Rearranging Algorithms
for Log2 ( N, 0, p) Switching Networks
with Even Number of Stages
Wojciech Kabaciriski, Senior Member IEEE
Janusz Kleban, Member IEEE, Marek Michalski
Mariusz Zal, Member IEEE
Poznan University of Technology,
Chair of Communication and Computer Networks,
ul. Polanka 3, 64-980 Poznan,
e-mail: (kabacins.jkleban.mmichal.mzal)@et.put.poznan.pl
Abstract-In this paper we consider the rearrangeable
multi-plane banyan-type switching fabrics, called also
log, (N, 0, p) switching networks, with even number of
stages. For such networks different rearranging algorithms
have been proposed for both: one-at-a-time and simultaneous connection models. In this paper we consider the oneat-a-time connection model, where connections arrive to the
system one-by-one, and in case of blocking rearrangements
are realized. To our knowledge, known algorithms require
several rearrangements, and the number of such rearrangements have not been considered in the literature. We
propose the new rearranging algorithm for the multi-plane
banyan-type switching fabric composed of even number
of stages. This algorithm leads to success using only one
rearrangement. We also introduce the modified version of
this new algorithm, in which rearrangement of an existing
connecting path can be realized without its interruption.
I. INTRODUCTION
Space-switching fabrics are classified according to
their blocking properties as blocking, rearrangeably nonblocking and strictly (or strict-sense) non-blocking. A
strict-sense non-blocking network can always connect
each of its idle inlets to an arbitrary idle outlet, independently of the already established connections. Therefore in strictly non-blocking fabrics, switching can be
performed fast without needing any complex routing
algorithms. This high degree of connecting functionality comes however at the cost of a significant hardware complexity: the number of Switching Elements
(SEs) required can be very high. Rearrangeable (nonblocking) networks can always connect each idle inlet
The work described in this paper was carried out with the support
of the BONE-project ("Building the Future Optical Network in Europe"), a Network of Excellence funded by the European Commission
through the 7th ICT-Framework Programme, and partially financed
from the research funding in years 2009-2012 as the research grant
3882/Btr02/2009/36.
Achille Pattavina, Senior Member IEEE
Guido Maier, Senior Member IEEE
Politecnico di Milano,
Dept. of Electronics and Information
Via Ponzio, 34/5-20133 Milano
e-mail:(pattavinac maierjts'elet.polimi.it
to an arbitrary idle outlet by applying a suitable internal
rearrangement of the connections already established.
Given a certain number of inlets and outlets, a strictly
non-blocking implementation is much more complex
and requires a much higher quantity of SEs than the
rearrangeable counterpart. Rearrangeable networks can
be an interesting alternative, especially in those cases in
which the number of SEs in the fabric is a critical parameter. This may occur for several reasons, such as cost,
mutual interference between the active connections in the
fabric (crosstalk), complexity of the control system, etc.
For instance, in the particular but very interesting case
of the photonic switching systems, a large number of
SEs implies accumulation of crosstalk and excess power
dissipation in most known technological implementations. Moreover, in integrated photonic technologies the
number of switching devices that can be included on
the same optical substrate is severely limited by the
maximum area allowed. Finally, not only is the total
number of SEs critical in photonic fabrics, but also the
maximum number of SEs crossed by an optical signal
that propagates through the fabric. The advantage of a
rearrangeable fabric in terms of cost and reduction of
signal degradation is paid by the drawback represented
by the need to rearrange the already established connections. The disruption of service due to rearrangements
implies a certain data-loss which depends on the fabric
reconfiguration speed and the frequency of reconfigurations. For example, in the case of an Optical Cross
Connect (OXC) in a circuit-switched WDM network
based on the current photonic technology, the dataloss ratio due to reconfigurations can sometimes be
higher than the data-loss ratio due to transmission errors.
For instance, considering a switching time of a few
milliseconds, typical of many today's slow switching
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Fig. 1.
The baseline multi-logylf switching fabric
devices (to be conservative), a reconfiguration period of
the connection in the axc of one day implies a data
loss ratio on the order of 10- 8 [1], [2], [3].
In order to face the issue of rearrangements, while
not giving up the advantages of rearrangeable switching
solutions, it is however possible to exploit the degrees
of freedom that are given by the control algorithms or
the architectures of the rearrangeable fabric. A number
of studies have already been developed in this direction:
the rationale is to design the rearrangement algorithms
or the architectures so to minimize (or eliminate) the
impact that rearrangements have in terms of disruption
of service. An example of such studies is the work by
EK. Hwang and others [4] in which Noninterruptive
Rearrangeable (NIR) versions of the Clos fabric are
presented. Rearrangeable logz(N, 0, p) switching fabrics
for unicast connections were considered by Lea [5]. For
the one-at-a-time connection model Paull's algorithms
proposed for Clos networks, can be used [5] and [6].
It is also possible to use algorithms for simultaneous
connection routing based on graph coloring or matrix
decompositions [7]. Authors of [8] show that these
algorithms sometimes lead to unnecessary rearrangements and propose the algorithm that eliminates them.
However, no analysis in case of these type of networks
have been done for the number of rearrangements.
Rearrangement algorithms can be divided into two
groups: single connection algorithms and simultaneous
connections algorithms. Single connection algorithms
are used to unblock the new connection when the path
searching algorithm fails. The most popular algorithms
(primary applied to three-stage switching fabrics) are
proposed by Slepian and Paull [6]. All of them are
based on the Paull matrix (called also the state matrix), which represents the state of the switching fabric.
Rows and columns correspond to the first and the last
stage of switches, respectively. Cells contain numbers
(or symbols) of switches in the middle sections which
are involved in connections set up between switches
in the first and the last stages. Unblocking of the new
connection is achieved by a sequence of rearrangements
in the state matrix.
In simultaneous connections we have a set of compatible connections and we have to choose a connecting
path for each call. In this group we can distinguish two
main types of algorithms: algorithms based on matrix
decomposition and algorithms based on graph coloring.
In the first type of algorithms a maximal assignment
is represented in a matrix. Each row (column) in this
matrix corresponds to one first stage (third stage) switch.
Cells contain information how many connections must
be set up between given switches in the first and the
last stages. The aim of these algorithms is to decompose
this matrix into a set of elementary permutation matrices.
Connections corresponding to each of these elementary
permutation matrices can be set up through one middle
stage switch [9]. In algorithms based on graph coloring
maximal assignments are represented by graphs. The
goal of these algorithms is to properly color vertices
or edges using the least number of colors possible (i.e.
not greater than the chromatic number or the chromatic
index). Connections corresponding to vertices or edges
colored with the same color can be realized using one
center stage switch. A control algorithm for rearrangeable logz(N,O,p) switching networks based on matrix
decomposition is presented in [5] and an algorithm based
on graph coloring is presented in [10].
An example of a single connection rearrangement
algorithm for logz(N, 0, p) switching fabrics is presented
in [8]. It also uses the state matrix. Additional matrices
called an input matrix and an output matrix are used in
this algorithm. Using these matrices it is possible to find
connections which cause blocking to a new connection,
and reswitch only these connections. In contrary, an
algorithm based on matrix decomposition [5] treats a
new connection and all existing connections as new
connections which are to be set up in the switching
network. It may result in reswitching more connections
than necessary.
In this paper we contribute to the topic by proposing new rearrangement algorithms that allow noninterruptive re-switching of existing connections. We will
target the class of rearrangeable space-switching fabrics
comprising the Banyan vertically-replicated networks.
These switching networks have N inlets/outlets and p
equal planes interconnected to the inlets (outlets) via
NIx N switching splitters (N x 1 combiners). Each
plane comprises logzN stages, each of N / 2 2 x 2 SEs,
interconnected by interstage sets of links characterized
by well-known regular patterns (e.g. baseline, switching
banyan, etc.). This paper, in particular, will consider only
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°
o
15t1-------"'_---""O---'t~
15
1
2
3
4
0
1
234
Fig. 2. A 16 x 16 baseline switching fabric (a) and its bipartite graph
representation (b)
those Banyan vertically-replicated networks in which
the number of SE stages in each plane is even. The
number of planes p is set by the necessary and sufficient
conditions for rearrangeability known by the classical
switching theory [10]: p = 2~ with n = 10gzN.
The paper is organized as follows: In section II
the switching fabric architecture and basic notation is
reminded after [7], [10], [11]. The number of rearrangements needed for nonblocking of a blocked connection is
derived in section III. In section IV the new rearranging
Algorithm is presented . The version of this algorithm
which enables non-interruptive rearrangements is also
discussed . At the end some conclusions are given.
II. NOTATION
A. General Information
The logz (N , 0, p) switching network is constructed of
p copies of logz (N, 0,1) switching networks connected
in parallel (Fig. 1). Each copy of logz (N , 0,1) switching
networks, called also a plane, is constructed of 2 x 2
switches arranged in n = logz N stages, N / 2 switches
in each stage. Several equivalent topologies, namely
banyan, baseline, omega, may be considered. In the
remaining part of this paper baseline topology will be
used. Stages are numbered 1,2, . .. , n from left to right,
inputs and outputs are numbered 0,1 , ... , N - 1 from
top to bottom , and switches in each stage are numbered
0,1 , . .. N / 2 - 1 from top to bottom.
The bipartite graph (e.g. the 2-chromatic graph whose
nodes can be colored with two colors') is often used to
represent the topology of the switching network [10],
[12]. In this representation inputs, outputs, and interstage links are represented by vertices (or nodes), while
crosspoints in 2 x 2 switches correspond to edges. Vertices are arranged in n + 1 stages numbered 0,1, ... , n
I In this graph nodes in odd numbered stages could be colored with
one color and nodes in even numbered stages could be colored with
the second color, so the graph is 2-chromatic.
from left to right, respectively. The nodes in stage
correspond to inputs of the switching network and the
nodes in stage n correspond to outputs of the switching
network. These nodes are numbered 0, 1, ... , N -1 from
top to bottom, respectively. The baseline architecture
of the log- lf switching network and its bipartite graph
representation are shown in Fig. 2a and b, respectively.
The 10gz(N,0,p) switching fabric is the single path
switching fabric, i.e. it has only one connecting path
between the given input-output pair. For each connecting
path we can construct a graph of paths which intersect
with the considered path in the node of stages from 1
to n - 1. This graph is called the graph of intersecting
paths, and it is shown in Fig. 3. (for n even) [10]. We
can also define sets SIi and SO i. Set SIi contains all
inputs for which their connecting paths intersect with the
considered paths for the first time in the node of stage
i. By analogy set SOi contains all outputs for which
connecting paths intersects with the considered paths for
the first time in the node of stage n - i. It should be
noted, that the cardinality of sets S I, and SO i is ISI, I =
ISOil = 2min(i - l ,n- i - l ).
B. Three Types of Blocking Connections
Let us consider a new connection between input x
and output y, and let us assume, that this connection is
blocked in the log, (N , 0, 2 ~) switching fabric. Connections which block this new connection can be divided
into three types: type A, type B, and type C. The
connection of type A is a connection which intersects
with the new connection in any node(s) of stages from
1 to ~ - 1. The connection of type B is a connection
which intersects with the new connection in any node(s)
of stages from ~ + 1 to n - 1. It should be noted that
some connections may intersect with the new connection
in both sets of nodes. We will consider such a connection
as the connection of type A. Finally, the connection of
Fig. 3.
The graph of intersecting paths for connection (0;0)
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type C is a connection which intersects with the new
connection only in the node of stage ~ . In other words
any connection from inputs in S I 1 to S I ~ -1 is the type
A connection. Any connection to outputs SOl to SO~-l
(except those from inputs Sl, to Sl~-d is the type B
connection, and a connection from an input in SI ~ to an
output in SO ~ is the type C connection. Considering the
cardinalities of sets SIi and SOi we can write that any
blocked connection can be blocked by at most 2~-1_1
connections of type A, at most 2~-1 -1 connections of
type B, and at most 2~-1 connections of type C.
Waiting for new connection
No
III . REARRANGEMENTS IN LOG2 (N; 0; p)
SWITCHING NETWORKS
A. Blocking Connections
Lemma 1. If the new connection (i,j) is blocked
in the log2(N, 0, 2 ~) switching fabric, then for this
connection there are blocking connections of type A,
Band C, i.e. connection (i, j) cannot be blocked by
connections of only two types.
Proof: The rearrangeable log, (N, 0, p) switching fabric is composed of p = 2~ planes. When the new
connection (i, j) is blocked, it means, that in each plane
there is at least one blocking connection and there are
at least 2 ~ blocking connections all together. But there
may be at most 2~-1 - 1 connections of type A and
B, and 2~-1 connections of type C. So connections
A and B may block at most 2~ - 2 planes. Similarly,
connections of types A and C (B and C) may block at
most 2~ -1 planes. So we will have at least one or two
planes available for the new connection. But since the
connection is blocked , it means that we can find in the
switching network connections of types A, B, and C.
B. Number of Rearrangements
Theorem 1: In the log2(N, 0, 2~) switching fabric
with even number of stages, a blocked connection can
be unblocked by rearranging only one of existing connections .
Proof: Let connection (i, j) is a new blocked connection. According to Lemma I, there are at least 2 ~
blocking connections, and these connections are of types
A, B, and C. Let us assume the new connection is
blocked by:
- x connections of type C, 2 :;::; x :;::; 2 ~-1
- y connections of type A, 1 :;::; Y :;::; 2~-1 - 1
- z co~nections of type B, 15 z :;::; 2~-1 - 1
and 2"2 :;::; x + Y + z :;::; 3 * 2"2- 1 - 2.
Let us consider type C connections. These connections
may be further blocked for rearrangements only by other
connections from inputs in SI~ and to outputs in SO~.
The number of such connection s can be no greater than
ISI~I - x + ISO ~I - x = 2~ - 2x , since x inputs
and outputs are already occupied by considered type C
Fig. 4.
Algorithm I - Algorithm for rearranging connections
connections. So 2~ -2x connections can block not more
than 2 ~ - 2x planes. But connections A and B block at
least 2~ - x planes. So the number of planes occupied
by connections A and B is greater than the number of
planes in which connections C can be blocked. It means
that we can always find such a connection C set up in
plane c that can be rearranged to plane a (or b) through
which connection of type A (or B) is set up, and the new
connection will be set up in plane c.
IV. THE NEW REARRANGING ALGORITHM FOR N
EVEN
A. The Control Algorithm
From the proof of Theorem 1 we can conclude
that to unblock a blocked connection we should move
one connection of type C to a plane through which
connection of type A or B is set up and we can always
find such a pair of connections. The control algorithm
can be as follows :
Algorithm 1:
Step 1: Wait for a new connection.
Step 2: Choose a plane for the new connection using
any existing path searching algorithm
(sequential, quasi-random, etc.).
Step 3: If a plane for a new connection is determined
- go to step 8, otherwise go to step 4.
Step 4: Construct following sets of planes:
SA = set of planes through which connections
of type A are set up.
SE= set of planes through which connections
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(0;0)
(1;8)
(1;8)
(2:2)
(2;2)
(3:3)
(3;3)
(4; 1)
--0 --{)
8
(4: 1)
--0 --{) 8
Fig. 7. The switching fabric's state with connection (0;0)
Fig. 5. The switching fabric's state, where four connections block
connection (0;0)
(1;8)
(3:3)
(4: 1)
--0 --{)
8
Fig. 6. The switching fabric's state after disconnection (2; 2)
of type B are set up.
Se = set of planes through which connections
of type C are set up.
Step 5: Find connection C e in plane c from Se which
can be moved to one of the planes in (SA + SB),
say it is plane a.
Step 6: Disconnect Ce .
Step 7: Connect Ce in plane a. Determine plane c for
the new connection
Step 8: Set up the new connection through the
determined plane .
Let us assume, that in log2(16, 0, 4) switching fabric
following connections are set up: from input 1 to output
8 in plane 1, from input 2 to output 2 in plane 2, from
input 3 to output 3 in plane 3, and from input 4 to output
1 in plane 4. This state is shown in Fig 5. In this state
there is no plane available for the new connection from
input 0 to output 0, and one of existing connections
must be re-switched to unblock connection (0; 0). For
this new connection, connection (1; 8) is the type A
connection, connection (4; 1) is the type B connection,
and connections (2;2) and (3; 3) are type C connections.
According to Algorithm 1, we are looking for a type C
connection «(2; 2) or (3; 3» , which can be re-switched
to another plane (i.e. this connection occupies a plane
which is not used by connection (1;8) or (4; 1) . The
connection chosen by Algorithm 1 for re-switching is
connection (2; 2), so it is disconnected in plane 2 (the
state of switching fabric after this disconnection is shown
in Fig 6), after this the same connection is set up in
plane 1. After such operation plane 2 is available for
connection (0; 0) and this connection can be set up. The
state after these rearrangements is shown in Fig. 7.
B. Non-Interruptive Rearrangements
In Algorithm 1 it is assumed, that connection C e is
first disconnected in plane c and then it is connected in
plane a. However, it is possible to set up connection Ce
in plane a without disconnecting it in plane c. This is
possible when de-multiplexers and multiplexers can set
up connections from one input to many outputs (at least
2) and from many inputs (at least 2) to one output. In
this case for some period of time connection Ce uses
two connecting paths. This state of the switching fabric
is shown in Fig. 8. After connection C; is set up in
plane a it can be disconnected in plane c. The modified
algorithm which allows non-interruptive rearrangements
is as follows :
Algorithm 2:
Step 1: Wait for a new connection.
Step 2: Choose a plane for the new connection using
any existing path searching algorithm
(sequential, quasi-random, etc .).
Step 3: If a plane for a new connection is determined
- go to step 8, otherwise go to step 4.
Step 4: Construct following sets of planes:
SA= set of planes through which connections
of type A are set up.
SB= set of planes through which connections
of type B are set up.
Se= set of planes through which connections
of type C are set up.
Step 5: Find connection Ce in plane c from Se which
can be moved to one of the planes in (SA + SB) ,
say it is plane a.
Step 6: Connect Ce in plane a. Determine plane c
for the new connection
Step 7: Disconnect Ce •
Step 8: Set up new connection through the determined
plane .
Let us assume, that Algorithm 2 works in the
log2(16; 0; 4) switching fabric , and an actual state of
such fabric is the same as shown in Fig 5, i.e. the set
of existing connections is the same as described earlier.
In this state the connection (0; 0) is blocked, i.e. there
is no plane available for this connection. To unblock
this connection Algorithm 2 will choose the connection
(2; 2) for re-switching. According to Algorithm 2, this
connection first will be set up in it's final plane (the state
of switching fabric after this operation is shown if Fig .
10), and then the same connection will be disconnected
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connections. It means that the number of rearrangements
needed in this switching fabric is equal to one. We have
also proposed the control algorithm for choosing this
o
0
(1;8)
(2;2)
(3;3)
(4;1)
Fig. 8.
Fig. 10.
Connection (i ,j) with two connecting paths
»r--:- (1_St;_
rl_I_
)
r - - - - - -_+(
--Q
--0 8
The switching fabric's state with duplicates of connection
(2;2)
connection for rearrangement. The modification is also
proposed which ensures, that this rearrangement can
be done in the switching fabric in the non-interruptive
manner. The similar algorithm for the switching fabric
with odd number of stages is under consideration.
___
Waiting for new connection
REFERENCES
A new
blocked?
connection
n_is
'0 3
4
y_esl
-r-r
No
[I] H.-Y. Ng, M. R. Wang, D. Li, X. Wang, J. Martinez , R.
[2]
[3]
Set up a new connection
in one of available planes
Fig. 9. Algorithm 2 - Algorithm for non-interruptive rearranging
connections
[4]
[5]
[6]
in its original plane (the final state of the switching
fabric is the same as in switching fabric controlled by
Algorithm 1, and it is shown in Fig. 7). In switching
fabrics controlled by Algorithm 2 the reswitched connection will not be interrupted. Moreover, sometimes
reswitched connections will have its duplicate routes in
different planes (such situation is shown in Fig. 8). Such
realization of rearrangements guaranties non-interruptive
transmission of information all the time.
V. CONCLUSIONS
In this paper we considered rearrangeable
l092(N, 0, p) switching fabrics with even number
of stages. We have proved, that the blocked connection
can be unblocked by rearranging only one of existing
[7]
[8]
[9]
[I 0]
[I I]
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