Distribution of Image Points in Random Mappings

Michèle Soria

Université Paris VI

Algorithms Seminar

November 10, 1996

[summary by Pierre Nicodème]

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Abstract
This talk presents a general theorem which can be used to identify the limiting distribution for a class of combinatorial schemata. For example, many parameters in random mappings can be covered in this way.

## 1   Methods

We consider the general working scheme ``Symbolic Structures A or { A,w} ® Generating Functions a(z) or a(u,z) ® an or an,k''. Then by Cauchy's formula, we get for structures A
a(z) =
 å a Î A
z
 |a|
|a|!
=
 å n³ 0
an
zn
n!
Þ
an
n!
=
1
2ip
ó
õ
a(z)
dz
zn+1
.
When considering marked structures with parameters { A,w}, (w is a mapping A ® N), we have
a(u,z)=
 å aÎ A
u
 w(a)
z
 |a|
|a|!
=
 å n,k
an,kuk
zn
n!
.
In this case, an,k can be obtained by double Cauchy inversion, or by Cauchy inversion and Continuity Theorem. Table 1 gives some examples of translation of marked combinatorial structures to generating functions. The mark is represented by character ``·'' and translated to parameter u.

Description Structure Generating Function
Degree at the root in Cayley trees A = Node ×Set(· A) a(u,z) = z exp(ua(z))
Random Mappings G = Set(Cycle( A)) g(z)=1/1-a(z)
--- by number of cycles G = Set(·Cycle( A)) g(u,z)= exp(ulog1/1-a(z))
--- by number of trees G = Set(Cycle(· A)) g(u,z) = 1/1-u a(z)

Table 1: Some examples of generating functions

By a classical theorem about characteristic functions (Xn) converges weakly to Y if and only if fXn(q) converges to fY(q) for all q, with fZ = E(eiq Z). We also have a(u,z)=ån,kuk zn/n! = ån pn(u)zn/n!, which gives the probability generating function of Xn as pn(u)/pn(1) = ån Pr(Xn =k)uk. We refer to  for the concept of (labelled) combinatorial structures and their translation to generating functions.

## 2   Trees and Random Mappings

A random mapping is an arbitrary mapping f: {1,...,n}®{1,...,n} such that every mapping has probability n-n. A mapping f can be identified to its functional graph Gf with vertices {1,...,n} and edges (i,f(i)), for 1³ i ³ n. Each component of Gf consists of a cycle and every cyclic point is the root of a tree.

The basic property for analysis is that solutions of functional equations usually have algebraic singularity of square-root type. For trees, we get a(u,z) = t(u,z) -h(u,z)(1-z/r(u))1/2. For sequences of trees, we get an expression of the form 1/(1-a(u,z)), and for random mappings an expression of the form
s(u,z) =
1
1-T(u,z)+h(u,z)(1-z/r(u))1/2
.
We recall that when we get an expression of the form 1/(1-uC(z)), the asymptotic distribution of the corresponding random variable depends on the value C(rc), where rc is the only singularity on the circle of convergence of C(z). If C(rc)>1, the limit law is normal; if C(rc)<1, the limit law is derivative of geometric, and if C(rc)=1 the limit law is Rayleigh.

## 3   Examples

##### Leaves.
For Cayley trees, we have a(u,z) = z ea(u,z) + z(u-1), for sequences of trees, s(u,z)=1/(1-a(u,z)), and for functional graphs
g(u,z) =
1
1-zea(u,z)
=
1
1-a(u,z)+z(u-1)
.

##### Nodes of arity r.
For trees,
a(u,z) = z æ
ç
ç
è
 å m¹ r
am(u,z)
m!
+ u
ar(u,z)
r!
ö
÷
÷
ø
= z ea(u,z)+z(u-1)
ar(u,z)
r!
.
For sequences of trees, we have s(u,z)=1/(1-a(u,z)), and for functional graphs,
g(u,z) =
1
1-a(u,z)+z(u-1) æ
ç
ç
è
ar-1
(r-1)!
-
ar
r!
ö
÷
÷
ø
.

##### Nodes at distance d from a cycle.
We have the recurrence
.
For functional graphs, this gives g(u,z) =1/(1-ad(u,z)).

##### Nodes with r pre-images in total.
For trees, we have a(u,z) = zea(u,z)+(u-1)ar+1zr+1, where ar+1 = (r+1)r is the number of trees of size r+1. For functional graphs, we have G=Set(Cycle( A)), which translates to g(z) = exp(åp³ 0ap(z)/p). This gives
g(u,z) =
1
1-zea(u,z)
exp æ
ç
ç
è
zr
r
 å p
(up-1)
rr-p
(r-p)!
ö
÷
÷
ø
=
K(u,z)
1-a(u,z)+(u-1)ar+1zr+1
.

##### Nodes d iterated.
(These nodes are at distance ³ d from a leaf.) For trees, we have
-(u-1)ld(z)     with    l0(z)=0,   ld+1(z)=ze
 ld(z)
.
For functional graphs, we have, for nodes at distance ³ d of a leaf of their sub-tree, sd(u,z) = 1/(1-ad(u,z)). For nodes at distance ³ d of a leaf, we have
gd(u,z) =
1
1-uze
=
1
.

## 4   A classification for limit laws of random mappings parameters

We begin with a proposition which applies to functional equations of trees.

Proposition 1   Let F(u,z,a(u,z)) be a power series in three variables with non-negative coefficients and F(0,0,0) = 0. Suppose that the system of equations {t=F(1,r,t), 1=F'a(1,r,t)} has positive solutions r and t such that F'z(1,r,t)¹ 0 and F''aa(1,r,t)¹ 0. Then, F(u,z,a)=0 has for solution
a(u,z) = t(u,z)-h(u,z)(1-z/r(u))1/2,
with t,h,r analytic,
t(1,r(1))=t(1)º t,   r(1)=r    and   h(1,r(1))=(
2r F'z(1,r,t)
F''aa(1,r,t)
)1/2.
We arrive to a general theorem which seems to be the proper theorem to discuss random mappings. We consider a generating function g(u,z)=ån,kgn,kuk zn corresponding for variables Xn to a probability distribution Pr(Xn=k)=gn,k/gn. We consider a local expansion in the neighbourhood of u=1,z=r(u), of the form
g(u,z)=
1
1-T(u,z)+h(u,z)(1-z/r(u))1/2
.
T, h and r are analytic and T(1,r)=1.
Theorem 1   With these hypotheses (T, h, r analytic and T(1,r)=1),
1. If r'(u) = 0 and T'u(1,r)>0, then Xn/(n)1/2® R(l), where l=1/2(h(r,1)/T'u(r,1))2 and R(l) is the Rayleigh distribution of density l x exp(-l/2x2). Moreover E(Xn)»(p n/2l)1/2 and Var(Xn) » (2-p/2)n/l.
2. If r'(1)¹ 0 and T'u(1,r)=0, then Xnn/(s2 n)1/2 ® N(0,1), where µ = -r'(1)/r(1) and s22+µ-r''(1)/r(1). Moreover E(Xn)» µ n and Var(Xn)» s2 n.
3. If r'(1) ¹ 0 and T'u(1,r)¹ 0, then Xn - µ n/(s2 n)1/2® N(0,1)* R(s2l), where µ and s are defined as in (2), l is defined as in (1) and the star operator represents the convolution operation.
Remark 1   If T(1,r) ¹ 1, then Xnn/(s2 n)1/2® N(0,1), (except if r'(u) = 0 and T(1,r)<1, in which case Xn ® d G, derivative of a geometric law).
The density and characteristic functions in these different cases are as follows.
1. R (Rayleigh) f R(l)(x)=l xe-l x2/2, and f R(q)= 1+iq(p/2)1/2e-q2/2(1-ierf(q/(2)1/2)).
2. N (Normal) f N(x)=1/(2p)1/2e-x2/2 and f N(q)=e-q2/2.
3. N* R (Normal conv. Rayleigh) f N* R(x)=(e-x2/4-e-x2/2)/(2p)1/2+xe-x2/4/2(2)1/2erf(x/2) and f N* R(q)= f N(qf R(q).
Proof.(Sketch) Let g(u,z) = ån³ 0pn(u)zn/n! with pn(1)=gn. The proof rests on the convergence of the corresponding characteristic functions to (1) f R(q), (2) e-q2/2, (3) e-q2/2× f R(q). For instance, in case (1), the characteristic function pn(eiq/(n)1/2)/gn converges to f R(q). The proofs in the different cases make use of Cauchy inversions along suitable contours of the complex plane .

## 5   Applications

We note Xn the law of Xnn/(s2 n)1/2.

##### Leaves.
We have a(z)=t(u,z)-h(u,z)(1-z/r(u))1/2. This gives {t=r et+(u-1)r, 1=r et}, which gives {t(1,r)º t(1)=1, r(1)=r}, and also by differentiation wrt u {t'=(r et)' + r+(u-1)r', 0=(r et)'}, these two last equations give {t'(1)=r, r'(1)=-r2¹ 0}. This gives for sequences of trees t(1,r)=1, r'(1)¹ 0, t'u(1,r)¹ 0, and therefore Xn ® N* R. This also gives for functional graphs, with T(u,z)=t(u,z)-(u-1)z, T(1,r)=1,r'(1)¹ 0, T'u(1,r)=t'(1)-r=0, and therefore Xn ® N.

##### Nodes with in-degree r.
As before, a(z)=t(u,z)-h(u,z)(1-z/r(u))1/2. We have {t = r et+r(u-1)tr/r!, 1=r et+r(u-1)tr-1/(r-1)!}. This gives t(1)=1 and r(1) = r. By differentiation wrt u, we obtain t'(1) = r(1/r!-1/(r-1)!) and r'(1) = -r2/r!¹ 0. For sequences of trees, we get t(1,r)=1, r'(1)¹ 0 and, if r³ 2, t'u(1,r)¹ 0, which implies Xn ® N* R. If r=1, the limit law is normal. For functional graphs, we have T(u,z)=t(u,z)-z(u-1)(ar-1/(r-1)!-ar/r!). We get T(1,r)=1,r'(1) ¹ 0, and T'u(1,r)=0, which implies that Xn ® N.

##### Nodes at distance d from a cycle.
We have ad(u,z) =td(u,z)-cd(u,z)(1-ez)1/2, with t0(z)=ug(z), td(u,z)=zetd-1(u,z), c0(z)=uk(z), cd(u,z)=td(u,z)cd-1(u,z). This gives r'=0, td(1,r)=1, t'd(1,r)=1. Applying this results to g(u,z)=1/(1-ad(u,z)), we get T(1,r)=1, T'u(1,r)¹ 0,r=Cst, which implies that Xn® R.

##### Nodes with in-degree r.
(Same method.) We have for sequences of Cayley trees xn® N* R, and for functional graphs Xn® N.

##### Nodes at distance ³d from a leaf.
(Same method.) From a leaf of their own subtree (sequences of Cayley trees), Xn® N* R. In the general case, Xn® N.

##### Nodes at distance d from a leaf.
(Same method.) If the path contains no cyclic edge, Xn® R* N (except if d=1, in which case Xn® N). If cyclic edges are allowed, for d£2, we have Xn® N. (Conjecture: this last result is true for all d.)

## References


Drmota (Michael) and Soria (Michèle). -- Images and preimages in random mappings. SIAM Journal on Discrete Mathematics, vol. 10, n°2, May 1997, pp. 246--269.


Vitter (Jeffrey Scott) and Flajolet (Philippe). -- Analysis of algorithms and data structures. In van Leeuwen (J.) (editor), Handbook of Theoretical Computer Science, Chapter 9, pp. 431--524. -- North Holland, 1990.

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