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\lecture{2}{Network Computing}{February 6, 1997}{Carlo Maley}

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The following lecture on network computing will focus mainly on
describing and analyzing butterfly networks and their related Bene\v{s}
networks.

\section{Ideal Networks}

In some sense, an``ideal'' data parallel computer may be depicted as in Figure~\ref{fig:ideal}.

\begin{figure}[h]
\centerline{\psfig{figure=ideal.ps}}
\caption{A serial central processor (CP) with its own memory (Mem) is
connected to an array of
processors (P).  Each processor is connected to all of the memory units (M).}
\label{fig:ideal}
\end{figure}


Each processor is connected to each memory unit.  This fully connected
network can be simulated
by a ``crossbar'' interconnection network, although there are some slight differences due to arbitration.

\begin{figure}[htb]
\centerline{\psfig{figure=mesh.ps}}
\caption{Processors along the bottom are connected to memory units along
the right hand side through a mesh of routing switches.}
\label{fig:mesh}
\end{figure}

But the problem with building a fully connected network is in the
constraints on the number of pins on the units.  The degree of a node in
a fully connected network grows linearly with the number of nodes, $n$.
Typically, however, we are constrained to a constant number of pins, or
at best $\log n$ pins.  This constraint
has driven a lot of interest in networks with low degree nodes.


\section{Butterfly Networks}

One of the most popular low degree networks is the ``butterfly network''
also known as the ``FFT network.''

\begin{figure}[htb]
\centerline{\psfig{figure=butterfly.ps}}
\caption{This 8-input butterfly network has 4 levels, or columns.  Each
row is labeled with a binary address.}
\label{fig:butterfly}
\end{figure}

It is possible to build a butterfly network recursively.  Two butterfly
networks can be composed into a larger butterfly network by connecting
line $i$ of the first network to line $i$ of the second network, for all
lines in the two networks.  Lines, running horizontally, are numbered
consecutively from top to bottom, for convenience.  This process
constructs the next ``level'' of nodes.  It may not be immediately
obvious but if you remove the first (2 by 2) level of nodes, the
resulting network consists of two interleaved but separate butterfly
networks.  The recursive structure of the butterfly network makes it a
convenient network for designing certain algorithms.

\begin{figure}[htb]
\centerline{\psfig{figure=interleaved-blfy.ps}}
\caption{If the first level of an 8-input butterfly network is removed,
the result is a pair of interleaved 4-input butterfly networks.  The
nodes of the two networks have been distinguished as either white or grey.}
\label{fig:interleaved-blfy}
\end{figure}

\subsection{Properties of an n-input Butterfly Network}
 
\defn{Diameter} of a network is the longest minimum distance between two
nodes.

The diameter of a butterfly network is $\Theta(\log n)$.  You can view a
butterfly network as a sort of binary tree.  Each step through the
network divides the number of accessible nodes in half.  In fact, if you
trace out the paths for one node on one side of the network to all the
nodes on the other side, you will trace out a fully binary tree.  This
works from either direction through the network.

The number of nodes in the network, $N = \Theta(n \log n)$ because there
are $\log n$ levels or columns of nodes in the net, and each level
consists of $n$ nodes.

\defn{Bisection width} is the number of wires you would need to cut in
order to divide the network in half.  

In general we would like large bisection widths in order to prevent
bottlenecks in the network.  The butterfly network has a $\Theta(n)$
bisection width.

\section{Routing}

Typically, there are two ways to organize the placement of the processor
and memory units.  The ``dance-hall'' organization puts all the
processors on one side of the network and all the memory units on the
opposite side.  In contrast, the ``distributed memory'' organization
merges a memory unit with each processor so that each processor has
fast access to its local memory.  Then the butterfly network must be
``wrapped around'' to give the processors access to non-local memory
units.  That is, the nodes on the opposite sides of the ordinary dance-hall
butterfly network are merged into a single column of nodes.

One nice aspect of butterfly networks is that it is easy to do the
routing through the network.  If we number the nodes in binary on either side of an
ordinary butterfly network, the number of the destination node determines the
route through the network.  If we start on the left side of the network
(the side with the simple 2 by 2 butterfly networks) then the path to a
node on the right is determined by taking the low order bits of the
destination node's number, one at a time, and interpreting a 0 as ``take
the upper path'' and a 1 as ``take the lower path'' at each intermediate
node.

\begin{example}
The route from node 011 on the left to node 101 on the right is
``down-up-down'' as illustrated in figure~\ref{fig:bfly-route}.

\begin{figure}[htb]
\centerline{\psfig{figure=bfly-route.ps}}
\caption{The route from node 011 to node 101 has been highlighted in
grey.  The choice of wires follows the bits of the destination node.
Here, 101 translates as the three choices of lower-upper-lower.}
\label{fig:bfly-route}
\end{figure}

\end{example}

This algorithm works in the opposite direction as well if the directions
are taken from the high order bits first, instead of the low order bits.

\defn{Minor cycles} are the small, 2 by 2 cycles in the network.

\defn{Major cycles} are the larger cycles connecting the minor cycles.

\subsection{Packet Routing}

Consider routing a packet from a node labeled with the bits
$x_1x_2\ldots x_d$ to the node $y_1y_2\ldots y_d$.  The route proceeds
by correcting one bit at a time starting from the major cycles and
progressing to the minor cycles as such.

\begin{gather*}
x_1x_2\ldots x_d \\
y_1x_2\ldots x_d \\
y_1y_2\ldots x_d \\
\vdots \\
y_1y_2\ldots y_d
\end{gather*}

Thus it takes $d$ steps to move the packet through the network.  This is
reasonably fast.  In practice, people often build 4-ary or 8-ary
butterfly networks to speed them up.

But the number of steps is not the whole story.  Congestion in the
network is a critical factor influencing the speed of the network.
Suppose we have $n$ packets on an $n$-input butterfly network.  If they
all have the same destination it will take $\Omega (n)$ time for all the
packets to reach their destination.  On the other hand, if the mapping
from source nodes to destination nodes is a permutation (no two packets
have the same destination) then it takes $\Omega (\sqrt{n})$ time in the
worst case.  This can be seen in the following example.

\begin{example}
Consider sending packets through an $n$-input butterfly network.  There
may be a subset of $\sqrt{n}$ of the packets, where each packet
has the form $x_1x_2x_3x_40000$ and is routed to the destination
$0000x_1x_2x_3x_4$.  We are assuming that the different packets have
unique combinations of bits $x_1x_2x_3x_4$
and so there is a one-to-one mapping between source and destination
nodes for these packets.  Halfway
through the network, all of these packets will go through the same node
on line
$00000000$ thus producing congestion of at least $\sqrt{n}$ packets.
\end{example}


\section{Bene\v{s} Networks}

Can we design a network to do better than $\Omega (\sqrt{n})$ time in the worst case?
Consider routing a packet through a butterfly network and then back
again the opposite direction.  That is, consider attaching another
butterfly network to the back of the first one as in
figure~\ref{fig:benes-net}.  This is a Bene\v{s} network.

\begin{figure}[htb]
\centerline{\psfig{figure=benes.ps}}
\caption{A Bene\v{s} network is composed of two butterfly networks.  The
minor cycles of one of the butterfly networks have been removed because
they do not increase the power of the network.}
\label{fig:benes-net}
\end{figure}

The second layer of 2 by 2 butterfly networks in the middle does not
give us any added power.  At the level of a 2 by 2 butterfly network,
either permutation of input to output nodes can be routed without any
congestion.

\begin{figure}[htb]
\centerline{\psfig{figure=bfly-2by2.ps}}
\caption{There are only two possible permutations to be routed on a 2 by
2 Bene\v{s} network.  Neither of them produce a conflict.  The routings
have been colored black.}
\label{fig:bfly-2by2}
\end{figure}

A Bene\v{s} network has the following nice properties:
\begin{enumerate}
\item One can determine the router switch setting for a permutation from
input to output nodes in $O(\log n)$ time offline.
\item There is no congestion!
\end{enumerate}

The algorithm for determining the router switch settings is essentially
a greedy strategy.  We work in recursively from the outside two layers
to the inner, smaller Bene\v{s} networks.  At any one recursive step,
each input node has two connections, one to each of two smaller,
independent Bene\v{s} networks.  We begin the selection of wires for
packets by arbitrarily choosing one of the two wires for the packet on
one of the input nodes.  This forces a series of routing decisions in
order to avoid any congestion.  At some point, another arbitrary
decision may arise.  Each time that happens, we will choose the upper
wire of the two wires.  Eventually, all the packets will be assigned a
route that does not conflict with any other packet's route.  We then
proceed by induction on the smaller two Bene\v{s} networks.

\begin{figure}[htb]
\centerline{\psfig{figure=benes-routing.ps}}
\caption{The grey rectangles represent abstractions of 4-input Bene\v{s}
networks.  The dashed lines show the beginnings of routing a
permutation.  The choice of routing the first step of the packet with
the destination of node 5 is arbitrary.  But once it has been chosen it
determines the routing of the packet with the destination of node 1, the
node which is paired to node 5 in the outer layer of this Bene\v{s}
network.  We work backwards from output node to input, routing through
the lower sub-network.  The next step would be to route the packet
destined for node 7 because that choice is forced by the routing of the
packet for node 1.  This process can be continued until all the packets
are routed.}
\label{fig:benes-routing}
\end{figure}

Why does this work?  Whenever we move from left to right, we go through
the upper box (inner Bene\v{s} network).  But whenever we move from right
to left we go through the lower box.  Upon returning to the left side,
the ``mate'' of that input node has either not yet been routed, or it
was routed on a move from left to right and so went through the upper
box.\footnote{This can be seen as a conflict graph with an even number
of nodes in each cycle.  The problem is then to do a two-coloring of the
nodes.}  In either case there is no conflict.  We then proceed by
induction on the smaller Bene\v{s} networks until we come to the 2 by 2
base case.

\defn{Bounded degree} means $O(1)$ degree.

{\corollary An $n$-input Bene\v{s} network can simulate any bounded
degree network on $n$ nodes in $O(\log n)$ time.}

{\proof The proof of this is left as an exercise}

Once the proper configuration for the switches has been determined,
setting the switches of the Bene\v{s} network takes $O(\log n)$ time.
In fact, if you are clever, you can pipeline the setting of the switches
with the routing of the packets.

We can simulate a mesh by doing each of the four directions separately.
The proof of the corollary requires generalizing this to non-uniform
networks.

The Bene\v{s} network is a ``universal'' network.  It can simulate an
ideal computer.  Unfortunately, the algorithm for setting the switches
is off-line.  We would like it to be done on-line.

\section{Congestion Bounds}

While the butterfly network performs poorly in the worst case, it turns
out to be pretty good in the average case.

{\theorem Consider the $N^N$ $N$-packet routing problems on an $N$-node
butterfly network.  These are the problems where every node has a packet
to send.  At least $N^N(1 - \frac{1}{N^{\Omega (1)}})$ of these problems
are routed in $O(\log N)$ time by the greedy algorithm.}

Note that there are $N^N$ of these problems because any node may need to
send its packet to any other node in the network.  Also note that we are
speaking of an $N$ is the number of nodes in the whole network while $n$
is the number of input nodes.  If we want to determine $n$, we know that
$N = n
\log n$ implies $\log N = \log n + \log \log n$ which implies that $\log
N = \Theta (\log n)$ which finally implies that $n = \Theta
(\frac{N}{\log N})$.

The theorem states that only a polynomially small fraction ($1/N$) of
these problems are bad for the butterfly network.  It turns out that you
can inflate this with any constant $k$ so that only $1/N^k$ problems are
bad, at the expense of multiplying the time needed for the ``good''
problems by a factor of $k$.

\medskip
{\proof We will only prove a congestion bound.

$N = n \log n$.  We will first consider an arbitrary node $x$ at level
$k$ in the network.  We first take $O(\log N)$ time to push all packets
along the horizontal wires until they are queued up on the left side of
the network.  We want to route them through to the other side of the
network in order to move each
packet to the line of its destination node.  Then we will push all the
packets back along the horizontal wires and allow the nodes to pick off
the packets as they reach their destinations.

We will first consider a random $N$-packet routing problem and show that
the probability that more than $O(\log N)$ packets go through any node
is $\leq \frac{1}{N^{\Omega (1)}}$.  We will do this by determining how
many packets use node $x$ at level $k$.  Then we will combine the
probabilities for the different nodes.

The number of packets that can reach node $x$ is $2^k \log n$.  We know
that each input node has $\log n$ packets queued up because each of the
$\log n$ nodes on that line contributed a single packet when they were
all pushed to the left.

The probability that a given packet passes through $x$ is $\leq 2^{-k}$.
Note that $k \leq \log n$ because there are only $\log n$ layers in the
network.

Consider any set of $r$ specific packets.  The probability that they all
pass through node $x$ is $2^{-kr}$.  Consequently, the
   probability that at least $r$ packets pass through node $x$ is
   at most

\begin{equation}
\text{Pr(at least $r$ packets pass through $x$)} \leq \begin{pmatrix} 2^k \log n \\ r \end{pmatrix}
2^{-kr}
\end{equation}

Note that this over counts.  If $r + \Delta$ packets pass
through $x$, this event will be counted $\begin{pmatrix} r + \Delta \\ r
\end{pmatrix}$ times within the $\begin{pmatrix} 2^k \log n \\ r
\end{pmatrix}$ ways of choosing $r$ packets.  Thus we have an upper
bound.  The rest is math.

We will use the simplification $\begin{pmatrix} a \\ b
\end{pmatrix} \leq \left( \frac{e a}{b}\right)^b$ from chapter 6 of CLR.

\begin{equation}
\begin{pmatrix} 2^k \log n \\ r \end{pmatrix}
2^{-kr} \leq \left( \frac{e 2^k \log n}{r}\right)^r 2^{-kr} = \left( \frac{e
\log n}{r} \right)^r
\end{equation}


So we need to pick $r > e \log n$ for our probability bound to be less
than 1.  We choose $r = 2e \log N$ which is $O (\log N)$.  

\begin{equation}
\left( \frac{e \log n}{r}\right)^r \leq (1/2)^{2e \log N}  =  N^{-2e}  \leq  N^{-5.4}
\end{equation}

Now we want to find an upper bound for any node.  We use Boole's inequality which states that the probability of the
union of events is less than or equal to the sum of the individual
probabilities of each event.  Therefore, the
probability that any node has $\geq 2e \log N$ packets routed through
it is $\leq N(N^{-5.4}) = N^{-4.4}$.

Therefore, at least $N^N(1 - \frac{1}{N^{\Omega (1)}})$ problems create
$\leq 2e \log N$ congestion.  \qedbox}
\medskip

By adjusting our choice of $r$ by a constant $k$ such that $r = (2e + k)
\log N$ we get an upper bound of $N^{-(4.4 + k)}$ bad problems.

It turns out that most of the typical permutations that we use, like
reversing the nodes or transposing a matrix, are ``bad'' routing
problems for a butterfly network.


The article by Ranade is an example of a delay sequence argument.  He
shows that you only need $O(1)$ queues at the nodes of an $N$-node butterfly
network in order to emulate a CRCW PRAM in $O(\log N)$ time.

Valiant has addressed the problem of the difficulties of efficiently
implementing the common forms of permutation.  Valiant suggests using
two butterfly networks back to back.  In order to implement a
permutation you route the input to a randomly chosen permutation on the
first butterfly network and then use the second butterfly network to
route the random permutation into the desired output permutation.  In
this way, both permutations are random and so the butterfly network will
almost always perform well.

Finally, Leighton, Maggs and Rao introduced the multibutterfly network.
This is essentially a butterfly network with randomized
interconnections.  The nodes of level $i$ are grouped into $2^i$
contiguous clusters.  So the zeroth level is all one cluster.  The nodes
of the next level are divided into the upper half and lower half of the
nodes, and so on.  Then the nodes of each level are randomly connected to a
constant $k$ number of nodes in each of the two clusters parallel with
the current node's cluster in the next level.  It is then possible to
prove that you never have congestion for permutations that take more
than $O(\log n)$ time.  There is a small chance that the
random connections generated a ``bad'' network.  Unfortunately, checking
to see if a network is ``bad'' is an NP-complete problem.

% your notes end here
\end{document}
