Wide-Sense Nonblocking Multicast
Log
2
(N; m; p) Networks
Frank K. Hwang and Bey-Chi Lin
Abstract—Recently, Tscha and Lee proposed a fixed-size
window algorithm for the multicastLog2( 0 ) network and expressed a desire to see its extension to theLog2( ) net-work. Later, Kabacinski and Danilewicz generalized the fixed-size window to variable size to improve the results. In this paper, we further extend the variable-size results from the Log2( 0 ) network to Log2( ). Note that this extension is difficult since each link in the channel graph of Log2( 0 ) has the same blocking effect, but not so in Log2( ). We also determine the optimal window size and optimal .
Index Terms—Channel graph,Log2( ) networks,
multi-cast, wide-sense nonblocking (WSNB) network, window algorithm.
I. INTRODUCTION
F
IG. 1 shows an inverse banyan network with ( ) stages and inputs (outputs), and also a with ( ) extra stages, which are mirror im-ages of the first stages.Lea and Shyy [5] first proposed the network, which consists of a vertical stacking of copies of , , sandwiched between and connected to an input stage and an output stage, each with (or ) cross-bars. As shown in Fig. 2, there are three copies of
sandwiched between the input and output stages.
A multicast network is strictly nonblocking if the current re-quest can always be connected regardless of how previous con-nections were routed, it is wide-sense nonblocking (WSNB) if the connection of the current request is assured only when all connections are routed according to a given algorithm.
Tscha and Lee [7] proved that is multicast strictly nonblocking if
for even for odd
However, Kabacinski and Danilewicz [4] pointed out that their proof using “windows” to split a multicast call implies a routing algorithm, hence, their result is WSNB instead of strictly nonblocking. Recently, Kabacinski and Danilewicz [4] extended the fixed window-size algorithm in [6] to variable window size. Tscha and Lee [7] stated in conclusion that whether their approach could be extended to
was unclear. Danilewicz and Kabacinski [2], [3] made such
Paper approved by P. E. Rynes, the Editor for Switching Systems of the IEEE Communications Society. Manuscript received November 29, 2001; revised December 20, 2002 and April 11, 2003.
The authors are with the Department of Applied Mathematics, Na-tional Chiao Tung University, Hsin Chu, Taiwan 30050, R.O.C. (e-mail: [email protected]).
Digital Object Identifier 10.1109/TCOMM.2003.818093
Fig. 1. (Extra-stage) inverse banyan networks.
Fig. 2. Log (8; 1; 3).
an attempt but encountered some difficulties. In this paper, we give such an extension for the variable window-size algorithm by adopting a channel graph blockage analysis first used by Shyy and Lea [6] on a single-cast network. The
network is much more difficult to analyze because of multipaths in the channel graph and each link having a different impact on blockage. We also determine the optimal window size for given
, and then compare the performance among different . II. GENERALAPPROACH
Define . Tscha and Lee [7] partitioned the out-puts of into windows, each containing the
outputs reachable from the same crossbar at stage
. Kabacinski and Danilewicz [4] extended the notion of window to -window, , which consists of the outputs reachable from the same crossbar at stage . In other words, if the outputs are labeled by binary sequences, then a -window consists of those outputs, which have the same most significant bits. Although an output can be reached by
Fig. 3. A 2-window ofBY (4; 2).
Fig. 4. A channel graph ofBY (n; m).
crossbars at stage , each such crossbar reaches the same window due to the well-known “buddy” property of banyan type networks. Fig. 3 shows that the outputs {0,1,8,9}, reachable from the first crossbar at stage five, form a 2-window of . We assume to avoid trivial cases.
A channel graph between an input crossbar and an output crossbar is the union of all paths between them (see Fig. 4). In , all channel graphs are isomorphic with the fol-lowing double-tree form (two binary trees with their leaves linked by paths in a one-to-one fashion).
The channel graph of a multicast call is simply the union of its point-to-point channel graphs.
Following Tscha and Lee [7], we split a multicast request into multicast requests if the involved outputs spread into win-dows, while each request must be routed through the same copy of . When we are discussing a multicast request with respect to a given -window, we refer to it as the
desig-nated -window. Further, a -window is designated if it
con-tains the designated -window. As Tscha and Lee [7] dealt only with , the connection from an input to an output is unique, and whether two connections intersect is determined. Therefore, an intersection graph among the connections within a designated -window can be defined, and its maximum degree plus one becomes the number of copies of suf-ficient for nonblocking.
For , the analysis is much more complicated as the connection between an input and an output is not unique. First of all, we have to be more specific about the window algo-rithm. We propose the delayed-splitting -window algorithm, which prescribes that a multicast connection to outputs in the same -window cannot be split before stage ( ).
Note that further delay is not always possible, since stage is the last stage where all outputs in the same window have common reachable crossbars. Also note that such an algo-rithm fixes only the relative routing of two outputs in the same -window, , but not the absolute routing to an output. Thus, whether two connections intersect is uncertain and the notion of an intersection graph used by Tscha and Lee [7] is not applicable. Instead, we adopt the method of channel graph blockage analysis, first proposed by Shyy and Lea [6] for single cast.
A link connecting stage and stage ( ) is called a
stage-link. Consider a -cast request in a -window. An intersecting connection is one which contains a link in the channel graph of
the request. We can count an intersecting connection either from its input end or its output end. An intersecting connection is an -intersecting connection if it first (last) intersects the channel graph in a stage- link when counted from the input (output) side.
We count all -intersecting connections,
, from the output side. Note that the outputs of these con-nections must all be in the designated -window. Thus, there are, at most, of such connections. Further, they have different impacts in blocking the paths in the channel graph, depending on . For example, for , an ( )-intersecting con-nection blocks a proportion of 1/2, since the channel graph has only two stage-( ) links, while an ( )-inter-secting connection blocks a proportion of 1/4, since the channel graph has four stage-( ) links.
On the other hand, we will count all -intersecting
con-nections, , from the input side.
Again, an -intersecting connection has a greater (or equality permitted) blocking impact than an ( )-intersecting call for . We will show that we never need to count from the input side over the stage . Therefore, we adopt the method used in [4] to count from small to large to maximize the blocking impact.
III. A NECESSARY ANDSUFFICIENTCONDITION FORNONBLOCKING
For any two stages in a multistage network, let and denote two crossbars at stage , and and be two sets of crossbars at stage that and can reach, respectively. Then the network is said to have the buddy property if either
or . It is well known [1] that
and many other networks have the buddy property. Note that in a buddy network, the set of inputs which can generate an in-tersecting connection to a multicast request is independent of the size of that request. To see this, consider a 2-cast call from input to two outputs and . Then an input can gen-erate a -intersecting connection (at a crossbar ) to the path from to if and only if it can generate a -intersecting con-nection (at a crossbar ) to the path from to , since the buddy property assures that if can reach , it can reach . Hence, increasing the size of the request does not increase the number of inputs which can generate intersecting connections, but the fact that these outputs are in the request makes them unavail-able as outputs to generate intersecting connections (see Fig. 5,
Fig. 5. Input 4 generates a 3-intersecting connection (4, 4) to (a) a 1-cast request (0, 0) and (b) a 2-cast request (0, {0, 8}).
for example). Further, each intersecting connection blocks one copy, so it is the number of intersecting connections that counts. Obviously, a 1-cast request maximizes that number.
For , although the same analysis on the number of intersecting connections applies, the -intersecting connec-tions block different fracconnec-tions of a copy, depending on . Since more outputs in a multicast request induce more -intersecting calls for larger , the worst case is not necessarily a 1-cast re-quest.
We consider two cases.
A.
The number of stage- links, , in the channel graph is constant, one for , and two for . Therefore, each intersecting connection has the same impact, regardless of which stage it intersects. The worst case occurs when there is a maximum number of intersecting connections, i.e., from the designated window, which cause a blocking of
copies.
B.
Let denote the part of the new request which goes to a des-ignated -window. Suppose is -cast and a 1-window contains
outputs in . Then it can block, at most if
only for the window which is in the designated window
if if
For instance, in Fig. 6, the first output crossbar corresponds to the case , and the third output crossbar corresponds to the
case .
Therefore, a 1-window can block, at most, 1/2 copy of the channel graph. Consequently, a -window can block, at most, copies, which is achieved by having either (each 1-window has ) or (half of the 1-window has
and half has ).
To count -intersecting connections for we consider two cases.
A.
The argument for this part is a straightforward extension of the argument in [4] for .
Fig. 6. Assume = 2 and (0, 0) is the request. r = 1 in the first output crossbar and connection (6, 1) blocks 1/2 copy, whiler = 0 in the third output crossbar and connections (4, 4) and (5, 5) each blocks 1/4 copy. Dotted lines indicate channel graph between the first input and the first output crossbar.
There are inputs which can generate an -intersecting connection. Further, an -intersecting connection can reach all windows for , and windows for . In the worst-case scenario, an -intersecting connection is a multicast connection going to one output in each window it can reach, except the designated window for . The reason for the exception is that all outputs in the designated window are already counted in the part concerning
. Since an -intersecting connection blocks copies for
and copies for , the total
blocking of up to stage is
for and
for
Note that these -intersecting connections, , use
up a maximum of outputs in a window.
Therefore, one ( )-intersecting connection can still fit in
if , or , which is the
case here. This ( )-intersecting connection reaches
windows for , and windows for ,
while each path to a window blocks copy.
To summarize, the number of blockings from the input side is for
for
B.
Then . Note that -intersecting connections for are counted from the output side. So the input side counts only up to stage (which is
Fig. 7. Connection (1, 8) blocks 1/2 copy if counted from the input side, but only 1/4 copy from the output side. Dotted lines indicate channel graph between the first input and the first output crossbar.
upper bounded by ). Thus, the number of blockings from the input side is
Since each intersecting connection counted from the output side blocks in the worst-case scenario, i.e., or , at least 1/4 copy, there is no reason for the counting from input side to go over stage , with one exception.
For , we can increase the blocking by allowing the unique 1-intersecting connection from the input side to also go to the designated window to reach an output blocking 1/4 copy (such an output exists when ). Then this inter-secting connection blocks 1/2 copy if counted from the input side, greater than its original value 1/4, as counted from the output side (see Fig. 7, for example). Note that no other such reversal of counting will bring any further increase, since the 1-intersecting connection is the only one which blocks more than 1/4 copy when counted from the input side. On the other hand, since all intersecting connections counted from the input side are before the middle stage, reversing them to the output side will only decrease their impact on blocking.
Combining the above, we have:
Theorem 1: is WSNB for broadcast under the -window algorithm if and only if is as shown in the equation at the bottom of the page.
Results for correspond to the results in [4]; results for 1, 2 correspond to the results in [2] and [3].
Note that is the Cantor network.
Corollary 2: The Cantor network is WSNB for broadcast
under the -window algorithm if and only if
(0 if ), for . IV. OPTIMIZATION
Let denote the maximum number of blockings re-quired in Theorem 1 for given and . In this section, we de-termine optimal for given and , and also compare the optimal solutions among different .
is decreasing in for . Hence, in that range. Since
for
we conclude for and , . It was shown
in [4] that is a better choice than . Since for has a unique minimum, we can start with and increase the window size until increases. In general, grows slowly with rate and can be quickly found.
is decreasing in for .
Since
for . Again, has a unique minimum, and is a good value to start the upward searching. Finally, for , we note that is increasing in for all . Since a larger implies more stages and larger cost, there is no reason to consider when it costs more but performs worse. For
for for for for for for for if for for
TABLE I
BESTCHOICE OFANDCORRESPONDINGVALUE OFpFORm = 2ANDSOMEn
The first equation is decreasing in in its range. Hence, .
Since
for . has a unique minimum and is a good value to start the upward searching.
We next compare the optimal solutions for 0, 1, 2. We will only compare the starting values in the search process.
Clearly, .
Furthermore
So does better in minimizing the number of copies required. However, we have to recall that a copy with or costs less. For all three values, the number of
crosspoints is about .
According to the above result, we choose , and com-pute the best choice of and the corresponding value of for each in Table I.
Note that for , two ’s yield the same -value. For larger in the table, we show the -values mainly for mathe-matical interest, not for practical use.
V. CONCLUSION
We extended the study of a multicast network in [2] and [5] to a multicast network by refining
their window algorithm. We obtain necessary and sufficient con-ditions on such that the network is WSNB. We also estimate the optimal window size.
Intuitively, one would expect the larger is, the more con-necting power the is, and hence, the fewer copies are needed for nonblocking. One would also expect the optimal grows with . We obtain the surprising result that is optimal universally. But this is a technical result, for which we have no insight into why it is so. Nonetheless, it is a very valu-able result, since regardless of how large is , we need only
to use moderate-size , i.e., , which
are relatively inexpensive to construct.
Like all routing algorithms, the delayed splitting algorithm restricts the scope of ways in connecting a multicast call. But it also restricts the scope of interference a multicast connection has on other requests. It is a tradeoff whose net value we do not know for sure. However, the delayed splitting algorithm sim-plifies routing to a degree that an analysis of the nonblocking condition becomes tractable.
REFERENCES
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[5] C.-T. Lea and D.-J. Shyy, “Tradeoff of horizontal decomposition versus vertical stacking in rearrangeable nonblocking networks,” IEEE Trans.
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[6] D.-J. Shyy and C. T. Lea, “Log (N; m; p) strictly nonblocking net-works,” IEEE Trans. Commun., vol. 39, pp. 1502–1510, Oct. 1991. [7] Y. Tscha and K.-H. Lee, “Yet another result on multi-log N networks,”
Frank K. Hwang received the B.A. degree from
Na-tional Taiwan University, Taipei, Taiwan, R.O.C., in 1960, and the Ph.D. degree from North Carolina State University, Raleigh, in 1968.
He worked at the Mathematics Center at Bell Labs from 1967–1996. He is now a Chair-Professor at National Chiao Tung University, Hsin Chu, Taiwan, R.O.C. He has published about 350 papers and written or coauthored the following books: The
Steiner Tree Problem (Amsterdam, The Netherlands:
North-Holland, 1992); Combinatorial Group Testing
and Its Applications (Singapore: World Scientific, 1993, 2nd edition, 2000); The Mathematical Theory of Nonblocking Switching Networks (Singapore:
World Scientific, 1998); and Reliabilities of Consecutive-k Systems (Norwell, MA: Kluwer, 2000).
Bey-Chi Lin received the B.S. degree in mathematics
education from National Teachers College, Taichung, Taiwan, R.O.C., in 1999. She is currently a Ph.D. student in the Department of Applied Mathematics, National Chiao Tung University, Hsin Chu, Taiwan, R.O.C.
Her research interests include network analysis and combinatorial mathematics.
