On the Diameter of the Symmetric Group: Polynomial Bounds

On the diameter of the symmetric group: polynomial bounds

L´aszl´oBabai∗, Robert Beals†, Akos´ Seress‡

Abstract which has G as its vertex set; the pairs g, gs : g G, s S are the edges. Let diam(G, S{{) denote} the∈ We address the long-standing conjecture that all per- diameter∈ } of Γ(G, S). mutations have polynomially bounded word length in terms of any set of generators of the symmetric group. The diameter of Cayley graphs has been studied The best available bound on the maximum required in a number of contexts, including interconnection word length is exponential in √n log n. Polynomial networks, expanders, puzzles such as Rubik’s cube and bounds on the word length have previously been estab- Rubik’s rings, card shuffling and rapid mixing, random lished for very special classes of generating sets only. generation of group elements, combinatorial group theory. Even and Goldreich [EG] proved that finding In this paper we give a polynomial bound on the the diameter of a Cayley graph of a permutation group word length under the sole condition that one of the is NP-hard even for the basic case when the group generators fix at least 67% of the domain. Words is an elementary abelian 2-group (every element has of the length claimed can be found in Las Vegas order 2). Jerrum [Je] proved that for directed Cayley polynomial time. graphs of permutation groups, to find the directed The proof involves a Markov chain mixing esti- distance between two permutations is PSPACE-hard. mate which permits us, apparently for the first time, No approximation algorithm is known for distance in to break the “element order bottleneck.” and the diameter of Cayley graphs of permutation As a corollary, we obtain the following average- groups. Strikingly, the question of the diameter of the case result: for a 1 δ fraction of the pairs of Rubik’s cube Cayley graph appears to be wide open generators for the symmetric− group, the word length (cf. [Ko]). We refer to [BHKLS] for more information is polynomially bounded. It is known that for almost about the history of the diameter problem and related all pairs of generators, the word length is less than results and to the survey [He] for applications of exp(√n ln n(1 + o(1))). Cayley graphs to interconnection networks. For G = Sn and G = An we conjecture that 1 Introduction diam(G, S) is polynomially bounded in n for all sets S of generators: Let G be a finite group and S a set of generators of G. We consider the undirected Cayley graph Γ(G, S) Conjecture 1.1. [BS1] There exists a constant C such that for all n and all sets S of generators of C G = Sn or An, diam(G, S) n . ∗Department of Computer Science, University of Chicago, ≤ 1100 East 58th Street, Chicago, IL 60637, and Mathemati- This conjecture would imply a quasipolynomial upper cal Institute of the Hungarian Academy of Science. Email: bound on the diameters of all Cayley graphs of all [email protected]. This research was partially supported by the NSF. transitive permutation groups [BS2]. †IDA Center for Communications Research - Princeton, 805 The Conjecture has been verified for very special Bunn Drive, Princeton, NJ 08540. Email: [email protected]. Department of Mathematics, Ohio State University, classes of generating sets. Driscoll and Furst [DF] ‡ 2 231 W. 18th Avenue, Columbus, OH 43210. Email: proved an O(n ) bound for the case when all gen- [email protected]. Partially supported by the NSF. erators are cycles of bounded lengths and McKen- zie [McK] gave a polynomial bound for the case when now able to overcome this obstacle. the generators have bounded degree. (The degree of a permutation is the number of elements actually Corollary 1.1. Fix k 2 and δ > 0. Let G be ≥ moved.) We note that McKenzie’s exponent is Θ(k) either Sn or An. Let S be a random subset of size k where k is the bound on the degree. in G and let G1 be the group generated by S. Then, with probability > 1 δ we have diam(G ,S) nC for We extend these results to a large class of gener- 1 a constant C = C(δ−). ≤ ating sets in the hope that this may lead to settling the Conjecture. 2 Proof of the worst-case bound: a Markov Theorem 1.1. Let be a positive constant. Let S be chain argument a set of generators for G = S or G = A . Assume n n We shall use the following fact. We explain the that S contains an element of degree n/(3+). Then concepts involved after the statement. diam(G, S) = O(nC ) where C is an≤ absolute constant (does not depend on ). Moreover, between any pair C Fact 2.1. Let X be a connected undirected graph on of elements of G, a path of length O(n ) can be found n vertices, with maximum degree ∆ and diameter d. in Las Vegas polynomial time. Then the strong mixing time for a lazy random walk on X (starting from any vertex) is O(d∆n log n). We did not attempt to optimize the exponent C; without effort, our proof yields a diameter up- A lazy random walk on the graph means a per bound of n7(log n)O(1). The bound reduces to 6 O(1) stochastic process where a particle moves from vertex n (log n) if the number of generators is bounded. to vertex; at each step, with probability 1/2 the The best upper bound known to hold for the particle does not move; the remaining probability 1/2 diameter of all Cayley graphs of An and Sn is is split evenly between the neighbors. On a connected exp(√n ln n(1 + o(1))). Except for the function im- graph, lazy random walks are ergodic and therefore the plicit in the o(1) term, this bound is the same as the distribution of the particle approaches the stationary largest order of elements of Sn [La]; indeed one of the distribution. operations used in the proof of the diameter bound is In the stationary distribution, each vertex is vis- raising an element to an arbitrary power, which makes ited with a frequency proportional to its degree. So the order of elements a bottleneck. for regular graphs (as will always be the case in our Theorem 1.1 seems to be the first result to over- applications), the stationary distribution is uniform. come the “element order bottleneck.” To achieve this, The strong mixing time of a random walk we have to avoid taking powers of permutations as a means the number of steps needed for the random degree-reduction tool. Instead we increase the degree- walk to reach a distribution close to the stationary reducing power of the commutator operation by ap- 1 distribution in the following sense: each state x is plying it to carefully chosen conjugates τ and π− τπ; reached with probability p(x)e± where p(x) is the the conjugating permutation π is obtained as the re- stationary probability. Here 0 < 1/2; the influence sult of a polynomial-length random walk. of the specific value of on the≤ strong mixing time Our worst-case result brings us close to proving a estimates is a factor of log(1/) which we can ignore polynomial upper bound for the “typical case.” First as long as is bounded away from zero. we need to recall Dixon’s celebrated result: almost The Fact remains true if we allow loops and all pairs of permutations in Sn generate either Sn or multiple edges. The Fact follows from a result of An [Dix] (cf. [Bo, Ba2]). Landau and Odlyzko [LO] stating that the eigenvalue 1 For random pairs of generators, the best diameter gap of the transition matrix is at least (d(∆ + 1)n)− . bound to-date states that the diameter is almost al- The eigenvalue gap refers to the quantity 1 max λ ways at most nln n(1/2+o(1)) [BH]. This expression also where the maximum is taken over all eigenvalues− λ =| 1| describes the order of almost all permutations [ET], so of the transition matrix. Note that 1 is a simple6 this result again suffers from the “element order bot- eigenvalue and the lazyness of the walk guarantees tleneck.” In 99% (but not in almost all) cases we are that λ < 1 for all other eigenvalues. | | By standard arguments, the Landau-Odlyzko Choose π with at most average value of A Aπ . eigenvalue gap implies an O(d∆n log n) strong mixing | ∩ | time. A permutation group G is t-transitive if for any The proof of Theorem 1.1 is based on Lemma 2.2 pair of ordered t-tuples of distinct elements of the below. We first state Lemma 2.1 in order to illustrate permutation domain, (x1, . . . , xt) and (u1, . . . , ut), σ the technique on a simple instance. there exists σ G such that for all i, x = ui. ∈ i Lemma 2.1. Let G = S be a transitive permutation Lemma 2.2. Let G = S be a (t + 1)-transitive group of degree n. Leth δi > 0 be fixed. Let A [n], permutation group of degreeh i n. Fix δ > 0. Let ⊂ A = k. Then there exists a word π of length (x1, . . . , xt) and (u1, . . . , ut) be t-tuples of distinct | | 2 1 O(n S log n) over S S− such that elements of [n]. Let A [n], A = k. Then there | | ∪ exists a permutation π ⊂G expressible| | as a word of π 2 2t+2 ∈ 1 A A (k /n)(1 + δ). length O(tn S log n) over S S− such that | ∩ | ≤ | | ∪ π 2 Note that if π is chosen at random uniformly from (a) A A t0 + (k /n)(1 + δ); G then the expected size of A Aπ is exactly k2/n | ∩ | ≤ ∩ π (cf. [BE]). The content of the Lemma is that we (b) xi = ui (i = 1, ..., t), can produce an intersection nearly this small by a permutation π which is a short word in the generators. where t0 is the number of pairs (xi, ui) which both We shall use a short random walk to find π. belong to A. First we note that without loss of generality, here Note that in our application below, t = 2. and later we may assume S < 2n by throwing away redundant generators. (It| is| shown in [Ba1] that the Proof. Let Γ be the following graph: the vertices of Γ longest subgroup chain in S has length < 2n). are the ordered (t+1)-tuples of distinct elements of [n]; n an edge corresponds to a transition between vertices Proof. Consider the connected undirected graph Γ 1 under some member of S S− . This is a connected (possibly with loops and multiple edges) consisting of undirected graph. ∪ the vertex set [n] and the edges i, iσ , where i [n] 1 { } ∈ Let s be the mixing time of the lazy random and σ S S− . Note that this graph is regular of degree∈ 2 S .∪ This graph is also connected because G walk on Γ, starting from any vertex with parameter is transitive.| | = ln(1 + δ)/2, so δ/2. By the Fact, s = O(tn2t+2 S log n) since Γ≈ is a regular graph of degree Let s denote the strong mixing time of the lazy 2 S with| n|(n 1) (n t) < nt+1 vertices). random walk on Γ. So the lazy random walk on | | − ··· − Let π be a lazy random word of length s over Γ, starting from any vertex, is -nearly uniformly 1 S S− . Let B denote the event that π satisfies distributed over [n] after s steps, i. e., each vertex ∪ condition (b). Then has probability (1/n)e± chance to be visited at time s. Let us choose := ln(1 + δ), so δ. We have s = O(n2 S log n) by the Fact. ≈ | | Prob(B) = (1/n(n 1) (n t + 1))e± , Let π be a lazy random word of length s over − ··· − 1 S S− . Then for any x [n], ∪ ∈ and for any x x1, . . . , xt and u u1, . . . , ut , 6∈ { } 6∈ { }

Prob(xπ A) (k/n)e = (k/n)(1 + δ). ∈ ≤ π Prob(B and x = u) = (1/n(n 1) (n t))e± . − ··· − Adding these probabilities over all x A, we ﬁnd that ∈ Therefore for the conditional probability we have

π 2 π 2 E( A A ) (k /n)(1 + δ). Prob(x = u B) = (1/(n t))e± . | ∩ | ≤ | − Hence, as in the proof of Lemma 2.1, we obtain the total word length bounded by n6 S (log n)O(1), the following bound on the conditional expectation of completing the proof of the diameter bound.| | A Aπ : | ∩ | The justification of the algorithmic claim in The- orem 1.1 follows the lines of the proof of the diameter bound. The first step is to discard redundant π 2 2 E( A A B) t0 + ((k t0) /n)e generators; this can be done in polynomial time us- | ∩ || ≤ − ing Sims’s algorithm ([Si, FHL, Kn], cf. Chaper 4 in 2 = t0 + ((k t0) /n)(1 + δ). − [Se]). Let ξ denote the intersection size produced by our random walk method. For any δ > 0, the proba- Therefore we may choose π to satisfy B as well as bility that ξ > (1 + δ)E(ξ) is less than 1/(1 + δ) by the inequality Markov’s inequality. Let us repeat the random walk until ξ (1+δ)E(ξ) holds; it follows from the preced- π 2 ≤ A A t0 + ((k t0) /n)(1 + δ). ing observation that we shall succeed in an expected | ∩ | ≤ − O(1/δ) trials. So the degree reduction will proceed at the same rate as in the proof of existence above. More precisely, we need to consider the conditional For a permutation σ we use supp(σ) to denote the π expectation of ξ conditioned on xi = ui (i = 1, 2); the support of σ, i. e., the set of elements moved by σ. 2 Note that supp(σ) is the degree of σ. probability that this happens is O(n− ), so the cost | | of taking this condition into account is an additional Proposition 2.1. Let σ, τ be permutations and A = factor of O(n2) in the running time. 1 supp(τ). Let x A and y = xτ . If xσ− A and 1 ∈ ∈ yσ− A then τ and τ σ do not commute. 6∈ 3 Average case 1 Proof. Let φ = τ σ. Since yσ− A, we have 1 1 Lemma 3.1. For every , δ > 0 there exists C = σ− τ σ− φ 6∈ τ τφ y = y , i.e., y = y. In other words, x = x . C(, δ) such that all but an fraction of the n! Assume now that τ and φ commute. We infer that 1 permutations of [n] have a set of at most C cycles xτ = xφτ and therefore x = xφ = xσ− τσ. Hence 1 1 1 whose combined length is greater than n(1 δ). xσ− = xσ− τ , i.e., xσ− A, contrary assumption. − 6∈ The proof of Theorem 1.1 now follows. We use Proof. Follow the argument of the proof of Lemma Lemma 2.2 with A = supp(τ), t = 2, x1, u1 A, τ ∈ 3.1 in [BH]; replace the reference to the Central Limit u2 := u1 A, and x2 A. Then, by Proposition 2.1, condition∈ (b) implies that6∈ τ and τ π do not commute. Theorem by Chebyshev’s Inequality. Note that we have t0 = 1. To prove Theorem 1.1, we use Lemma 2.2 repeat- edly, starting from some τ of degree < n/(3 + ). In Let now σ and τ be two random permutations each round we replace τ by the commutator [τ, τ π] and let G1 be the group generated by σ and τ. Setting where π satisfies conditions (a) and (b) with A = δ = 0.33, we find that by the Lemma, with probability supp(τ); we choose xi and ui as in the preceding para- > 1 , the permutation σ has cycles of lengths graph. Condition (b) guarantees that the new τ is not − n1, . . . , nk where k C for some constant C = C() the identity. Condition (a) implies that the degree of τ ≤ C such that i ni 0.67n. Let N = ni < n decreases rapidly: if the degree of the old τ is k and the P N≥ Q and let σ1 = σ . Now σ1 has degree 0.33n, degree of the new τ is ` then `/n < 3/n+3(k/n)2(1+δ) ≤ and almost surely σ1 = 1 because the order of σ is for an arbitrarily small fixed δ. Since at the start, almost surely nln n(1/2+6 o(1)) [ET]. So with probability k/n < 1/(3 + ) for a fixed > 0, in O(log log n) 1 o(1), the Cayley graph Γ(G1, σ, σ1, τ ) rounds, we shall reach k = 3. has≥ polynomialy− − bounded diameter. But{ obviously} Each round increases the word length by qua- diam Γ(G1, σ, τ ) N diam Γ(G1, σ, σ1, τ ), and drupling it and adding a fixed polynomially bounded the right-hand{ side} ≤ is polynomially{ bounded.} This amount (O(n6 S log n) from Lemma 2.2). This keeps completes the proof of Corollary 1.1. | | 4 Possible extension to almost all pairs of graphs of the symmetric group. J. Combinatorial generators Theory-A 49 (1988), 175–179. [BS2] L. Babai, A.´ Seress: On the diameter of permu- For a permutation σ, let `2 denote the total length tation groups. Europ. J. Comb. 13 (1992), 231–243. of its first two cycles. For a random permutation, [Bo] J. D. Bovey: The probability that some power of a the expected value of `2 is 3(n + 1)/4. Let kn permutation has small degree. Bull. London Math. → ∞ arbitrarily slowly. Let us pick a set S of kn generators Soc. 12 (1980), 47–51. J. D. Dixon: at random and let `2 denote the average of `2(σ) [Dix] The probability of generating the symmetric group. Math. Z. 110 (1969), 199–205. over σ S. Then the standard deviation of `2 is o(n) and∈ therefore, by Chebyshev’s inequality, almost [DF] J. R. Driscoll, M. L. Furst: Computing short generator sequences. Info. and Comput. 72 (1987), surely `2 0.7n. Therefore, by the argument of ≥ 117–132. the preceding section, diam Γ(Sn,S) is almost surely [ET] P. Erdos,˝ P. Turan:´ On some problems of a sta- polynomially bounded. tistical group theory I., Z. Wahrscheinlichkeitstheorie This argument assumes no more than pairwise verw. Geb. 4 (1965), 175–186. independence of the members of S. So if out of a [EG] S. Even, O. Goldreich: The minimum length generator sequence is NP -hard. J. Algorithms 2 (1981), random pair of permutations we could construct kn pairwise independent permutations as short words, 311–313. [FHL] M. L. Furst, J. Hopcropft, E. M. Luks: Poly- we would have proven that for almost all pairs of nomial time algorithms for permutation groups. In: generators, the diameter is polynomially bounded. Proc. 21st IEEE FOCS, IEEE Computer Society, Pairwise independence is surely too much to ex- 1980, pp. 36–41. pect (in fact one can prove that it cannot be at- [Je] M. R. Jerrum: The complexity of finding minimum tained by more than three words in the generators), length generator sequences. Theoretical Computer but it seems plausible that if σ, τ S are in- Science 36 (1985), 265–289. n M-C. Heydemann: dependent random permutations then∈ the permuta- [He] Cayley graphs and interconnection networks. In: Graph Symmetry (Montreal, tions σ, στ, στ 2, . . . , στ kn are pairwise “nearly inde- 1996), NATO Adv. Sci. Inst. Ser. C, Math. Phys. pendent.” What we would need is a concept of near- Sci. 497, Kluwer Acad. Publ., Dordrecht 1997, pp. independence of pairs of permutations appropriate for 167–224. this plan to go through. [Kn] D. E. Knuth: Notes on efficient representation of perm groups. Combinatorica 11 (1991), 57–68. References [Ko] R. E. Korf: Finding optimal solutions to Rubik’s Cube using pattern databases. In: Proc. 14th Nat. Conf. on Artificial Intelligence (AAAI-97), Amer. [Ba1] L. Babai: On the length of subgroup chains in Assoc. for Artificial Intelligence, 1997, pp. 700-705. the symmetric group. Communications in Algebra 14 [La] E. Landau: Handbuch der Lehre von der Verteilung (1986), 1729–1736. der Primzahlen. Bd. I, Teubner, Leipzig, 1909. [Ba2] L. Babai: The probability of generating the sym- [LO] H. J. Landau, A. M. Odlyzko: Bounds for eigen- metric group. J. Comb. Theory – A 52 (1989), 148– values of certain stochastic matrices. Linear Algebra 153. and its Appl. 38 (1981), 5–15. [BE] L. Babai, P. Erdos:˝ Representation of group ele- [McK] P. McKenzie: Permutations of bounded degree ments as short products, in: “Theory and Practice of generate groups of polynomial diameter. Info. Proc. Combinatorics” (A. Rosa, G. Sabidussi, J. Turgeon, Lett. 19 (1984), 253–254. eds.), Ann. Discr. Math. 12 (1982), 21-26. [Se] A.´ Seress: Permutation Group Algorithms. Cam- [BH] L. Babai, G. L. Hetyei: On the diameter of ran- bridge Univ. Press, 2003. dom Cayley graphs of the symmetric group. Combi- [Si] C. C. Sims: Computation with permutation groups. natorics, Probability, and Computing 1 (1992), 201- In: Proc. Second Symposium on Symbolic and Alge- 208. braic Manipualtion, ACM Press 1971, pp. 23–28. [BHKLS] L. Babai, G. Hetyei, W.M. Kantor, A. Lubotsky, A.´ Seress: On the diameter of finite groups. 31st IEEE FOCS, St. Louis MO, 1990, pp. 857–865. [BS1] L. Babai, A.´ Seress: On the diameter of Cayley