is an instance of G n,d/n and w e tak e S as large as indicated in Theorem A. The, the algorithm is successful with p robabilit y at least 1 − ( n − 0 . 1 + 2 n − 0 . 25 ) = 1 − O ( n − 0 . 1 ). F rom no w on, w e assum e that the input G , b elongs to this set of instances of G n,d/n that the algorithm is su cce ssful (whic h includes almost all instances, for suﬃciently large S ). What remains to b e sho wn is the b ound for the total v ariation distance b et ween the distri- butions of the colouring that is returned by the algorithm and the uniform ov er all th e prop er S -colouring of the inpu t grap h G . 27 First w e sh o w that (1) is v alid. Consider tw o systems, i.e. S 1 = P C S ( G, S ) an d S 2 = P C S ( G, S ), eac h indep endent of the other. Assu me that, in b oth s y s te ms, we ﬁx the colour assignmen ts of the ve rtices in A i , according to prob ab ility measure µ ( · ). Clearly , we hav e for b oth systems that P r [ X A i = C ( A i )] = µ ( X A i = C ( A i )), for C ( A i ) ∈ [ S ] A i . After ﬁxing the colour assignmen ts of the v ertices in A i w e lo ok at the colour assignments of the ve rtices at graph distance, at least, ⌊ ǫ log n ⌋ from the ve rtex v i , in b oth systems. Let V ′ b e the v ertex set whose colo ur in g is reve aled and let C ( V ′ ) b e the colouring w e see in S 1 and let C ′ ( V ′ ) b e the colouring w e see in S 2 . By the la w of total probab ility , it is easy , for one, to see that, in S 1 , the probability for v i to b e assigned colouring c , ∀ c ∈ [ S ], is equal to µ ( X v i = c | C 1 ( V ′ )). S imila rly in S 2 , the the probabilit y for v i to b e assigned colouring c , ∀ c ∈ [ S ], is giv en by the measur e µ ( X v i = c | C 1 ( V ′ )). Note that in the ab o ve w e ha ve ﬁxed the colourings of the vertice s according to µ ( · ) , but w e do not ha v e seen what are the actual colourings. W e see, only , the colourings of the ve rtices in V ′ . By Theorem A and the discussion at the end of the s ection 1.2, w e ha v e that the tota l v ariation d istance b et we en the probabilit y measures of interest, is b ounded as follo ws: d T V ( µ ( X v i | C 1 ( V ′ )) , µ ( X v i | C 2 ( V ′ ))) ≤ S D ( v i , ⌊ ǫ log n ⌋ ) ≤ n − 1 . 25 (15) Whic h sh o ws that (1 ) is v alid. T h u s , the asymptotic indep endenc e b et ween the colouring of the vertex v i and the colouring of the v ertex set V ′ remains when w e colour the ve rtices in A i according to distribu tion µ ( · ). Consider the follo wing coupling of our algorithm an d an ideal algorithm that giv es a p er- fect uniform sample b y colouring v er tices one by one in some w a y . At eac h rep etition, b oth algorithms assign a colour to some (the same) v ertex in G n,d/n . Consider a sp eciﬁc step of the coupling where the v ertex v is to b e coloured. By Theorem A and (1), we can ha v e a su ﬃcien tly large S suc h that, conditioning on the f ac t that all v ertices u n til no w are iden tically coloured by the tw o algorithms, the pr obabilit y for v to ha ve a diﬀerent colour assignmen t in the coupling is at most n − 1 . 25 . T h us , the pr ob ab ility for the couplin g to end with no disagreemen t is at least (1 − n − 1 . 25 ) n > 1 − n − 0 . 25 . The theorem follo ws. ⋄ Theorem C The time c omplexity of the sampling algorithm is w.h.p. asymptotic al ly upp er b ounde d by O ( n 2 ) , wher e n is the numb er of vertic es of the input gr aph. Pro of: First, we note that the algorithm will return failure if an y of the graph s B v i is n eit her unicyclic, nor a tree. Th e num b er of step the algo rithm n ee ds, in the case of failure, is at most equal to the num b er of steps that are need in the case of nonfail. Thus, time complexit y of the nonfailing execution is an u pp er b ound of the time complexit y of the algorithm. The algorithm needs O ( n ) steps to create the graph B v i , at the i -th iterati on of its for lo op. The graph B v i can b e cr eated by u sing an y tr a v ersal algorithm, e.g. d epth ﬁrst search. This time b oun d f o llo ws by th e fact th a t the num b er of vertice s and the n umber of ve rtices in B v i are upp er b ounded by the n um b er of v ertices and the ed ge s of the input graph. Usin g the Chernoﬀ b ounds (see [6]) it is d irect to sho w that w.h.p. the num b er of edge s in an instance of G n,d/n is O ( n ). Implemen ting a colouring of v i according to µ ( X v i | C ( A i ∩ V i )), is equiv alen t to generating a random list colouring of B v i and keeping only the colour assignment of v i from th is colouring. In the list co louring prob lem ev ery v ertex u ∈ V i has a set L ( u ) of v alid colours, where L ( u ) ⊆ [ S ] and u can only recei ve a colour in L ( u ). As argued in [4], for a tree on l v ertices w e can exactly compute the n umb er of k -colourings in time l · k . Therefore we can also generate a random 28 list colouring of the tree. Also, for a un icyc lic comp onen t we can simp ly consider all the ≤ k 2 colourings of the endp oints of th e extra edge and then recurse the r emaining tree . I.e. the time we need to colour v i according to µ ( X v i | C ( A i ∩ V i )), is at most O ( n ). Th e theorem follo ws by noting that w e need to colour n v ertices. ⋄ Ac kno w ledgemen t. W e w ish to thank D. Ac hlioptas for comment s on our wo rk and also E. Mossel and A. Sly for the exchange of comments ab out our an d their p ro ving m et ho dologies. W e wish to thank previous anonymous referees for comments that allo wed us to str u cture the pro of argumen ts b etter. References [1] D. Ac hlioptas and A. Naor. The two p ossible values of the chr omatic nu mb er of a r andom gr aph . In p r oc of 36th An n ual ACM STOC, pp 587-5 93, 2004. [2] D. Aldous. , R andom walks of ﬁnite gr oups and r apid ly mixing Markov chains. In : S ´ eminaire d e Pr obabilit ´ es XVI I 1981/8 2, Sp ringer-V erlag, Berlin. pp. 243-29 7. [3] N. Alon and J. H. Sp encer. The Pr ob abilistic Metho d , Second Edition, Wiley & Sons, Inc, 2000. [4] M. Dyer, A. Flaxman, A. M. F rieze an d E. Vigod a. R andomly c oloring sp arse r andom gr aphs with fewer c olors than the maximum de gr e e Rand om Stru ctures and Algorithms 29, pp 450-465 , 2006. [5] C. Efth ymiou and P . G. Sp irakis. R andomly Colouring Sp arse R an- dom Gr aphs usi ng a Constant N u mb er of Colours . DELIS-TR-0501, h ttp://delis.upb.de/do cs/tr-abst-050 1.h tml . [6] S. Jans o n, T. Luczak and A. Ruci ´ nski. R andom Gr aphs Wiley & S ons In c., 200 0. [7] M. R. Jerrum, L. G. V alian t and V. V. V azirani. R andom gener ation of c ombinatorial strucutr es fr om a uniform distribution . Theoretical Compu te r Science, 43, pp 169-1 88, 1986. [8] M. Jerru m. A very simple algorithm for estimating the numb er of k -c olourings of a low- de gr e e gr aph , Random S tructures and Algorithms 7, pp 157-165, 1995. [9] F. Martinelli, A. Sinclair and D. W eitz. The Ising mo del on tr e es: Boundary c onditions and mixing time . T ec hnical Rep ort UCB//CSD-03-1256 , UC Berk eley , Ju ly 2003. Also in pro c. of the 44th Symp osium on F ound ations of Computer Science (F OCS 2003), IE EE Computer So ciet y , pp 628-639, 2003. [10] M. Mollo y . Glaub er dyna mics on c olorings of a gr aph with high girth and maximum de g r e e , SIAM Journal on C omputing 33 p p 721 -737, 200 4. [11] A. Mon tanari and D. Shah. Counting go o d truth assignments for r andom k -SA T formulae . In Pro c. the Eighte enth Annual A CM-SI AM Symp osium on Discrete Algorithms, S OD A 2007, pp 1255-126 4. [12] E. Mossel and A. S ly . Gibbs R apid ly Samples Colourings of G n,d/n . P ersonal comm un ic a- tion, also in http://arxiv.or g /a bs/0707.3241 . 29 [13] S . R oss. Sto chastic Pr o c esses , 2nd Ed iti on. J o hn Wiley and Sons, Inc. , 1995. [14] L. V alian t. The c omplexity of Enumer ation and R eliability Pr oblems . In SIAM Journal on Computing 8, pp 410-421, 1979. [15] D. W eitz. Counting indep endent sets up to the tr e e thr eshold. In pr oc of 38th A CM Sym- p osium on Theory of Computing (STO C 06),pp 140 -149, 2006. [16] M. W ain wright and M. Jordan. Gr aphic al M o dels, exp onental families and v ariational infer enc e . T ec h nical Rep ort 649 , Dept. of Statistics U.C. Berk eley 2003. 30

Random sampling of colourings of sparse random graphs with a constant number of colours

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