%Letter to the Editor, Inf. Proc. Letters


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\begin{document}
\begin{center}
{\LARGE Letter to the Editor} \\
\big
%{\bf Comments to ``Reducing Conflict Resolution Time for Solving 
%Graph Problems in Broadcast Communications''} \\
%by {\sl Chang-Biau Yang\/}, {\sl Information Processing Letters\/} 
%{\bf 40}, 295--302, 1992.
{\bf A Note on Binomial Recurrences Arising in the Analysis of Algorithms}
\end{center}
\par
\bigskip
\noi
\begin{tabular}{l@{\hspace{0.7in}}l}
Helmut Prodinger\footnotemark&Wojciech Szpankowski\footnotemark\\
Department of Algebra\&Discrete Mathematics&Department of Computer Science\\
Technical University of Vienna&Purdue University\\
Vienna A-1040&W. Lafayette, IN 47907\\
Austria&U.S.A.\\
\end{tabular}
\footnotetext[1]{This work is supported by Fonds zur F\"{o}rderung der
Wissenschaftlichen Forschung Project P7497-TEC}
%\footnotetext{This work is supported by Fonds zur F\"{o}rderung der
%Wissenschaftlichen Forschung Project P7497-TEC}
\footnotetext{This research was supported by the NSF Grants INT-8912631, 
CCR-8900305 (its extension CCR-9201078) and NCR-9206315,
in part by AFOSR grant 90-0107, and in part by grant R01 LM05118 from the 
National Library of Medicine.}
\par
\bigskip\bigskip

Chang-Biau Yang in 
``Reducing Conflict Resolution Time for Solving
Graph Problems in Broadcast Communications"
{\it Information Processing Letters\/},
{\bf 40}, 295--302, 1992 analyzed the following recurrence
(we replace the authors notation $C^n_k$ for a binomial coefficient
by the more common $\binom nk$)
%Replacing the unpleasant notation $C^n_k$ for a binomial coefficient
%by the more common $\binom nk$, the author studies (Eq. 2) the recursion
$$
g_n=1+\sum_{k=0}^n2^{1-n}\binom nk g_k \eqno(1)
$$
for $n\ge 2$ and $g_0=g_1=1$.

The author presents a lengthy derivation for 
the following bounds on $g_n$:
$$
\frac 83\le\frac{g_n}{n}\le 3
$$

However, such recursions were intensively studied since the appearance of 
Knuth's books \cite{knuth} in the context of {\it tries\/}, 
and have gained more attention in the past few years in the context of 
{\it conflict resolution protocols} \cite{mf}, \cite{spa1},
analysis of extendible hashing \cite{flajolet}, \cite{spa3}, and
so forth.

The exact answer to (1) is (cf. \cite{knuth}, \cite{spa1})
$$
g_n=1+\sum_{k=2}^n (-1)^k {n \choose k} {2k-2 \over 1-2^{1-k}} ~. \eqno(2)
$$
Much more can be said. The asymptotic expansion of (2) leads to
(cf. \cite{mf}, \cite{knuth}, \cite{spa1}, \cite{spa3})
$$
g_n=\frac{2}{\log 2}n+ n\delta(\log_2n)+ 
{\it smaller\ order\ terms} ~, \eqno(3)
$$
where
$\delta(x)$ is a periodic function of period 1, small amplitude, mean 0
and known Fourier coefficients. Note that $2/\log 2= 2.8853\dots\ $.

Computing the variance of the conflict resolution session 
(the quantity analyzed by C. Yang) is much harder, 
but it can be done using a general methodology explained briefly below. 
The difficulty arises from a new sort of recurrence that must be solved.
This recurrence (for the second factorial moment of $g_n$) is
of the following type
$$
h_n=2ng_n+2^{1-n}\sum_{k=0}^n{n \choose k} g_kg_{n-k} -n(n+1)+
2^{1-n}\sum_{k=0}^n{n \choose k} h_k ~. \eqno(4)
$$
The above recurrence is difficult to solve mainly due to the 
appearance of $g_n$ (cf. (2)) in the second term above.
But, even the first term is not quite trivial to handle.
In  Kirschenhofer, Prodinger and Szpankowski \cite{kps1,kps2} asymptotic
expansions of $h_n$ and the variance (i.e., $h_n+g_n-g_n^2$) are
presented. Such an analysis involves
non trivial identities from the theory of modular functions and Ramanujan's
identities (cf. \cite{kp2}). 
The interested reader is referred to the new book of
Hosam Mahmoud \cite{mahmoud} which treats this subject 
in a more detailed way.

In this short note we only indicate how (2) and (3) can be obtained.
Consider a general  recurrence equation of the following type
$$
x_n=a_n+2^{1-n}\sum_{k=0}^n {n \choose k} x_k \ \ \ \ \ \ n\geq 2 ~,
\eqno(5)
$$
given $x_0$ and $x_1$, where $a_n$ is any sequence.
(For (1) we need to assume that $a_n=1$ and $x_0=x_1=1$.) 
In \cite{knuth,spa1,spa2} it is shown that
such a recurrence possesses the following solution
$$
x_n=x_0+\sum_{k=0}^n (-1)^k{n \choose k}{k\bar{x}_1+\hat{a}_k -\bar{x}_0
\over 1-2^{1-k}} \eqno(6)
$$
where $\bar{x}_0=a_0+x_0$ and $\bar{x}_1=a_1+x_0$. The inverse relationship
$\hat{a}_n$ is defined as follows (cf. Knuth \cite{knuth})
$$
\hat{a}_n =\sum_{k=0}^n{n \choose k} (-1)^k a_k ~. \eqno(7)
$$
For example, if $a_n=1$, then $\hat{a}_n=\delta_{n,0}$. Using this,
we can easily obtain solution (2) of the recurrence (1).

The asymptotic expansion of the alternating sum (6) requires some
tools from  complex analysis \cite{henrici}. 
Consider a general alternating
sum of the following type
$$
x_n=\sum_{k=2}^n (-1)^k{n \choose k} f_k ~,
\eqno(8)
$$
An asymptotic expansion of the above sum can be either obtained by
Rice's method as popularized by Knuth \cite{knuth} and
Flajolet and Sedgewick \cite{fs} (see also \cite{kp})
or by the Mellin-like transform
as shown by Knuth \cite{knuth} and generalized by Szpankowski
\cite{spa2}. We discuss below only the Mellin-like approach.
In Szpankowski \cite{spa2} it is proved that the alternating sum (8) can
be evaluated by the following complex integral
$$
x_n={1\over 2 \pi i} \int_{-3/2-i \infty}^{-3/2+i\infty}
\Gamma(z) n^{-z} f(-z) dz + \hbox{{\it smaller order terms}} ~, \eqno(9)
$$
where $\Gamma(z)$ is the gamma function \cite{henrici}, 
$f(z)$ is an analytical continuation
of $f_k$, that is, $f(k)=f_k$. The above formula holds under some 
mild conditions on the behaviour of $f(z)$ at infinity. In passing, we note
that the integral (9) is usually {\it very easy} to evaluate 
asymptotically by 
Cauchy's residue theorem \cite{henrici}.

In the case of the recurrence (1) with the explicit solution (2),
we apply (9) to obtain
$$
g_n\sim - {1\over 2 \pi i} \int_{-3/2-i \infty}^{-3/2+i\infty} 
{(2z+2) \Gamma(z) n^{-z} \over 1-2^{1+z}}\, dz ~.
\eqno(10)
$$
Noting that $z_0=-1$ is a double pole of the function 
under the integral and computing the residue at $z_0$, we obtain the
first term in (3). The second term (oscillating function 
$\delta(\log_2 n)$) is derived from the infinite number
of poles of the form
$z_k=-1+2\pi k i/\log 2$ of the denominator. 
Details can be found in the references already
mentioned above.

In passing, we note that in fact much more was done in this area.
We refer to seminal works %of such prominent researchers as
Knuth and Flajolet,
and some others, like (in alphabetical order) Jacquet, Kirschenhofer,
Prodinger, R\'egnier, Schmid, Sedgewick and Szpankowski.
\big
{\bf Added in the Proof}. Yet another binomial recurrence was consider by
Ressler in his recent paper ``Random List Permutations in Place'',
{\it Information Processing Letters}, {\bf 43}, 271-275, 1992. Namely,
for $n\geq 0$ 
$$
T_{n+1}=n+2^{1-n}\sum_{k=0}^n {n \choose k} T_k
\eqno(11)
$$
with $T_0=0$, and Ressler proved that $T_n\leq n\log_2 n$.
This recurrence does not fall exactly under
our general scheme (5). In fact, it is harder to solve, and
it appears in the analysis of another digital tree, namely, {\it digital
search tree} (cf. \cite{flajolet2}, \cite{fs}, \cite{fr}, 
\cite{knuth}, \cite{spa5}). 

More generally, as in \cite{spa5} consider the following recurrence
for $n\geq 1$
$$
x_{n+1}=a_{n+1}+2^{1-n}\sum_{k=0}^n {n \choose k} x_k 
\eqno(12)
$$
with $x_0=0$. It is proved in \cite{spa5} that the above recurrence
has the following solution
$$
x_n=- \sum_{k=2}^n (-1)^k {n \choose k} \hat{x}_{k-2} 
\eqno(14)
$$
where
$$
\hat{x}_n = Q_n \sum_{i=1}^{n+1} (\hat{a}_i-\hat{a}_{i+1}-a_1)/Q_{i-1}
\eqno(15)
$$
with $Q_n=\prod_{k=1}^n(1-2^{-k})$.
Note that the solution (14) has the form of the alternating sum as
in (8), hence the methodology discussed above can be directly applied
to obtain an asymptotic expansion of (14).

In particular, for the recurrence (11) studied by Ressler we can easily prove
that $T_n=\sum_{k=2}^n (-1)^k {n \choose k} Q_{k-2}$, and 
$$
ET_n=n\log_2n+n\left({\gamma-1\over \log 2}+{1\over 2} 
-\alpha+\delta_1(\log_2n)\right) +O(\log n) ~,
\eqno(16)
$$
where $\gamma=0.57721\ldots$ is Euler's constant and
$\alpha=\sum_{n\geq 1}1/(2^n-1)=1.60669\ldots~$, and
$\delta_1(x)$ is a continuous periodic function
of period $1$, mean $0$ and very small amplitude ($< 10^{-6}$).
For more details, the reader is referred to the literature cited above.

\begin{singlespace}

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\end{singlespace}

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