Wide-sense nonblocking for multi-log
d
N networks under various
routing strategies
F.H. Chang, J.Y. Guo
∗, F.K. Hwang
Department of Applied Mathematics, National Chiaotung University, Hsinchu, ROC Taiwan, 300
Received 27 July 2005; received in revised form 1 October 2005; accepted 12 October 2005
Communicated by D.-Z. Du
Abstract
Chang et al. showed that the number of middle switches required for WSNB under strategies: save the unused, packing, minimum index, cyclic dynamic, and cyclic static, for the 3-stage Clos network C(n, m, r) with r 3 is the same as required for SNB. In this paper, we prove the same conclusion for the multi-logdN network. We also extend our results, except for the minimum index strategy, to a general class of networks including the 3-stage Clos network and the multi-logdN network as special cases. © 2005 Elsevier B.V. All rights reserved.
1. Introduction
The symmetric 3-stage Clos network C(n, m, r) which has r switches of size n × m in the first stage, m switches of size r × r in the second(middle) stage, and r switches of size n × m in the third stage (see Fig.1).
The multi-logdN network, first proposed by Lea [7], is composed of p copies of logdN network connected in parallel (see Fig. 2). Each copy of the logdN network, also called banyan-type networks, is constructed of d × d switches arranged in n stages, N = dn, labeled 1, 2, . . . , n from left to right. Each stage has dn−1d × d switches. In each copy, there is exactly one path between an arbitrary input and an arbitrary output. There are many varieties of logdN networks, such as banyan, Omega, baseline, . . . , but they are all equivalent in the sense that the connection property is invariant under a permutation of switches in the same stage.
A request is an (input, output) pair seeking connection. A set of requests can be routed if there exists connecting paths not intersecting each other in a node.
A multi-logdN network is said to be strictly nonblocking (SNB) if a request can always be routed regardless of how the previous pairs are routed. It is said to be wide-sense nonblocking (WSNB) with respect to a routing strategy A if every request is routable under A. It is said to be rearrangeable nonblocking (RNB) if every request can be connected provided routing paths of existing connections can be rearranged (rerouted).
For convenience of analysis, we transform a logdN network to a digraph by converting each link, including the inputs and the outputs, to a node, while a crosspoint connecting two links in the network becomes an arc in the digraph
∗Corresponding author. Tel.: +886 937699688.
E-mail address:[email protected](J.Y. Guo). 0304-3975/$ - see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.tcs.2005.10.048
log216
log216
log216
Fig. 1. C(3, 5, 4). Fig. 2. A multi-log216 network with 3 copies of log216 networks.
0 1 15 0 1 15 (a) (b)
Fig. 3. (a) A 16× 16 binary baseline network and (b) its graph model.
(see Fig.3). Nodes are arranged in n + 1 stages labeled 0, 1, . . . , n from left to right. The nodes in stage 0 correspond to inputs and the nodes in stage n correspond to outputs. The restraint that no two paths in the original network competes for the same link is translated to that no two paths in the derived network(digraph) competes for the same node.
For the 3-stage Clos network, a routing strategy deals with the choice of a middle switch to route the request when many are available. Five routing strategies have been proposed in the literature (see [2] for a survey):
(i) Save the unused (STU). Do not route through an empty middle switch unless there is no choice. (ii) Packing (P). Choose a busiest, yet available, middle switch.
(iii) Minimum index (MI). Label all middle switches from M1to Mp. For each request, route in the order M1, M2, . . . ,
until the first available one emerges.
(iv) Cyclic dynamic (CD). If Mkwas used last, try Mk+1, Mk+2, . . . , until the first available one emerges.
(v) Cyclic static (CS). If Mkwas used last, try copy Mk, Mk+1, . . . , until the first available one emerges.
The existence of a WSNB network was first demonstrated by Beneš[1] for the symmetric 3-stage Clos network. He proved that C(n, m, 2) is WSNB under packing if and only if m3n/2 which is the only positive result. Smith [9] proved that C(n, m, r) is not WSNB under P or MI if m < 2n − n/r, which was improved to 2n − (n/(2r − 1)) in Du et al. [3] and extended to all five strategies. For P, Yang and Wang [11] gave a linear programming formulation of the problem and ingeniously found the closed-form solution m2n − n/F2r−1 where F2r−1is the 2r − 1st Fibonacci
number, as a necessary condition for C(n, m, r) to be WSNB. Note that for r large, all the above negative results show that 2n − 1 middle switches are required for WSNB. Tsai et al. [10] culminated this line of results by giving a unifying proof for all possible strategies, not just the listed five.
For finite r, Du et al.[3] proved that for r 3 C(n, m, r) is WSNB for P or STU if and only if it is SNB, namely, m2n − 1, with a complicated proof. Chang et al. [2] simplified the proof and extended it to the other three strategies for r 2; thus severely dashing the hope that a clever strategy can save hardware from SNB networks and yet preserve nonblockingness. We can translate the five routing strategies to the multi-logdN network by replacing “choosing a middle switch” to “choosing a copy (of logdN)”. In Section 2, we prove a similar conclusion that these five strategies require the same number of copies as SNB does. In Section 3, we extend our results to a general class.
Presumably, one can ask the same question for RNB, namely, how many middle switches are required for RNB if a certain routing strategy is followed. This has not been studied in the literature, not even for C(n, m, r). The reason is because RNB can also be interpreted as nonblocking if all requests are to be routed simultaneously [1,6]. Then there exist better routing strategies yielding the requirements of n middle switches for the 3-stage Clos network [4] and dn/2 copies for the multi-logdnetwork [8]; showing that the cost of RNB is much less than that of SNB.
2. Main result
Shyy and Lea [8] proved the following theorem for d = 2 and Hwang [5] extended it to the d-nary version. Theorem 1. Multi-logdN network is strictly nonblocking if p p(n), where
p(n) =
(d + 1) × dn2−1− 1 for n even,
2× dn−12 − 1 for n odd.
A request from input x to output y, represented by (x, y), has a unique path in a logdN network. Hence two intersecting
paths must be routed through different copies of logdN network.
Theorem1 was stated in [5] only as a sufficient condition. We need prove that it is also necessary. Theorem 2. Multi-logdN network is strictly nonblocking only if p p(n).
Proof. For any request = (x, y), assume that the path of consists of links L0, L1, . . . , Ln. For n odd, let I1(O2) be
the set of inputs(outputs), except x(y), which can reach L(n−1)/2, then|I1| = d(n−1)/2− 1 and |O2| = d(n+1)/2− 1.
Let O1(I2) be the set of outputs(inputs), except y(x), which can reach L(n+1)/2. Then |O1| = d(n−1)/2− 1 and
|I2| = d(n+1)/2 − 1. Note that cannot be routed through the same copy with any request from I1 to O2 or I2
to O1. Suppose p = p(n) − 1 while |I1| requests from I1to O2\ O1 and|O1| requests from O1 to I2\ I1 have
already been connected in different copies. In this case, they can occupy|I1| + |O1| = p(n) − 1 = p copies, with
no copy left for. For n even, let I1(O2) be the set of inputs(outputs), except x(y), which can reach Ln/2−1, then
|I1| = dn/2−1− 1 and |O2| = dn/2+1− 1. Let O1(I2) be the set of outputs(inputs), except y(x), which can reach
Ln/2+1. Then|O1| = dn/2−1− 1 and |I2| = dn/2+1− 1. Let I3(O3) be the set of inputs(outputs), except x(y), which
can reach Ln/2. Then|I3| = |O3| = dn/2− 1. Note that cannot be routed through the same copy with any request
from I1to O2, I2to O1, or I3to O3. Suppose p = p(n) − 1 while |I1| requests from I1to O2\ O3,|O1| requests from
O1to I2\ I3, and|I3\ I1| requests from I3\ I1to O3\ O1have already been connected in different copies. In this
case, they can occupy|I1| + |O1| + |I3\ I1| = |I1| + |O1| + |O3\ O1| = p(n) − 1 = p copies, with no copy left for
. Hence p must be greater than or equal to p(n).
We call such a set of p(n) − 1 requests blocking the maximal blocking configuration (MBC), denote by M(n, ). Note that if a network is SNB, then it is also WSNB. i.e. multi-logdN is WSNB if p p(n). Therefore, we only
need to prove necessity in the following proofs. In all these proofs, we assume that the network carries no traffic at the beginning.
We consider strategy CD first.
Theorem 3. Multi-logdN network is WSNB under CD if and only if p p(n).
Proof. Suppose p < p(n). Consider a sequence of p +1 requests with p requests from M(n, ) followed by the request . By the property of strategy CD, these p requests will be routed in p copies. Then we cannot route any more. Hence p must be greater than or equal to p(n).
For strategy CS,
Theorem 4. Multi-logdN network is WSNB under CS if and only if p p(n).
Proof. Suppose p < p(n). For a request and any p requests of M(n, ), say 1, 2, . . . , p, route1in copy 1, then
route in copy 2(because 1blocks in copy 1). Then disconnect and route 2in copy 2. Then route in copy 3.
Again disconnect it and route3in copy 3. Doing this iteratively untilpis routed in copy p. Then cannot be routed any more. Hence p must be greater than or equal to p(n).
For strategies P or STU, we introduce a lemma.
Lemma 5. For any request and M(n, ), there exists a request which does not block or any request in M(n, ) in the logdN network.
Proof. Use the graph model of the baseline network as an example. Without loss of generality, let = (0, 0). For all
requests (i, j ) in M(n, ), we obtain i < N/d and j < N/d. Hence = (N − 1, N − 1) will satisfy our claim.
Theorem 6. Multi-logdN network is WSNB under P or STU if and only if p p(n).
Proof. Suppose to the contrary, p < p(n). For any request and any p requests of M(n, ), say 1, 2, . . . , p, we
route1in copy 1 first. Then route in copy 2 and route in copy 2 (because copy 1 are as busy as copy 2, we can
choose copy 2). Now, we disconnect and route 2in copy 2. Then disconnect. Similarly, we route in copy 3 and
in copy 3, then disconnect and route 3in copy 2. Finally, we routepin copy p. Then cannot be routed any more.
Hence p must be greater than or equal to p(n).
MI is more complicated. We first introduce a result in[2].
Theorem 7. The 3-stage Clos network C(n, m, r) for r 2 is WSNB under MI if and only if m2n − 1.
In the following theorem, only the baseline architecture will be considered. However, the theorem is also true for other equivalent logdN network.
Theorem 8. Multi-logdN network is WSNB under MI if and only if p p(n). Proof. We discuss two cases:
(i) n is odd. Select two subset I1and I2of inputs and two subset O1and O2of outputs. Set I1= O1= {0, 1, 2, . . . ,
d(n−1)/2− 1}, I2= O2= {d(n−1)/2, . . . , 2 × d(n−1)/2− 1}. See Fig.4. By the configuration of baseline network, every request from I1to O1∪ O2must intersect node 0 in stage (n − 1)/2 and every request from I2to O1∪ O2
must intersect node 1 in stage (n − 1)/2. Therefore, for i = 1 or 2, all requests from Ii to O1∪ O2must use
different copies. Similarly, every request from I1∪ I2to O1must intersect node 0 in stage (n + 1)/2 and every
request from I1∪ I2to O2must intersect node d(n−1)/2in stage (n + 1)/2. Therefore, for i = 1 or 2, all requests
from I1∪ I2 to Oi must use different copies. Now, we match this to a 3-stage Clos network C(d(n−1)/2, 1, 2), where Iiis the ith input switch, Oiis the ith output switch, for i = 1 or 2, and the complete bipartite graph induced by nodes 0 and 1 of stage (n − 1)/2 and nodes 0 and d(n−1)/2 of stage (n + 1)/2 is the middle switch. Then a request (i, j ) in C(d(n−1)/2, p, 2) routed through the kth middle switch under MI corresponds to a request (i, j ) in the multi-logdN using copy k. Therefore, by Theorem 7, the network is not WSNB if
p < 2 · (dn−12 ) − 1 = 2 × dn−12 − 1 = p(n).
(ii) n is even. Select four subset I1, I1, I2and I2 of inputs and four subset O1, O1, O2, and O2 of outputs. Set I1=
O1= {0, 1, 2, . . . , dn/2−1− 1}, I1 = O1 = {dn/2−1, . . . , dn/2− 1}, I2= O2= {dn/2, . . . , (d + 1)dn/2−1− 1},
I1 I1 I2 O1 O2 I2 O1 O2 stage 2 n-2 2 n-1 2 n+2 2 n+2 0 2 n-1 d 2 n-1 d 2 n-1 d 0 1 0 0
Fig. 4. The left figure is an induced graph of the graph model of a multi-logdN network, for n odd. And the right figure is its correspondence to a
3-stage Clos network.
intersect node 0 in stage n/2 − 1, every request from I2to O1∪ O2must intersect node d in stage n/2 − 1, every
request from I1∪ I2to O1must intersect node 0 in stage n/2 + 1, and every request from I1∪ I2to O2must
intersect node dn/2in stage n/2 + 1. Similar to case (i), we can treat I1, I2, O1, O2as the inputs and outputs of
C(dn/2−1, 1, 2), and the subgraph sketch in bold line in Fig.5 is the middle switch. Therefore, by Theorem 7, the network is not WSNB if
p < 2 · (dn2−1) − 1. (1)
Besides, we observe that, for i, j = 1, 2, every request from Ii to Oj must block every request from Iito Oj in
the same node in the stage n/2. Therefore, if we connect all (d − 1)dn/2−1requests in Iito Oj in copy 0 to copy
(d − 1)dn/2−1− 1 before every time we connect a request from I
i to Ojand disconnect them after connected,
then we can force the copy chosen to route begin at least (d − 1)dn/2−1th copy. Hence (1) can be enlarged to
p < 2 × (dn2−1) − 1 + (d − 1)dn2−1= p(n).
Note that, in Theorem8, it does not need to consider all inputs and outputs, because I1∪ I2and O1∪ O2are enough
to force p p(n) which is the bound of SNB. 3. Some generalizations
We extend our results to a class of networks including the 3-stage Clos networks, the multi-logdN and the logd(N, k, m) networks as special cases.
A vertical-copy network V consists of an input stage of r1(n1×m)-crossbars, an output stage of r2(m×n2)-crossbars
and a middle stage of m copies of a network with r1 inputs and r2outputs. There exists exactly one link between
each input(output) crossbar and each copy of. When is the r1× r2crossbar, V is a 3-stage Clos network. When
n1= n2= 1 and is the logdN network, V is a multi-logdN network. When n1= n2= 1 and is the k-extra-stage
logdN network, then V is the logd(N, k, m) network. In particular, if k = n − 1, then V is the Cantor network. Suppose that the necessary and sufficient condition for to be SNB is known. Consider p = p(n) − 1. For any request, there must be a state s such that is blocked in each of the p(n) − 1 copies 1, 2, . . . , p(n)−1. Let Ribe the
set of all requests routing throughi in s and M(, ) = {Ri| i = 1, 2, . . . , p(n) − 1}. i.e., V is SNB if and only if the number of copies is larger than|M(, )|. Let “Route Riinj” mean “Route all requests in Ri injconsecutively”. Theorem 9. A vertical-copy network V is WSNB under the CS routing if and only if V is SNB.
I1 I2 O1 O′ O2 O′ 1 2− n 1 2− n 2 n 1 2+ n 0 0 0 d 2 n d 1 d-1 d d+1 2d-1 1 0 0 I1′ -1 d -1 2 1 2 n d -1 2 n d 2 n 2d -1 I′2 2 n 2 n 2 n 2 n 2 n 2 n 2 n 2 n 2 n 2 n
Fig. 5. This is an induced graph of the graph model of a multi-logdN network, for n is even.
Proof. Suppose there are p < p(n) copies 1, 2, . . . , pin V . For a request , we route R1in1, then route in 2
( is blocked in 1). Then disconnect and route R2in2. Then route in 3. Again disconnect it and route R3in3.
Doing this iteratively until Rpis routed inp. Then cannot be routed in any copy. Hence p must be greater than or
equal to p(n).
For CD, we use another argument.
Theorem 10. A vertical-copy network V is WSNB under the CD routing if and only if V is SNB.
Proof. First, we claim every request can be routed in kfor a given k. Route in i. If i = k, then disconnect and route it again ini+1. Similarly, if i + 1 = k, then disconnect and route it again in i+2until is routed in k. Note
that if i = p, then we let i + 1 be 1. Therefore, if p < p(n), then we can route Riini for i = 1 to p as we want. Then cannot be routed in any copy. Hence p must be greater than or equal to p(n).
For STU, if there exists a requesti which does not block{} ∪ Ri for all i, Theorem6 remains true if M(n, ) is replaced by M(, ) and iis replaced by Ri. But we use a different argument for P.
Theorem 11. Suppose there exists a requesti which does not block{} ∪ Ri for all i. A vertical-copy network V is WSNB under the P routing if and only if V is SNB.
Proof. It suffices to prove the “only if” part. Suppose there are only p = p(n) − 1 copies 1, 2, . . . , pin V . For
the request = (0, 0), without loss of generality, suppose Ri = {i,j| j = 1, . . . , i} and 12 · · · p. Let|i|
denote the number of connections ini. For a given k, let s(k, B) be a state satisfying the following conditions: (i) |k| < k,
(ii) Connections ini are those from Ri,
(iii) |i| = |k| + 1 or |i| = i if i ∈ B ≡ {i | |i| > |k|}
Let S(k) denote the state that i contains Rifor all 1i k. We make two claims: Claim A. We can add another connection of Rkink in state s(k, B).
a b c d a b c d
Fig. 6.and M(V ,) = {a, b, c, d} in C(3, 4, 2).
We prove both claims by induction on k. For k = 1, then B = ⭋. Clearly, we can add to 1, and keep on adding
other connections untilicontains Ri. So consider general k > 1. From s(k, B) we can obtain the state s∗(k, B), which
differs from s(k, B) by having icontaining Rifor all 1i k − 1, by applying induction to claim B(with k = k − 1).
In state s∗(k, B), must be routed in k. Now delete all connections in s∗(k, B) \ s(k, B) so that |k||i| for all i.
Thenkcan be routed ink. Delete and route in k. Deletekand Claim A is proved. Also, we can keep on adding all remaining connections of Rktokto prove Claim B.
Setting k = p in Claim B, then cannot be routed in any of the p copies. Hence at least p(n) copies are needed.
Example 1. For simplicity, we will represent a state by its||-sequence. To help clarify the state, let |i|∗denote the fact that is in the i,|i|the fact thatis and|i|the fact that both are. Suppose p = 3 and we want to reach the state S(3) = (1, 2, 3) = (2, 3, 4). The the ||-sequence of our construction in Theorem11 would be:
(0, 0, 0) ⇒ (1, 0, 0) ⇒ (2, 0, 0) ⇒ (2, 1∗, 0) ⇒ (1, 1∗, 0) ⇒ (1, 2, 0) ⇒ (1, 1, 0) ⇒ (1, 2, 0) ⇒ (1, 1, 0) ⇒ (2, 1, 0) ⇒ (2, 2∗, 0) ⇒ (2, 3, 0) ⇒ (2, 2, 0) ⇒ (2, 3, 0) ⇒ (2, 2, 0) ⇒ (2, 3, 0) ⇒ (2, 3, 1∗) ⇒ (1, 1, 1∗) ⇒ (1, 1, 2) ⇒ (1, 1, 1) ⇒ (1, 1, 2) ⇒ (1, 1, 1) ⇒ (2, 1, 1) ⇒ (2, 2∗, 1) ⇒ (2, 3, 1) ⇒ (2, 2, 1) ⇒ (2, 3, 1) ⇒ (2, 2, 1) ⇒ (2, 3, 1) ⇒ (2, 3, 2∗) ⇒ (2, 2, 2∗) ⇒ (2, 2, 3) ⇒ (2, 2, 2) ⇒ (2, 2, 3) ⇒ (2, 2, 2) ⇒ (2, 3, 2) ⇒ (2, 3, 3∗) ⇒ (2, 3, 4) ⇒ (2, 3, 3) ⇒ (2, 3, 4) ⇒ (2, 3, 3) ⇒ (2, 3, 4) Therefore, we obtain the state S(3).
Corollary 12. logd(N, k, m) is WSNB under any of CS, CD, STU, and P if and only if it is SNB, i.e.,[5],
m >
k + 3 · 2n−k2 −1− 2 for n − k even,
k + 2n−k+12 − 2 for n − k odd.
Proof. Note that logd(N, k, m) is a vertical copy network. Then the results for CS and CD follow from Theorems9 and 10. For P and STU, it is easily verified thati = (N − 1, N − 1) does not block any request in {} ∪ Ri for all i. Then the results follow from Theorems 11.
What packing is a good routing strategy has been a folklore for a long time and documented in literature [1]. One motivation for that folklore is that C(n, m, 2) is WSNB under P if and only if m3n/2 [1], while it is SNB if and only if m2n − 1. The seemingly discrepancy between the m3n/2 result and Theorem 11 is explained by the fact thatdoes not exist in C(n, m, 2) since M(V , ) occupies both input switches (see Fig. 6).
For r 3, it was proved[2] that C(n, m, r) is WSNB under P if and only if it is SNB. Thus the saving of C(n, m, 2) under P seems to be a fluke rather than a testimony of its goodness. In this paper, again we showed that in the worst-case scenario, P does not help. Instead, MI is the only routing strategy which is still not ruled out to be useful.
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