- Email: [email protected]
On the number of predecessors in constrained random mappings
t--
y, s
t~
;-.' jJl
ELSEVIER
IJI'Y Statistics & Probability Letters 36 (1997) 29-34
On the number of predecessors in constrained random mappings Bernhard Gittenberger Department of Algebra and Discrete Mathematics, TU Wien, WiednerHauptstrasse 8-10/118, A-1040 Wien. Austria Received 1 April 1996; received in revised form 1 December 1996
Abstract We consider random mappings from an n-element set into itself with constraints on coalescence as introduced by Arney and Bender. A local limit theorem for the distribution of the number of predecessors of a random point in such a mapping is presented by using a generating function approach and singularity analysis. © 1997 Elsevier Science B.V.
Keywords: Random mappings; Generating functions
1. Introduction For Mn = {1,2,...,n} let F, denote the set of all functions from Mn into itself equipped with the uniform distribution. Then an element of F, is called a random mapping. Each random mapping f can be represented by a functional graph, i.e. the graph consisting of the nodes 1,2 . . . . . n, and of the edges ( i , f ( i ) ) , i = 1..... n. Various characteristics of random mappings have been studied. See e.g. Amey and Bender (1982), Drmota and Sofia (1995, 1997), Flajolet and Odlyzko (1989), Harris (1960), Kolchin (1986), Mutafciev (1989) and Stepanov (1969). Arney and Bender (1982) examined a more general model: They considered mappings such that the number of preimages of every point lies in a given set D of nonnegative integers (with 0 E D) or, equivalently, the degrees of the nodes of the functional graph have to be in D. Let F,° denote the set of those mappings on Mn. Arney and Bender derived the distributions of many random mapping characteristics for Fn° mainly by means of combinatorial counting arguments. In this paper we will study the distribution of a further random mapping characteristic using a generating function approach. Let x E M~ and f C Fff. y E M~ is called a predecessor of x if there exists a j > 0 such that the j-th iterate of f applied on y yields x. Denote the number of predecessors of x by co(x). Then we will show Theorem 1. Let dp(x)=~ksoxk/k! and fl be the positive root of fldp'(fl)= dp(fl). Furthermore, define 2 := r2 ~p"(fl)/c~(fl) and d : : gcd{k: k c D}. Then for a randomly chosen point of a mapping of F~ the expected 0167-7152/97/$17.00 (~ 1997 Elsevier Science B.V. All rights reserved PH S0167-7152(97 )00045-X
B. Gittenberger / Statistics & Probability Letters 36 (1997) 2 9 - 3 4
30
number o f predecessors equals ~ v / - ~ 2 2 and moreover a local limit theorem holds." I f dlr then d
P[co = r] = V/2~2r3( 1 _ ( r / n ) ) ( 1 + o(1)) f o r r ---+oo and n - r ---* oo. Otherwise P[o) = r] = 0
Remark. For the special cases D = {0, 1,2 .... } and D = {0,k} the above result was obtained by Rubin and Sitgreaves (1954).
2. Combinatorial background The basic concept which our method relies on is that of combinatorial constructions described e.g. in Vitter and Flajolet (1990): Let ~¢ be a set of combinatorial objects where each object ~r C J has a size ]~r[. If an denotes the number of objects in d having size n, then .'~(z) = E
an zn
n~>O
is called the (exponential) generating function (GF) of d . The advantage of the generating function approach is the fact that there is a direct correspondence between operations on combinatorial constructions and on GFs (see Vitter and Flajolet, 1990). Using the graph representation of random mappings it is easy to see that they may be regarded as multisets of cycles of Cayley trees, i.e. labelled rooted trees. Let d denote the set of Cayley trees. Then d can be defined recursively by the symbolic equation d
= o * mset(~4),
where o represents a node. Analogously, we get for the set of random mappings, ~ : = mset(cycle(d))
This implies the following equations for the corresponding GFs: The GF of Cayley trees is given by a(z ) =- ze a(z)
and the GF of random mappings
1 f ( z ) - 1 - a(z)"
As we want to study the random mappings with constraints on coalescence, we have to modify a(z) properly and obtain a(z) =zc~(a(z)),
where q~(z) = E kED
zk
B. Gittenberoer I Statistics & Probability Letters 36 (1997) 29-34
31
and the GF of constrained random mappings is given by 1 f ( z ) - 1 - b(z)'
where b(z)=zck'(a(z))
since the root of each tree has an additional predecessor coming from the cycle. Let C'nk -{r) denote the number of mappings in Fff which have exactly k points x satisfying o~(x)= r, where ~(X) is the number of predecessors of the point x. The number of all points in all mappings of F~ that satisfy this equation is denoted by b~r). Hence we have 1
IF~[ n b~r)
=
P { o = r}.
Thus for establishing the local limit theorem we have to compute b (r). Obviously the relation b(n~)= Z " KCnk (r) k~l holds. Therefore b~r) can be calculated by
b~r)=
~.
(Cr)u(Z, 1),
(1)
where [zn]f denotes the coefficient of z n in the power series of f and (er)u denotes the partial derivative with respect to u. Cr(Z, U) is defined by
n,k >~O and can be obtained by marking the nodes we are keeping track of in the functional graphs which corresponds to introducing a further variable in the GF. Further examples of these marking techniques in combinatorial constructions can be found in Drmota and Sofia (1995, 1997). (1) shows that in order to get the asymptotic behaviour of b~~) we have to evaluate the coefficients of certain power series. In order to do this we will use the following theorem by Flajolet and Odlyzko (1990). Theorem 2. Let f ( z ) be analytic in the domain
A = {z/Izl
+ . , larg(z - s)l >14J},
where s, q are positive real numbers and 0 < c~ < x/2. Furthermore let ~( u ) = u ~ log/~ u, ~ ~ {0, - 1, - 2 .... }. Then the Taylor coefficients o f f satisfy f(z)~a
for z---~s in A ~
[ z " ] f ( z ) ~ snnF(~ ).
Analogous formulas hold for (9 and o instead o f ~.
3. Proof of Theorem 1 For convenience, define ar = [zr]a(z), bmr = [zr]b(z) m and assume that d = 1. b, = [z"]l/(1 - b(z)). First of all, let us set up the GF cr(z,u). Therefore we have to mark all nodes that are roots of a tree with size r. This leads to the following modification of a(z): t,.(z, U) = z~(tr(Z, U)) + (U -- 1 )ar zr.
B. Gittenberger / Statistics & Probability Letters 36 (1997) 2 9 - 3 4
32
Moreover, we have to take into account that the set of predecessors of a node which belongs to a eycle consists of all nodes in the component. Hence in a component of size r all nodes in the cycle have to be marked. For components containing a cycle of length m this yields the GF (z~)t(tr(Z, U) ) ) m
m
1--(urn _ 1)zrbmr. m
Thus we have
1
Cr(Z,u)= 1 - zr~t(tr(Z,U)) exp
Zr
--bmrm
1)
m=l
and consequently
(Cr)u(Z, 1 ) = arzr+l cy'(a(z)) q
r b mr zr ~-'~m=l
Note that bmr
=
(2)
1 - b(z)
(1 - b(z)) 3
0 if m = 0 or m > r and thus 1
bmr = [zr] 1 - b(z-------~=
bmr ~m=l
br.
m=O
Hence the functions a(z) and b(z) contain the information we need. a(z) has a singularity p on the circle of convergence which is determined by
a(p) = pc~(a(p)),
1 = pq~'(a(p)).
To proceed we will need the following lemma which is an immediate consequence of (Evgrafov 1966, Theorem 7.1):
Lemma 1. Let F(z, y) be continuous in {(x,y): [z-zo] < r l , lY-Y0[
Thus in the vicinity of p a(z) admits the following local expansion:
a(z)~fl-V-~
1--=fl-p
~--
1--,p
asz---~p,
where fl=a(p) and 2 = fl2~"([3)/~([3). By expanding ~b(z) we immediately obtain b(z),-~l-p~/249(fl)c~"(fl) ~1
. .z . .
1 x/~ ~l
P
z
a s z---+p
P
and thus
1 1 ( I ~ '/2 ,-~ ~ \ ~ . ] 1 b(z)
as z --~ p.
(3)
B. Gittenberger / Statistics & Probability Letters 36 (1997) 29-34
33
Now we are able to apply Theorem 2 and get fl
.r ~ Pr
r3
and
b~
1
(4)
as r --+ oo.
Using this we finally get
b~r) 1 IFD[n -- nbn [z"](C,)u(2, 1) pnv/2"~ ( f l [zn_r] n ,--,.a -
1 ( 1 - b(z)) 3
+tz l n - r
1
tz r
,
-
n
p ~- -
p,_r 2V/~-~
1
\
,
+ pr~
)
pn-rV/2n2( n _ r)
1
V/27t2r3(1
-
r/n)
What remains to be done is the calculation of the mean value/~n: By (2) we have 1
n
#n : -~n [Z ] Z r ( C r ) u ( ~, l ) r~O
l{~b'(a(Z))z2a'(z)+l(1)'
= nbn[Z n]
~k(]~b--~3
,,
Using the functional equation of
a'(z) =
z(1
a(z)
~ z
1-b(z)
) "
we immediately get
a(z) b(z))" -
This implies #"=
nTn
{ 2z~"(a(z))a(z)
[z'] \
(1
-b(z))
Now observe that for z ~ p Theorem 2 #,=
4
z~'_(a(z ))
+ (1 - b ( z ) ) 3 ] "
we have
zq~"(a(z))a(z)~ 2.
Thus we get by using (3) and (4) and applying
1 +C
and the proof is complete. Remark 1. In an analogous way it is possible to reobtain Amey and Bender's (1982) results by the method presented here or to derive local limit theorems for other random mapping characteristics. Note, for instance, that recently Baron et al. (1996) derived the missing distribution of Arney and Bender (1982, Table II) by using a generating function approach. Remark 2. It should be mentioned that the coefficient of coalescence 2 which occurs in the distributions of several random mapping characteristics has a simple probabilistic interpretation, as Aldous pointed out: If we consider the trees which build up the random mappings as representations of Galton-Watson branching processes, then the offspring distribution is determined by the tree function ~b(z) and 2 is equal to its variance.
34
B. Gittenberoer / Statistics & Probability Letters 36 (1997) 29 34
References Amey, J., Bender, E.A., 1982. Random mappings with constraints on coalescence and number of origins. Pacific J. Math. 103, 269-294. Baron, G., Drmota, M., Mutafchief, L., 1996. Predecessors in random mappings. Combin. Probab. Comput. 5, 317-335. Drmota, M., Sofia, M., 1995. Marking in combinatorial constructions: generating functions and limiting distributions. Theoret. Comput. Sci. 144, 67-99. Drmota, M., Sofia, M., 1997. Marking in combinatorial constructions: images and preimages in random mappings. SIAM J. Discrete Math. 10(2), 246-269. Evgrafov, E.A., 1966. Analytic Functions. Dover, New York. Flajolet, P., Odlyzko, A.M., 1990. Random mapping statistics. In: Quisquater, J.-J., Vandewalle, J., (Eds.), Advances in Cryptology, Lecture Notes in Computer Science, vol. 434, Springer, Berlin, pp. 329-354. Proceedings of Eurocrypt '89, Houtalen, Belgium, April 1989. Flajolet, P., Odlyzko, A.M., 1990. Singularity analysis of generating functions. SIAM J. Discrete Math. 3(2), 216-240. Hams, B., 1960. Probability distributions related to random mappings. Aim. Math. Statist. 31(2), 1045-1062. Kolchin, V.F., 1986. Random Mappings. Optimization Software, Inc., New York. Mutafciev, L.R., 1989. The limit distribution of the number of nodes in low strata of random mappings. Statist. Probab. Lett. 7, 247-251. Rubin, H., Sitgreaves, R., 1954. Probability distributions related to random transformations of a finite set. Technical Report No. 19a, Applied Mathematics and Statistics Laboratory, Stanford. Stepanov, V.E., 1969. Limit distributions of certain characteristics of random mappings. Theory Probab. Appl. 14, 612-626. Vitter, J.S., Flajolet, P., 1990. Average-case analysis of algorithms and data structures. In: van Leeuwen, J. (Ed.), Handbook of Theoretical Computer Science, vol. A: Algorithms and Complexity. North-Holland, Amsterdam, ch. 9, pp. 431-524.
y, s
t~
;-.' jJl
ELSEVIER
IJI'Y Statistics & Probability Letters 36 (1997) 29-34
On the number of predecessors in constrained random mappings Bernhard Gittenberger Department of Algebra and Discrete Mathematics, TU Wien, WiednerHauptstrasse 8-10/118, A-1040 Wien. Austria Received 1 April 1996; received in revised form 1 December 1996
Abstract We consider random mappings from an n-element set into itself with constraints on coalescence as introduced by Arney and Bender. A local limit theorem for the distribution of the number of predecessors of a random point in such a mapping is presented by using a generating function approach and singularity analysis. © 1997 Elsevier Science B.V.
Keywords: Random mappings; Generating functions
1. Introduction For Mn = {1,2,...,n} let F, denote the set of all functions from Mn into itself equipped with the uniform distribution. Then an element of F, is called a random mapping. Each random mapping f can be represented by a functional graph, i.e. the graph consisting of the nodes 1,2 . . . . . n, and of the edges ( i , f ( i ) ) , i = 1..... n. Various characteristics of random mappings have been studied. See e.g. Amey and Bender (1982), Drmota and Sofia (1995, 1997), Flajolet and Odlyzko (1989), Harris (1960), Kolchin (1986), Mutafciev (1989) and Stepanov (1969). Arney and Bender (1982) examined a more general model: They considered mappings such that the number of preimages of every point lies in a given set D of nonnegative integers (with 0 E D) or, equivalently, the degrees of the nodes of the functional graph have to be in D. Let F,° denote the set of those mappings on Mn. Arney and Bender derived the distributions of many random mapping characteristics for Fn° mainly by means of combinatorial counting arguments. In this paper we will study the distribution of a further random mapping characteristic using a generating function approach. Let x E M~ and f C Fff. y E M~ is called a predecessor of x if there exists a j > 0 such that the j-th iterate of f applied on y yields x. Denote the number of predecessors of x by co(x). Then we will show Theorem 1. Let dp(x)=~ksoxk/k! and fl be the positive root of fldp'(fl)= dp(fl). Furthermore, define 2 := r2 ~p"(fl)/c~(fl) and d : : gcd{k: k c D}. Then for a randomly chosen point of a mapping of F~ the expected 0167-7152/97/$17.00 (~ 1997 Elsevier Science B.V. All rights reserved PH S0167-7152(97 )00045-X
B. Gittenberger / Statistics & Probability Letters 36 (1997) 2 9 - 3 4
30
number o f predecessors equals ~ v / - ~ 2 2 and moreover a local limit theorem holds." I f dlr then d
P[co = r] = V/2~2r3( 1 _ ( r / n ) ) ( 1 + o(1)) f o r r ---+oo and n - r ---* oo. Otherwise P[o) = r] = 0
Remark. For the special cases D = {0, 1,2 .... } and D = {0,k} the above result was obtained by Rubin and Sitgreaves (1954).
2. Combinatorial background The basic concept which our method relies on is that of combinatorial constructions described e.g. in Vitter and Flajolet (1990): Let ~¢ be a set of combinatorial objects where each object ~r C J has a size ]~r[. If an denotes the number of objects in d having size n, then .'~(z) = E
an zn
n~>O
is called the (exponential) generating function (GF) of d . The advantage of the generating function approach is the fact that there is a direct correspondence between operations on combinatorial constructions and on GFs (see Vitter and Flajolet, 1990). Using the graph representation of random mappings it is easy to see that they may be regarded as multisets of cycles of Cayley trees, i.e. labelled rooted trees. Let d denote the set of Cayley trees. Then d can be defined recursively by the symbolic equation d
= o * mset(~4),
where o represents a node. Analogously, we get for the set of random mappings, ~ : = mset(cycle(d))
This implies the following equations for the corresponding GFs: The GF of Cayley trees is given by a(z ) =- ze a(z)
and the GF of random mappings
1 f ( z ) - 1 - a(z)"
As we want to study the random mappings with constraints on coalescence, we have to modify a(z) properly and obtain a(z) =zc~(a(z)),
where q~(z) = E kED
zk
B. Gittenberoer I Statistics & Probability Letters 36 (1997) 29-34
31
and the GF of constrained random mappings is given by 1 f ( z ) - 1 - b(z)'
where b(z)=zck'(a(z))
since the root of each tree has an additional predecessor coming from the cycle. Let C'nk -{r) denote the number of mappings in Fff which have exactly k points x satisfying o~(x)= r, where ~(X) is the number of predecessors of the point x. The number of all points in all mappings of F~ that satisfy this equation is denoted by b~r). Hence we have 1
IF~[ n b~r)
=
P { o = r}.
Thus for establishing the local limit theorem we have to compute b (r). Obviously the relation b(n~)= Z " KCnk (r) k~l holds. Therefore b~r) can be calculated by
b~r)=
~.
(Cr)u(Z, 1),
(1)
where [zn]f denotes the coefficient of z n in the power series of f and (er)u denotes the partial derivative with respect to u. Cr(Z, U) is defined by
n,k >~O and can be obtained by marking the nodes we are keeping track of in the functional graphs which corresponds to introducing a further variable in the GF. Further examples of these marking techniques in combinatorial constructions can be found in Drmota and Sofia (1995, 1997). (1) shows that in order to get the asymptotic behaviour of b~~) we have to evaluate the coefficients of certain power series. In order to do this we will use the following theorem by Flajolet and Odlyzko (1990). Theorem 2. Let f ( z ) be analytic in the domain
A = {z/Izl
+ . , larg(z - s)l >14J},
where s, q are positive real numbers and 0 < c~ < x/2. Furthermore let ~( u ) = u ~ log/~ u, ~ ~ {0, - 1, - 2 .... }. Then the Taylor coefficients o f f satisfy f(z)~a
for z---~s in A ~
[ z " ] f ( z ) ~ snnF(~ ).
Analogous formulas hold for (9 and o instead o f ~.
3. Proof of Theorem 1 For convenience, define ar = [zr]a(z), bmr = [zr]b(z) m and assume that d = 1. b, = [z"]l/(1 - b(z)). First of all, let us set up the GF cr(z,u). Therefore we have to mark all nodes that are roots of a tree with size r. This leads to the following modification of a(z): t,.(z, U) = z~(tr(Z, U)) + (U -- 1 )ar zr.
B. Gittenberger / Statistics & Probability Letters 36 (1997) 2 9 - 3 4
32
Moreover, we have to take into account that the set of predecessors of a node which belongs to a eycle consists of all nodes in the component. Hence in a component of size r all nodes in the cycle have to be marked. For components containing a cycle of length m this yields the GF (z~)t(tr(Z, U) ) ) m
m
1--(urn _ 1)zrbmr. m
Thus we have
1
Cr(Z,u)= 1 - zr~t(tr(Z,U)) exp
Zr
--bmrm
1)
m=l
and consequently
(Cr)u(Z, 1 ) = arzr+l cy'(a(z)) q
r b mr zr ~-'~m=l
Note that bmr
=
(2)
1 - b(z)
(1 - b(z)) 3
0 if m = 0 or m > r and thus 1
bmr = [zr] 1 - b(z-------~=
bmr ~m=l
br.
m=O
Hence the functions a(z) and b(z) contain the information we need. a(z) has a singularity p on the circle of convergence which is determined by
a(p) = pc~(a(p)),
1 = pq~'(a(p)).
To proceed we will need the following lemma which is an immediate consequence of (Evgrafov 1966, Theorem 7.1):
Lemma 1. Let F(z, y) be continuous in {(x,y): [z-zo] < r l , lY-Y0[
Thus in the vicinity of p a(z) admits the following local expansion:
a(z)~fl-V-~
1--=fl-p
~--
1--,p
asz---~p,
where fl=a(p) and 2 = fl2~"([3)/~([3). By expanding ~b(z) we immediately obtain b(z),-~l-p~/249(fl)c~"(fl) ~1
. .z . .
1 x/~ ~l
P
z
a s z---+p
P
and thus
1 1 ( I ~ '/2 ,-~ ~ \ ~ . ] 1 b(z)
as z --~ p.
(3)
B. Gittenberger / Statistics & Probability Letters 36 (1997) 29-34
33
Now we are able to apply Theorem 2 and get fl
.r ~ Pr
r3
and
b~
1
(4)
as r --+ oo.
Using this we finally get
b~r) 1 IFD[n -- nbn [z"](C,)u(2, 1) pnv/2"~ ( f l [zn_r] n ,--,.a -
1 ( 1 - b(z)) 3
+tz l n - r
1
tz r
,
-
n
p ~- -
p,_r 2V/~-~
1
\
,
+ pr~
)
pn-rV/2n2( n _ r)
1
V/27t2r3(1
-
r/n)
What remains to be done is the calculation of the mean value/~n: By (2) we have 1
n
#n : -~n [Z ] Z r ( C r ) u ( ~, l ) r~O
l{~b'(a(Z))z2a'(z)+l(1)'
= nbn[Z n]
~k(]~b--~3
,,
Using the functional equation of
a'(z) =
z(1
a(z)
~ z
1-b(z)
) "
we immediately get
a(z) b(z))" -
This implies #"=
nTn
{ 2z~"(a(z))a(z)
[z'] \
(1
-b(z))
Now observe that for z ~ p Theorem 2 #,=
4
z~'_(a(z ))
+ (1 - b ( z ) ) 3 ] "
we have
zq~"(a(z))a(z)~ 2.
Thus we get by using (3) and (4) and applying
1 +C
and the proof is complete. Remark 1. In an analogous way it is possible to reobtain Amey and Bender's (1982) results by the method presented here or to derive local limit theorems for other random mapping characteristics. Note, for instance, that recently Baron et al. (1996) derived the missing distribution of Arney and Bender (1982, Table II) by using a generating function approach. Remark 2. It should be mentioned that the coefficient of coalescence 2 which occurs in the distributions of several random mapping characteristics has a simple probabilistic interpretation, as Aldous pointed out: If we consider the trees which build up the random mappings as representations of Galton-Watson branching processes, then the offspring distribution is determined by the tree function ~b(z) and 2 is equal to its variance.
34
B. Gittenberoer / Statistics & Probability Letters 36 (1997) 29 34
References Amey, J., Bender, E.A., 1982. Random mappings with constraints on coalescence and number of origins. Pacific J. Math. 103, 269-294. Baron, G., Drmota, M., Mutafchief, L., 1996. Predecessors in random mappings. Combin. Probab. Comput. 5, 317-335. Drmota, M., Sofia, M., 1995. Marking in combinatorial constructions: generating functions and limiting distributions. Theoret. Comput. Sci. 144, 67-99. Drmota, M., Sofia, M., 1997. Marking in combinatorial constructions: images and preimages in random mappings. SIAM J. Discrete Math. 10(2), 246-269. Evgrafov, E.A., 1966. Analytic Functions. Dover, New York. Flajolet, P., Odlyzko, A.M., 1990. Random mapping statistics. In: Quisquater, J.-J., Vandewalle, J., (Eds.), Advances in Cryptology, Lecture Notes in Computer Science, vol. 434, Springer, Berlin, pp. 329-354. Proceedings of Eurocrypt '89, Houtalen, Belgium, April 1989. Flajolet, P., Odlyzko, A.M., 1990. Singularity analysis of generating functions. SIAM J. Discrete Math. 3(2), 216-240. Hams, B., 1960. Probability distributions related to random mappings. Aim. Math. Statist. 31(2), 1045-1062. Kolchin, V.F., 1986. Random Mappings. Optimization Software, Inc., New York. Mutafciev, L.R., 1989. The limit distribution of the number of nodes in low strata of random mappings. Statist. Probab. Lett. 7, 247-251. Rubin, H., Sitgreaves, R., 1954. Probability distributions related to random transformations of a finite set. Technical Report No. 19a, Applied Mathematics and Statistics Laboratory, Stanford. Stepanov, V.E., 1969. Limit distributions of certain characteristics of random mappings. Theory Probab. Appl. 14, 612-626. Vitter, J.S., Flajolet, P., 1990. Average-case analysis of algorithms and data structures. In: van Leeuwen, J. (Ed.), Handbook of Theoretical Computer Science, vol. A: Algorithms and Complexity. North-Holland, Amsterdam, ch. 9, pp. 431-524.