Generating random regular graphs quickly

Transcription

1 Generating raom regular graphs quickly A. S T E G E R a N. C. W O R M A L D Institut für Informatik, Technische Universität München, D-8090 München, Germany Department of Mathematics a Statistics, University of Melbourne Parkville, VIC 305, Australia Received We present a practical algorithm for generating raom regular graphs. For all d growing as a small power of n, the d-regular graphs on n vertices are generated approximately uniformly at raom, in the sense that all d-regular graphs on n vertices have in the limit the same probability as n. The expected runtime for these d s is O.. Introduction There are various algorithms known for generating graphs with n vertices of given degrees uniformly at raom. Unfortunately, none of them is of practical use for all degree sequences, even for those with all degrees equal. In this paper we examine an algorithm which, although it does not generate uniformly at raom, is provably close to a uniform generator when the degrees are relatively small. Moreover, it is easy to implement a quite fast in practice. The most interesting case is the regular one, when all degrees are equal to d = dn say. Moreover, methods for the regular case of this problem usually exte to arbitrary degree sequences, although the analysis can become more complicated a it may be needed to impose restrictions on the variation in the degrees such as is analyzed by Jerrum et al. [4]. The first algorithm for generating d-regular graphs uniformly at raom was implicit in the paper of Bollobás [] a also in the approaches to counting regular graphs by Beer a Canfield [] a in [3] see also [4] for explicit algorithms. The configuration or pairing model of raom d-regular graphs is as follows. Start with points even in n groups, a choose a raom pairing of the points. Then create a graph with an edge from i to j if there is a pair containing points in the i th a j th groups. If no duplicated edge or loop i.e., a pair of points in the same group occurs, Research supported by the Australian Research Council

2 A. Steger a N.C. Wormald the resulting d-regular graphs occur uniformly at raom. For graphs on n vertices this takes expected time of the order of e d /4 per graph, at least for d up to n /3, so is not polynomial time unless d = O log n. In [9] a polynomial expected time uniform generation algorithm was given for d = On /3, but the expected running time per graph is On d 4. This is rather complicated to implement. Also, it applies to arbitrary degree sequences. Another algorithm was given specifically for regular graphs which reduces the time to O 3, however this is prohibitively difficult to implement, a again it only applies for d = On /3. The need for generating such graphs can also be met by simpler algorithms which do not generate the graphs uniformly at raom. For example, Tinhofer [] gives one. However, these algorithms are not easy to analyse, a the resulting probability distribution can be virtually unknown. As discussed in [], one can achieve uniformity by an accept/reject procedure, but the inherent difficulties in analysis mean that no such algorithms are yet known which are of practical use for uniform generation. On the other ha, Jerrum a Sinclair [5] provided an approximately uniform generation algorithm, which runs in time polynomial in n a /ɛ, where all graphs have probabilites varying by a factor of at most + ɛ. They do not precisely analyse the running time of there algorithm, nor do they claim that their algorithm is of practical use. There are two ways to describe the algorithm in the present paper. It can be regarded as a modification of the pairing algorithm described above. First, we define two points to be suitable if they lie in different groups a no currently existing pair contains points in the same two groups. Our algorithm is the following. Algorithm Start with points {,,..., } even in n groups. Put U = {,,..., }. U denotes the set of unpaired points. Repeat the following until no suitable pair can be fou: Choose two raom points i a j in U, a if they are suitable, pair i with j a delete i a j from U. 3 Create a graph G with edge from vertex r to vertex s if a only if there is a pair containing points in the r th a s th groups. If G is d-regular, output it, otherwise return to Step. However, the algorithm actually arose from extensions of the algorithm examined in []. There, one begins with n vertices a continually selects a raom edge to add, subject to keeping all vertices of degree at most d a no multiple edges. It was shown in [] that for fixed d a dn even, this almost surely produces a d-regular graph as n. In [6] this process is modified by selecting the edges non-uniformly. One interesting choice of the non-uniformity studied there gives the following algorithm. Algorithm Start with a graph G with n vertices {,,..., n} a no edges. Repeat the following until the set S is empty: Let S denote the set of pairs of vertices of G which are non-adjacent a both have degree at most d. Choose a raom

3 Generating raom regular graphs quickly 3 pair {u, v} in S with probability proportional to d dud dv where d denotes the degree in G. Add the edge {u, v} to G. 3 If G is d-regular, output it, otherwise return to Step. Note that the probabilities of edges in Algorithm are exactly the probabilities of points being chosen between the correspoing groups in Algorithm, so the two algorithms are equivalent. Our results in Section show that Algorithm generates graphs with nearly uniform probability distribution, in the sense that as n, provided d does not grow too quickly with n, the probabilities of all graphs are asymptotically equal. For larger d, but still not very large, we show the algorithm generates d-regular graphs with a distribution which is close to uniform, in the sense that the probability of any event is different from that in the uniform space by o. In Section 4 we show that the expected time for Step of Algorithm is O + d 4 for all d. Moreover, the probability it produces a regular graph which is then accepted at Step 3 is o for d = on/log n 3, a so the number of repetitions of Step required is O for such d. In fact, it is quite possibly boued for all d n/... Notation a Preliminary Results. We use log x to denote the natural logarithm, a assume d throughout the paper. By Rn, d we denote the set of all labelled d-regular graphs on n vertices. We can view Step of Algorithm as the raom selection of a sequence x,..., x k of edges on n vertices. We call such a sequence a path because it is a possible path for the course of the algorithm. For each path P, we can consider the probability that one run of Step of Algorithm produces P. This iuces a probability space whose elements are paths, which we call the A-model. Exactly the same space is produced by Algorithm. Note that the edges in such a path determine a maximal graph with no vertices of degree greater than d. We need to compare this with the pairing model, which can be defined similarly to Algorithm, but with Step accepting two points i a j even when they are not suitable. By the U-model we mean the probability space of paths iuced by one run of Step in this modified algorithm. Actually, pairs of points are chosen in Step, a the correspoing edges should be determined by the mechanism in Step 3 of Algorithm, at which stage loops a multiple edges may be formed. Note that in practice one could restart the algorithm as soon as unsuitable points were fou, but for reasons of taste in this analysis we permit Step to run its course in the modified algorithm until all points are paired. Let G be a d-regular graph. By P athsg we denote the set of all orderings of the set of edges of G; that is, paths whose edges are precisely the edges of G. Then For a path P P athsg we use P athsg =!. P A [P] resp. P U [P] to denote the probability of P in the A-model resp. in the U-model. Note that in the

4 4 A. Steger a N.C. Wormald U-model all paths P P athsg have, in fact, the same probability. Namely, P U [P] = d! n.! On the other ha, denote the subgraph of G consisting of the first m edges of P by G m P. If a path P P athsg has edges x,..., x dn/ where edge x i joins vertices u i a v i then P A [P] = / d d m u i d d m v i d dm ud d m v where the sum in the denominator is over all u v V G such that {u, v} / EG m, a d m denotes the degree in G m. For comparison, in the pairing model for generating regular graphs uniformly, after m pairs have been added there are m unmatched points remaining. Also, the number of pairs correspoing to a valid edge {u, v} is d d m ud d m v. Hence, the probability in the U-model can be written as P U [P] = / d d m u i d d m v i. m For a subgraph H of G we denote by H the difference in the normalizing denominators in a for the factors due to G m = H. That is, we let where a H := H = H + H, H := v V G {u,v} EH d dh v. 3 d d H ud d H v 4 For brevity we use m P for G m P. Observe from a a by considering the number of acceptable pairs of points in Step of Algorithm that P A [P] P U [P] = m m m P. 5 We will fi it useful to know the value of P U [G] approximately. The following estimate of the size of Rn, d was fou first in the special case of boued d by Beer a Canfield []. Bollobás [] proved it for all d log n by considering the probability of obtaining no loops or mulitiple edges in the pairing model. Then McKay analysed the same model for substantially larger d. Theorem.. McKay [8] Assume d = on /3. Then Rn, d = + o e 4 d!! d!. n

5 Generating raom regular graphs quickly 5 Correction terms are required if the upper bou on d is relaxed further see McKay a Wormald [0] for the formula when d = o n. The formula above was obtained by estimating the probability in the configuration model of having no loops or multiple edges, so although it is working back-to-front we can deduce the following. Corollary.. Assume d = on /3. Then P U [G] = + o e 4 d / Rn, d. Proof. We just observe that P U [G] = P U [P] P P athsg a P U [P] = d! n /! for all paths P P athsg.. Main Results Our first result shows that the probabilities of all graphs generated by Algorithm are asymptotically equal, provided d grows slowly enough with n. Theorem.. Assume that d = On 8. Then there exists a function fn, d = o such that all d-regular graphs G satisfy P A[G] Rn, d < fn, d Rn, d. Our proof actually yields the same result for a slightly larger d. But we believe it is true even for much more quickly growing d. The next result is useful if Algorithm is to be used for estimating probabilities by simulation. The conclusion of this theorem is equivalent to the assertion that the total variation distance between the distribution of graphs given by Algorithm, a the uniform distribution, goes to 0 as n. Theorem.. Assume that d = on/log n 3. Then there exists a function fn, d = o a a subset X Rn, d such that a P A [G] = + Ofn, d Rn, d X = fn, d Rn, d. for all graphs G X

6 6 A. Steger a N.C. Wormald Observe that Theorem. implies that one can experimentally determine the probability of an event E defined on the set of d-regular graphs by simulation using Algorithm. We formulate this in a slightly more general setting: Corollary.. Assume that d = on/log n 3 a let X be a boued function of graphs defined on the sets Rn, d for all n a d. Then XG P U [G] XG P A [G] = o. G Rn,d G Rn,d That is, one can compute the expectation to within o error by simulation using Algorithm. For practical use of the algorithm it is comforting to know the following. Theorem.3. Assume that d = on/log n 3. Then the probability that Step of Algorithm produces a regular graph is asymptotic to as n. This result is in a similar direction to the main theorem of [], which was only proved for boued d, the process in that paper being rather different. 3. Proof 3.. Outline. Fix a d-regular graph G on n vertices. In the U-model P U [P] is constant for all paths P P athsg. This is obviously not true in the A-model. Here the probability P A [P] depes strongly on the order of the edges. We can however estimate an expected or average probability by considering a path chosen uniformly at raom from all possible paths. For now just assume that avg denotes such an estimate. In Section 3.3 we will assign a particular value to avg. Our proof strategy is as follows. We partition the set P athsg into suitable subsets P athsg = M G M G AvG such that i the sets M i G, which contain in some sense misbehaving paths, are small a ii paths in AvG have a probability which is roughly equal to avg. We now make these ideas precise. Lemma 3.. Let d = dn a let gn, d be a positive function with gn, d. Assume there exists a positive function avg such that for all d-regular graphs G there is a partition of the set P athsg as above such that: P A [P] e gn,d avg for all P P athsg, 6 a M k G = oe gn,d P athsg, 7 EP A [P] = + o avg 8

7 Generating raom regular graphs quickly 7 for P chosen uniformly at raom from AvG. Then P A [G] = + o avg P athsg. 9 Proof. P A [G] = P A [P] = P P athsg P A [P] + P A [P] P M G M G P AvG = o P athsg avg + + o avg AvG = + o avg P athsg. As we know that P athsg =!, Theorems. a. will follow easily from Lemma 3. if we can show that avg is iepeent of the graph G a has the correct order of magnitude. This we will show in Section 3.3. Throughout the remaining part of this section we assume that G is an arbitrary, but fixed d-regular graph. 3.. Some Useful Lemmas. In this section we collect some helpful lemmas a facts that will eventually enable us to bou the ratio given in 5 for a raomly chosen path P P athsg. Lemma 3.. Let H be a subgraph of G a let m = EH. Then H d m. Proof. By Jensen s inequality the sum in 3 is boued from above by m d d. Lemma 3.3. Let H be a subgraph of G a let m = EH. Then H d m. Proof. {u,v} EH d d H u d d H v = x u:{x,u} EG\EH v Γ H u d G\H v. Observe that d G\H v d for all v Γ H u a that there at most d such v s. Corollary 3.. All paths P in P athsg satisfy m P d m for all 0 m.

8 8 A. Steger a N.C. Wormald The following tail bou on the sum of iepeent raom variables is due to Hoeffding [3]. Hoeffding s inequality. Let X,..., X n be iepeent raom variables with 0 X i for all i n, a let X = n i= X i, p = nex a q = p. Then for 0 t < q p+t q t n p q P[X pn tn]. p + t q t A corollary of this will turn out to be very useful. Corollary 3.. Let X,..., X n be iepeent variables with 0 X i for all i n, a let X := n i= X i. Then e δ EX { e 4 δex if δ 4 5 P[X > + δex] + δ +δ otherwise e 4 δex a P[X < δex] e δ EX. Proof. The first factor in Hoeffding s inequality can be written as + δ +δp a the seco as + δp/ p δp p δp < e δp. From here, the first inequality can be verified by comparing logarithms noting that we can assume t q because X n always. The seco is by McDiarmid [7, 5.6], which he attributes in the binomial case to Angluin a Valiant. In fact by [7, 5.5], it is possible to improve the factor /4 in the exponent in the case δ 4/5 to /3, but we do not need this Average Paths. Our aim in this section is to estimate m P for a uniformly raomly chosen path P P athsg. Observe that this just correspos to the value of m H for a raom subgraph H of G with m edges. To simplify calculations, however, we don t choose m edges raomly from G, but instead choose edges with probability p = pm = m. We denote such a raom subgraph by RG p. Note that we are not claiming that the expectation of m, which we are about to calculate, is the same for both models. In the following we simply calculate that expectation with respect to the model where the edges of G are chosen with probability p = pm. Later we draw conclusions about m P from this calculation. To simplify notation we abbreviate in the rest of the paper E RG p where p = m by µ m. Similarly, we let µ i m := E i RG p for p = m a i =,. Lemma 3.4. µ m = m d.

9 Generating raom regular graphs quickly 9 Proof. For every path of length in G we introduce a 0/ variable X i which is equal to iff both edges of the path do not belong to RG p. As EX i = p a there are n d such paths we have µ m = E i X i = n d p = n d m = d m. Lemma 3.5. µ m = m md n d. Proof. For every ordered tuple x, u, v, y such that {x, u}, {u, v}, a {v, y} are edges of G we introduce a 0/ variable Y i which is equal to one iff the edge {u, v} belongs to RG p a both edges {x, u} a {v, y} do not belong to RG p. As EY i = p p a there are d such tuples we have µ m = E i Y i = d p p = d m m. Corollary 3.3. µ m = d m + d m n d. After these preliminaries we are now in a position to precisely define avg, the probability of an average path in the A-model. Namely, we assume that the value m occurring in the denominators of the edge probabilities of the path is exactly equal to µ m. To make the formula slightly nicer, we define it in terms of the probability of a path in the U-model, which we know is iepeent of the graph G a the actual path P P athsg: Lemma 3.6. avg := P U [P] m m µm. 0 The value avg is iepeent of the graph G a, if d = on /3 then avg = + o e 4 d P U [P] = + o e 4 d P U [G] P athsg. Proof. The iepeence of avg from G follows immediately from the fact that the values µ m are iepeent of G. avg µ m = + P U [P] µm = + m d + d m / n d m O d n

10 0 A. Steger a N.C. Wormald = exp d log + + d m / n d m O d n As log + x = x Ox for all x, we can bou the sum for all d = on /3 as follows: = = d log + + d m / n d log + d + d m d n d + O n d + d m n d = d d + n d = d + 4 d + o = 4 d + o. m O d n m + d n d + O n + d n m d + O n + d n log 3.4. The Misbehaving Sets. In this section we define the sets of misbehaving paths. They will be characterized in terms of the deviation of their value m P from µ m. Recall that we defined µ m to represent, in a somewhat vague sense, the value of m P for a raom path in P athsg. So we expect that most paths have values m P which are close to µ m. In this section we will quantify this. Let M G := set of all paths P P athsg such that µ m m P max{d 4 4µ m log n, 4d 5 log n} for some 0 m, M G := set of all paths P P athsg such that µ m m P max{d 5 4µ m log n, 4d 6 log n} for some 0 m, a Lemma 3.7. For n sufficiently large, M G e 5d log n P athsg. Proof. We consider an arbitrary but fixed 0 m. Clearly, it suffices to show that for this m we have P[ m P µ m max{d 4 4µ m log n, 4d 5 log n}] e 5d log n /.

11 Generating raom regular graphs quickly If we choose a path P P athsg raomly, then G m P correspos to a raom subgraph of G with m edges. That is, if we denote with RG m the raom subgraph of G with m edges, we observe that is equivalent to P[ RG m µ m max{d 4 4µ m log n, 4d 5 log n}] e 5d log n /. Instead of the raom graph RG m we will work with the raom subgraph RG p of G with edge probability p = m. Observe that in this model every subgraph of G with m edges is equally likely a P[ ERG p = m] = p m p m e log n m For m a m teing to infinity the inequality follows easily from Stirling s formula, for the remaining m s observe first that it suffices to consider the small m s, then use m m m a the fact that m e log n for m small enough. P Hence, it suffices to show [ RG p µ m max }] {d 4 4µ m log n, 4d 5 log n e 5d log n log n /. This we will now do. Observe that by Brooks theorem, for example we can color the vertices of G with d + colors in such a way that no two vertices with the same color are adjacent. We hale each color class separately. Let X i denote the number of tuples x, u, y such that x y, {x, u} a {u, y} belong to EG \ ERG p a u belongs to the ith color class. Let X = d+ i= X i. Then EX is equal to RG p a we therefore have to show that P[ X EX max{d 4 4µ m log n, 48d 5 log n}] e 5d log n log n /. Clearly, it suffices to show that for all i d + P[ X i EX i max{d 3 4EX log n, 4d 4 log n}] e 6d log n. Let n i be the number of vertices in the ith color class. As X i can be written as the sum of n i iepeent variables with values in the interval [0, dd ] we may apply the Hoeffding bous for the raom variables Y i := X i /d a Y := X/d. First we hale those m s for which EY 4d log n. Here we apply Corollary 3. with δ = d 4EY log n/ey i : P[Y i EY i + d 4EY log n] = P[Y i + δey i ] a { e 4 4d EY log n/ey i e 6d log n if δ 4 5 e d 4 4EY log n e 6d log n otherwise P[Y i EY i d 4EY log n] = P[Y i δey i ] { e 4d EY log n/ey i e d log n if EY i d 4EY log n 0 otherwise.

12 A. Steger a N.C. Wormald For m s such that EY 4d log n we just apply the first of the inequalities in Corollary 3. with δ = 4d log n/ey i to deduce P[ Y i EY i 4d log n] = P[Y i EY i + 4d log n] = P[Y i + δey i ] e 6d log n. Lemma 3.8. M G e 5d log n P athsg. Proof. Proceeding as in the proof of Lemma 3.7 we first observe that it suffices to show that P[ RG p E m max{d 5 4E m log n, 4d 6 log n}] e 5d log n log n /. Observe that we can color the edges of G with dd + colors in such a way that any two edges of the same color are not connected by an edge in G, i.e. every color class is the edge set of an iuced matching in G. To see this consider the graph where every edge of G correspos to a vertex a every such vertex is connected to all vertices correspoing to edges which have to be colored differently. As the maximum degree of this graph is dd the claim follows from Brook s theorem. With this fact in ha we can now again proceed similarly to the proof of Lemma 3.7. Let X i denote the number of tuples x, u, v, y such that {x, u} a {v, y} belong to EG \ ERG p, while {u, v} belongs to the ith color class of G a to RG p. Let X = d + i= X i. Then EX is equal to RG p a it therefore suffices to show that for all i d + P[ X i EX i max{d 3 4EX log n, 4d 4 log n}] e 6d log n. By the choice of the coloring we again have that each X i is the sum of iepeent variables with values in the interval [0, d ]. An application of Corollary 3. similar to the one in the proof of Lemma 3.7 thus concludes the proof of Lemma Proof of the Main Theorems. Our aim in this section is to verify the remaining hypotheses of Lemma 3., with misbehaving sets M i G defined as in the previous section. We start with some auxiliary lemmas. Lemma 3.9. Let ξ = ξn, d be an arbitrary function such that ξ d +. For all paths P P athsg: m µm m P log4ξd 4. ed m= ξ m

13 Generating raom regular graphs quickly 3 Proof. Observe first that m µm m d m P µ m m Using Corollary 3. twice we furthermore observe that m Hence, m P m P m P d m m m m m= ξ 4 µm m P for all d m. for all ξ m d +. d d d m P + m m= ξ m P d d 4 d exp m P m m= ξ m P ξ expd logd 4 exp d i i=d + expd logd 4 + d logξ +. Corollary 3.4. Assume that d = on /3. Then P A [P] e 4d log n avg Proof. The expression in Lemma 3.9 for ξ = is P A [P] avg expd log 5. for all paths P P athsg. Lemma 3.0. All paths P AvG satisfy m P µ m m P 00d/ log n + 00d6 log n n m m for all m d +. m Proof. First we observe that Corollary 3. implies that m m P 4 m for all m d +. Consider an arbitrary path P AvG. From the definition of M G a Lemma 3.4 we know { 4d m P µ 5 log n if µ m 4d log n m 4d 4 m log n/n otherwise.

14 4 A. Steger a N.C. Wormald Similarly, the definition of M G a Lemma 3.5 imply Hence, m P µ m m P µ m m P m 4 { 4d 6 log n if µ m 4d 5 m m log n/n m P µ m + m P µ 4 otherwise. m m 4 + 4d 6 log n m + 4d4 log n 9d6 log n m + 9d/ log n n m. 4d log n + 4d5 m log n n m n m Corollary 3.5. Assume that d = on /3 a let ξ = ξn, d be an arbitrary function such that ξ d 4 log n. All paths P AvG satisfy ξ m m µm m P = expod6 log n/ + ξ log n. Proof. Let P be an arbitrary path in AvG. Using Lemma 3.0 we deduce ξ m ξ µm m = + mp µ m m P m m P ξ + mp µ m m m P ξ exp m P µ m m m P exp O i=ξ d / log n n i + d6 log n i expod 6 log n/ + ξ log n. For the proof of the lower bou observe first that the assumption on ξ together with Lemma 3.0 implies that for sufficiently large n all m ξ satisfy m P µ m m P. m Using the fact that x e x for all x we deduce that + mp µ m m P mp µ m m P exp mp µ m, m P m m m

15 Generating raom regular graphs quickly 5 from which the desired inequality follows in the same way as above. Lemma 3.. e 5dξ n Assume that d = on /3. Then for all ξ : m= ξ µ m m µm Proof. From Corollary 3.3 we deduce that m µ m d m n n m= ξ m m= ξ m= ξ µ m m e ξ n. for all 0 m. Using that e x x e x for all x which is most easily verified by differentiation we therefore conclude µ m e ξ n n a m= ξ µ m m µm m= ξ d n d n e 4dξ n d e 5dξ n. Corollary 3.6. for all paths P AvG. Assume that d = on /3. Then P A [P] e O d d 7 log n 3 /n avg Proof. Let P be an arbitrary path in AvG a let ξ = 5 log n. Observe that for m ξ we trivially have + mp µ m m m P if m P µ m µ m otherwise. m µ m Thus, using Corollary 3.5 a 3. we deduce P A [P] = + mp µ m avg m m P ξ m µm m P exp O m d log n 3 n + d6 log n ξ m= ξ + dξ n µ m m µm

16 6 A. Steger a N.C. Wormald exp O d d7 log n 3. n Lemma 3.. Assume that d = On 8 a ξ d 6 log n. Then for a path P selected uniformly at raom from AvG, the expected value of m m m= ξ µm m P is asymptotic to as n. Proof. Let p denote this expected value. We compute with a path P selected uniformly at raom from P athsg, a let p 0 denote the expected value of for such P. Later we deduce what we need about p in the restricted space. Note that p 0 is determined by the last ξ edges in the ordering of the edges of G correspoing to P, which form a raom ordered ξ-subset S = SP of the edges of G. Let S denote the probability space of these subsets, determined by all P P athsg. Let rs denote the number of edges in S which have distance at most from a later edge in S where by distance at most we mean that one vertex of the new edge a one vertex of a later edge in S are at distance. Let R r denote the set of S for which rs = r. Then for S S, PS R r = r ξ d 3 ξ r ξ d ξ r O. 3 n Consider an arbitrary path P a assume that S = SP R r. We first aim at proving upper bou on m P for all m ξ. In order to do so we imagine the sequential generation of S starting with the last edge. Let H k denote the subgraph of G where the last k edges of S are removed from G. Observe that, trivially, H 0 = 0. Now consider what happens if we change from H k to H k, that is, if we remove the kth edge. An important observation is that in fact H k = H k if the kth edge has distance at least from all previously removed edges. That is, in this respect we only have to consider the edges which have distance at most from some later edge. In order to obtain the type of bous we need, we will, however, instead consider all edges which have distance at most from a previously removed edge separately. Consider the ith such edge, say edge {u, v}, a assume it is the kth edge in total. We want an upper bou for H k H k. One easily checks that H k H k i a we claim that H k H k 4i. To see this observe that by removing edge {u, v} only the summas correspoing to edge {u, v} itself a those of the adjacent edges can contribute. Edge {u, v} contributes a negative term, so we don t have to worry about it. On the other ha, an edge adjacent to {u, v}, say {u, w}, adds to the difference exactly

17 the degree of w in G H k. That is, H k H k Generating raom regular graphs quickly 7 x N Hk u d G Hk x + y N Hk v d G Hk y. Observe that any two edges in G H k which contribute to the sum for u are at distance from each other, a there are at most i such edges. Each such edge can count up to two times for u once at each of its es a up to two times for w. This shows the desired bou of 4i. Summarizing, we have for all k ξ r r + H k H ξ 5i 5 i= implying that 5 r+ is also an upper bou on m P for m ξ. Also note that m P < t for m = t. Hence for SP Rr, m m m= ξ m P t0 t= t ξ t=t 0+ = Or t 0 exp ξ t=t 0+ t t 5 r+ r O = Or t 0 4 provided t 0 is chosen such that t 0 = Or a t 0 > 5 + ɛ r+ for some ɛ > 0. Recalling Lemma 3. that e dξ m n d m m= ξ µm e ξ n, 5 we see that p 0 o, a by 3 a 4 with t 0 = 8 5 r > 5 r, ξ r 56/5 d ξ r p 0 + O = + o 6 n r= since r ξ. This was all for P selected uniformly at raom from P athsg; note that 5 similarly implies p o. Furthermore, since is always positive, we have p 0 p PM G M G + o by Lemmas 3.7 a 3.8 a 6. Proof of Theorem.. Using 5 a the definition 0 of avg we obtain for a path P chosen uniformly at raom from AvG EP A [P] = AvG P A [P] P AvG t

18 8 A. Steger a N.C. Wormald = avg AvG P AvG m m µm m P. From Corollary 3.5 a Lemma 3. using ξ = d 6 log n we obtain 8. Hence from Lemmas 3.7 a 3.8 a Corollary 3.4 we deduce from Lemma 3. a Lemma 3.6 P A [G] = + o avg P athsg = + o e 4 d P U [G]. As we know from Corollary. that P U [G] = +o e 4 d / Rn, d, the theorem follows. Note that the little oh terms do not depe on G. Proof of Theorems. a.3. Combining Lemmas 3.7 a 3.8 a Corollary 3.6 we deduce from Lemma 3.6 a Corollary. that for d = on/log n 3 P A [G] + o avg P athsg = + o e 4 d P U [G] = + o Rn, d. As the sum P A [G] over all d-regular graphs G can be at most, we have that this sum is asymptotic to, a the claims of Theorems. a.3 follow Concluding Remarks. The following example shows that Theorem. is in some sense best possible. More precisely, this example shows that the probability distribution of two d-regular graphs G are not exactly equal. Consider the following two -regular graphs on 6 vertices: a C 6 a b two disjoint copies of a C 3. For each graph there are 6! = 70 different paths. With the help of a small computer program one easily checks that P A [C 6 ] = while P A [ C 3 s] = It is also possible to prove that infinitely many larger examples exist, by showing that the primes occurring in the denominators of the probabilities for some graphs will not occur in others. But it seems annoyingly difficult to show what we believe is true, that graphs usually occur with different probabilities. We also do not have good examples of d-regular graphs which are considerably more or less likely than most of the d-regular graphs, which would limit the possible strengthenings of Theorem.. A good caidate for this might be the graph which consists of the union of pairwise disjoint K d s. 4. Performance of the Algorithm Theorem 4.. For all d = On / Step of Algorithm can be implemented so as to require expected time O, a space O.

19 Generating raom regular graphs quickly 9 Proof. It can very easily be implemented in three phases. In the first phase, keep a list L of all the points, as an array with those in U at the front, a the other points in pairs afterwards, as well as another array I whose i th entry is the position in L of the point i. Then two points i a j can be chosen raomly in U in constant time assuming the number of digits in n is not a problem, a assuming a perfect constant-time raom number generator. Moreover, they can be checked for suitability in time Od, since all d points in the same group can be fou in L in time Od using I, a for each such point, if it is in a pair, its mate is next to it on a known side in L. This process is repeated until a suitable pair has been fou. Then, if they are suitable, update L by swapping the chosen points with the last two listed in U, which are m + a m + from the e of L if m pairs have already been added a update I for the up to four points so moved. This takes constant time. Hence, the running time of phase is Od times the total number of points i a j checked for suitability throughout this phase. Phase stops when the number of points in U first falls below d. Note that each point has at most d other points in U with which it does not make a suitable pair. Hence, in phase, when there are k > d points left in U, the expected number of trials of two points before a suitable pair is fou is at most. The sum of this over d < k < is O, giving the bou O on the total time to reach this stage. On the other ha, when the number of points in U first falls below d, phase begins. Instead of choosing a pair at raom, choose a raom pair of groups i.e. vertices of the graph G, say u a v, which do not yet have all points matched. If {u, v} is already an edge of G, repeat this step. Then raomly choose a pair of points i a j in the groups correspoing to u a v respectively. If these points i a j are in U, they are a raomly chosen suitable pair; if not, repeat the choice of u, v, i, j again. Phase lasts until the number of available groups drops below d. The list of available groups can be maintained just like the list of available points, with negligible time required. Thus, choosing an available pair of groups takes constant time. The probability they form a non-edge of G is at least / as there are at least d groups a each has an edge to at most d others. Also, for each such u a v, the probability of fiing a pair of points in U is at least /d, a so for the Od pairs added this stage takes expected time Od 4. When the number of available groups falls below d, phase 3 begins. Construct in time Od the graph iuced by all vertices of G of degree less than d, a form its complement H on these same vertices. Then work as in phase, but instead of choosing u a v at raom from the vertices of H, choose an edge {u, v} of H uniformly at raom a accept it with probability x uv /d, where x uv denotes the product of the number of suitable points in u times the number of suitable points in v. Given a list of edges of H, fiing the next edge requires expected time at most Od. Note that H can be updated in constant time each time an edge has been chosen, with the help of suitable data structures. The algorithm stops when no more suitable points exist, that is, H has no more edges. Hence phase 3 requires time Od 4. This establishes the time

20 0 A. Steger a N.C. Wormald requirements; the space requirements are several lists of the points O a the space required for H Od. The theorem gives the running time for one trial of Step : if the algorithm fails due to no regular graph being produced, it has to start again. By Theorem.3 the probability of failure of Step in this sense is o for d = on/log n 3, a so the expected time for the algorithm to produce a regular graph is O O for such d. Moreover, experiments with n ranging from 50 to 400 a d ranging from 0.05n to 0.5n strongly suggest that the probability of succeeding on one trial is at least 0.3 for such n a d. In fact, it appears roughly equal to 4d/3n up to d = 0.75n. Thus, the algorithm seems usable, expecting a boued number of repetitions of Step per graph produced, for all d up to n/ though for such d we do not know much about the resulting probability distribution!. For larger d, one can of course generate the complementary graph. Note that for d s larger than n the proof of Theorem 4. only gives a complexity of Od 4, but this can be improved by the use of more clever data structures. We stress that we have no proof that asymptotically for say d n/ the expected number of repetitions required is boued. In addition, we have reasons to believe that for d n / the resulting probability distribution will no longer be approximately uniform. Clarifying these possible extensions of Theorem.3 would be interesting. References [] E. A. Beer a E. R. Canfield. The asymptotic number of labeled graphs with given degree sequences. J. Combinatorial Theory, Series A, 4:96 307, 978. [] B. Bollobás. A probabilistic proof of an asymptotic formula for the number of labelled regular graphs. European Journal of Combinatorics, :3 36, 980. [3] W. Hoeffding. Probability inequalities for sums of boued raom variables. J. Am. Stat. Assoc., 58:3 30, 963. [4] M.R. Jerrum, B.D. McKay, a A.J. Sinclair. When is a graphical sequence stable? In A. Frieze a T. Luczak, editors, Raom Graphs Vol., pages Wiley, 99. [5] M.R. Jerrum a A.J. Sinclair. Fast uniform generation of regular graphs. Theoretical Computer Science, 73:9 00, 990. [6] T. Luczak a N.C. Wormald. Phase transition for raom graph processes. In preparation. [7] C. McDiarmid. On the method of boued differences. In Surveys in combinatorics, Lo. Math. Soc. Lect. Note Ser. 4, pages 48 88, 989. [8] B.D. McKay. Asymptotics for symmetric 0- matrices with prescribed row sums. Ars Combinatorica, 9A:5 5, 985. [9] B.D. McKay a N.C. Wormald. Uniform generation of raom regular graphs of moderate degree. Journal of Algorithms, :5 67, 990. [0] B.D. McKay a N.C. Wormald. Asymptotic enumeration by degree sequence of graphs with degrees on /. Combinatorica, :369 38, 99. [] A. Ruciński a N.C. Wormald. Raom graph processes with degree restrictions. Combinatorics, Probability a Computing, :69 80, 99. [] G. Tinhofer. On the generation of raom graphs with given properties a known distribution. Appl. Comput. Sci., Ber. Prakt. Inf., 3:65 97, 979. [3] N.C. Wormald. Some problems in the enumeration of labelled graphs. Ph. D. thesis, University of Newcastle, 978. [4] N.C. Wormald. Generating raom regular graphs. Journal of Algorithms, 5:47 80, 984.