Compositions of Random Functions on a Finite Set

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1 Compositios of Radom Fuctios o a Fiite Set Aviash Dalal MCS Departmet, Drexel Uiversity Philadelphia, Pa. 904 ADalal@drexel.edu Eric Schmutz Drexel Uiversity ad Swarthmore College Philadelphia, Pa., 904 Eric.Joatha.Schmutz@drexel.edu Submitted: July, 00; Accepted: July 9, 00 MR Subject Classificatios: 60C05, 60J0, 05A6, 05A05 Abstract If we compose sufficietly may radom fuctios o a fiite set, the the composite fuctio will be costat. We determie the umber of compositios that are eeded, o average. Choose radom fuctios f,f,f 3,... idepedetly ad uiformly from amog the fuctios from [] ito []. For t >, let g t = f t f t f be the compositio of the first t fuctios. Let T be the smallest t for which g t is costat(i.e. g t (i) =g t (j) for all i, j). We prove that E(T ) as,wheree(t ) deotes the expected value of T. Itroductio If we compose sufficietly may radom fuctios o a fiite set the the composite fuctio is costat. We ask how log this takes, o average. More precisely, let U be the set of fuctios from [] to[]. Let A be the elemet subset of U cosistig of the costat fuctios: g A iff g(i) =g(j) for all i, j. Let f,f,f 3,...be a sequece of radom fuctios chose idepedetly ad uiformly from U.Letg = f,adfort>letg t = f t g t be the compositio of the first t radom maps. Defie T ( f i i= ) to be the smallest t for which g t A. (If o such t exists, defie T =. It is ot difficult to show that Pr(T = ) =0.) Our goal i this paper is to estimate E(T ). It is atural to restate the problem as a questio about a Markov chai. The state space is S = {s,s,...,s }. For t>0adr [], we are i state s r if ad oly if g t has exactly r elemets i its rage. With the covetio that g 0 is the idetity permutatio, we start i state s at time t = 0. The questio is how log (i.e. how may compositios) it takes to reach the absorbig state s. For m>, let τ m = {t : Rage(g t ) = m} betheamoutoftimewearei state s m.thust = τ m.lett cosist of those states that are actually visited: the electroic joural of combiatorics 9()(00), #R6

2 for m>, s m T iff τ m > 0. The visited states T are a (o-uiform) radom subset of S that icludes at least two elemets, amely s ad (with probability ) s. We prove later that T typically cotais most of the small umbered states ad relatively few of the large umbered states. This observatio forms the basis for our proof of Theorem E(T )=( + o()) as. We should metio that there is a stadard approach to our problem usig the trasitio matrix P ad liear algebra. Let Q be the matrix that is obtaied from P by strikig out the first row ad colum of P.TheE(T ) is exactly the sum of the etries i the last row of (I Q). See, for example, chapter 3 of [5]. This fact is very coveiet if oe wishes to compute E(T ) for specific small values of. A aoymous referee cojectured that E(T ) = 3 + o() after observig that, for small values of, E(T ) +3. This cojecture is plausible, but we are owhere ear a proof. The Trasitio Matrix The trasitio matrix P ca be determied quite explicitly. Suppose g t has i elemets i its rage, How may fuctios f have the property that f g t has exactly j elemets i its rage? There are ( j) ways to choose the j-elemet rage of f g t,ads(i, j)j! waystomapthei-elemet rage of g t otoagivej elemet set. (Here S(i, j) is the umber of ways to partitio a i elemet set ito j disjoit subsets, a Stirlig umber of the secod kid.) Fially, there are i elemets i the complemet of the rage of g t,ad i ways to map them ito []. Thus there are ( j) S(i, j)j! i fuctios f with the desired property, ad for i, j, the trasitio matrix for the chai has i, j th etry P (i, j) = ( j ) S(i, j)j! i. () The statioary distributio π assigs probability to s. The trasitio matrix has some ice properties. It is lower triagular, which meas the eigevalues are just the diagoal etries: for m, λ m = P (m, m) = m k=0 ( k ). () For future referece we record two simple estimates for the eigevalues, both of which follow easily from (). Lemma ad ( m ) λ m = + O(m4 ) ( ) m λ m exp( /). the electroic joural of combiatorics 9()(00), #R6

3 3 Lower Boud The proof of the lower boud requires a estimate for the Stirlig umbers S(m, k). The literature cotais may precise but complicated estimates for these umbers. Here we prove a crude iequality whose simplicity makes it coveiet for our purposes. Lemma 3 For all positive itegers m ad k, S(m, k) (k) m. Proof: The proof of this lemma will be doe by iductio usig the recurrece S(m, k) =S(m,k ) + ks(m,k). Whe k =, we kow that S(m, ) = ad (k) m = m. So clearly the iequality holds true for k = (for all positive itegers m). Now let φ m deote the followig statemet: for all k>, S(m, k) (k) m. It suffices to prove that φ m is true for all m. Form =,S(,k)=0k for all k>. Now let k>ad assume, iductively, that φ m is true (i.e. S(m,k) (k) m for k>.) Thewehave S(m, k) =S(m,k ) + ks(m,k) ((k )) m + k(k) m { } =(k) m (k )m + k m. Realize that the quatity iside the large braces is less tha oe. With lemma 3 available, we ca proceed with the proof that E(T ) (+o()). Sice T = τ m,wehave E(T )= Pr(s m T)E(τ m s m T). (3) Obviously a lower boud is obtaied by trucatig this sum. To simplify otatio, let l = log log. The l E(T ) Pr(s m T)E(τ m s m T). (4) To estimate the secod factor i each term of (4), ote that Applyig lemma, we get E(τ m s m T)= t= tλ t m ( λ m )= λ m. (5) E(τ m s m T)= ( m )( + O( m )). (6) To estimate the first factor of each term i (4), we make the followig observatio: if s m T, the there is a trasitio from s m+d to s m j for some positive itegers d ad j. Hece, the electroic joural of combiatorics 9()(00), #R6 3

4 Pr(s m T )= m d= m j= Pr(s m+d T) P (m + d, m j). (7) ( λ m+d ) (The factor ( λ m+d ) = i=0 P (m + d, m + d) i is there because we remai i state s m+d for some umber of trasitios i 0beforemovigotostates m j.) Let σ := m m S(m+d,m j) λ m j d= j= j+d λ m+d. Puttig () ad Pr(s m+d T) ito (7), we get Pr(s m T ) m d= m j= ( ) S(m + d, m j)(m j)! m j m+d = σ. (8) ( λ m+d ) A first step i boudig σ is to ote that > ( )=λ λ 3 λ 4... λ > 0, ad therefore λ m j =. λ m+d λ m+d λ Hece m m S(m + d, m j) σ ( ) d d= j= j. Applyig lemma 3 to each term of the iside sum, we get m j= S(m + d, m j) j m j= ((m j)) m+d j Hece σ ( ) l(l)l m(m )m+d m d= < l(l)l+d. ( l )d = O( (l)l+ )=o(). Thus Pr(s m T) o() for all m l, Puttig this ad (6) back ito (4), l ad usig the fact that ( m ) = l ( m m )= l, we get the lower boud E(T ) ( + o()). 4 Upper Boud If Rage(g t ) = m, the the restrictio of f t to Rage(g t ) is a radom fuctio from a m elemet set to []. Before provig that E(T ) ( + o()), we gather a simple lemma about the size of the size of the rage for such radom maps. Lemma 4 Suppose h : [m] [] is selected uiformly at radom from amog the m fuctios from [m] ito [],ad let R be the cardiality of the rage of h. The the mea ad variace of R are respectively E(R) = ( )m ad Var(R) = {( )m ( )m } + {( )m ( )m }. the electroic joural of combiatorics 9()(00), #R6 4

5 Proof: Let U = R = I i,wherei i is if i is ot i the rage of h, ad i= otherwise I i is zero. The E(R) = E(U), ad Var(R) =Var(U). E(U) =E(I )=( )m. (9) E(U )= i j E(I i I j )+E(U) = ( )( )m + E(U). Therefore { Var(U) = ( )m ( } )m + {( )m ( } )m. The ext corollary shows that there are gaps betwee the large states i T. Let ξ = log,adletβ = β() = (ξ + ( )ξ ). Although β is quite large (β ) all we really eed for our purposes is that β as. log 4 Corollary 5 Pr(s m δ T for δ β s m T)= o() uiformly for ξ m. Proof: Suppose we are i state s m at time t ad select the ext fuctio f t. Let h be the restrictio of f t to the rage of g t,adletr be the cardiality of the rage of h, adletb = m R. Observe that if B>βthe the ext β states are missed: s m δ T for δ β. Note that E(B) =m + ( )m > β. Applyig Chebyshev s iequality to the radom variable B, weget Pr(B β) Pr(B 4Var(B) E(B)) (E(B)). (0) For ξ m,wehavee(b) =m +( )m ξ +( )ξ log 4. (A calculus exercise shows that E(B) is a icreasig fuctio of m.) To boud Var(B) otethat, Therefore (0) yields ( )m ( )m = O( m ). Pr(B β) =O( m log8 )=o(). Now we proceed with the proof of the upper boud E(T ) ( + o()). Split the sum (3) ito three separate sums as follows. Let ξ = log,adletξ = log, so that (3) becomes the electroic joural of combiatorics 9()(00), #R6 5

6 E(T )= ξ + ξ m=ξ + + m=ξ + The first sum i () is estimated usig (5), lemma, ad the fact that Pr(s m T ) : ξ Pr(s m T)E(τ m s m T) = ξ ( m ) =(+O( ξ )) ξ m4 + O( ) ξ λ m ( m ) =( + o()). The secod sum i () is estimated usig a crude boud o the eigevalues. For ξ <m ξ,wehaveλ m λ ξ = log + O( ). Hece the secod log sum i () is at most ξ m=ξ + λ m λ ξ ξ m=ξ = O(ξ log ) =O( log ). For the last sum i (), we ca o loger get away with the trivial estimate Pr(s m T ). However ow the size of the eigevalues ca be hadled less carefully: m=ξ + Pr(s m T) λ m The first factor i () is easily estimated usig (): max = m ξ λ m λ ξ () ( )( ) max Pr(s m T). () m ξ λ m m=ξ exp( ( ξ ) /) for all sufficietly large. To deal with the secod factor i () we use Corollary 5. The idea is that there caot be too may hits (visited states) simply because every time there is a hit it is followed by β misses. To make this precise, defie V = χ m,whereχ m m=ξ is if s m T ad 0 otherwise. Thus the secod factor i () is just E(V ). Also cout large umbered states that are ot i T with W = ( χ m )sothat m=ξ W + V = + ξ ad E(V )= + ξ E(W ). If a state s m is i T,adif the ext β possible states s m,s m,...,s m β are ot i T, the those β missed states together cotribute exactly β to W. the electroic joural of combiatorics 9()(00), #R6 6

7 β If we let J m = χ m ( χ m δ ), the W β J m. But the δ= m ξ E(W ) β E(J m )= m ξ β Pr(s m T)Pr(s m,s m,...s m β T s m T). m ξ By Corollary 5, Pr(s m,s m,...s m β T s m T)= o(). Hece But the E(W ) β( + o()) Pr(s m T)=(+o())βE(V ). m=ξ which implies that E(V )= + ξ E(W ) + ξ β( + o()))e(v ), E(V ) + ξ +β( + o()) = O(log4 ). Thus the secod factor of () is o(), which meas that the third sum i () is egligible. Refereces [] D.Aldous ad J.Fill, Reversible Markov Chais ad Radom Walks o Graphs [] P.Diacois ad D.Freedma, Iterated Radom Fuctios, SIAM Review 4 No., p [3] J.C.Hase ad J.Jaworski, Large Compoets of Radom Mappigs, Radom Structures ad Algorithms 7 (000) [4] J.Kemey, J.L.Sell, ad A.W.Kapp, Deumerable Markov Chais, Va Nostrad Co., 966. [5] J.G.Kemey, J.L.Sell, Fiite Markov Chais, Spriger Verlag, 976. [6] J.Jaworski, A Radom Bipartite Mappig, Aals of Discrete Math., (985). [7] V.F.Kolchi, Radom Mappigs, Optimizatio Software, 986. [8] V.F.Kolchi, B.A.Sevastyaov, ad V.P.Chistaykov, Radom Allocatios, Wisto, 978. [9] J. S. Rosethal, Covergece Rates for Markov Chais, SIAM Review the electroic joural of combiatorics 9()(00), #R6 7