Kim and Du extended a routing algorithm R(l), first used by Gao and Hwang in multirate point-to-point model (see Corollary 2.4.4), to broadcast traffic.
It is the algorithm of reserving l middle switches only for large requests (which can overflow to other middle switches) where a request is called large if its weight ω > 1/(q + 1), q = b1/Bc. Notice that, the f -cast results in this subsection are all for the closed-end traffic.
Theorem 2.4.1. A C(n, m, r) network is WSNB under R(l) and a k-limited algorithm for β[b, B] multirate broadcast assignments if
m >
βn(q + 1)(Bq + B + q − 1)
q2 (k + r1/k) for B < 23/32, (15βn
8 + n − 1)(k + r1/k) for B ≥ 23/32.
and
l = d(βn(Bq + B − 1)(q + 1)/q2)(k + r1/k)e.
where q = bB1c.
We now extend the above theorem to f -cast asymmetric model and the proof is given in Appendix.
Theorem 2.4.2. A C(n1, r1, m, n2, r2) network is WSNB under R(l) and a k-limited algorithm for β[b, B] f -cast broadcast assignments if
m ≥ s + l, where
s > (βn1 − b)(q + 1)
q k + (βn2 − b)(q + 1) q f1/k, l =
&
[(βn1− q+11 )k + (βn2− q+11 )f1/k − s(1 − B)](q + 1) q
'
and q = bB1c.
Our result slightly improves over [8] which replace the two b terms in s and two q+11 terms in l by 0. To get the point-to-point result, we set f = k = 1.
For b → 0 and the symmetric case, we obtain
Corollary 2.4.3. A C(n, m, r) network is WSNB under R(l) for β(0, B]
point-to-point assignments if
m ≥ 2βn(q + 1)(Bq + B + q − 1)
q2 − 2
q. and
l = d2βn(Bq + B − 1)(q + 1)/q2−2 qe.
For this model, Gao and Hwang’s result [5] proved:
Corollary 2.4.4. A C(n, m, r) network is WSNB under R(l) for β(0, B]
point-to-point assignments if
m ≥ 2βn(q + 1)(Bq + B + q − 1)
q2 .
and
l = d2βn(Bq + B − 1)(q + 1)/q2e.
3 Rearrangeable Nonblocking Networks
For the multirate-multicast model, Kim and Du gave a rearrangeable algo-rithm. The algorithm orders the requests by their weights and routes each of them using at most k middle switches. The requests are routed in the order from heavy to light. To route the next heaviest request, the algorithm would not disturb the heaviest requests which were already routed. It continues to route the other requests until the lightest request is successfully routed.
Let us introduce a multirate model, the recursive half channel model. In this model, there are h rates ω1, · · · , ωh with ω1 > ω2 > · · · > ωi−1 > 1/2 ≥ ωi > ωi+1> · · · > ωh and ωj divides ωj−1 for i + 1 ≤ j ≤ h.
Kim-Du [8] (1998) proved
Theorem 3.0.5. C(n, m, r) is rearrangeable for the (0, 1] recursive half chan-nel model and broadcast assignments if
m > (n − 1) min
1≤k≤min(n−1,r)(k + r1/k).
We extend Theorem 3.0.5 to (i) f -cast, (ii) asymmetric Clos network and give a proof in Appendix.
Theorem 3.0.6. C(n1, r1, m, n2, r2) is rearrangeable for the (0, 1] recursive half channel model and f-cast assignments if
m > min
1≤k≤min(n2−1,f )
©(n1− 1)k + (n2− 1)f1/kª .
Setting f = k = 1 in the above theorem, we get the point-to-point result.
And it is the same with the result given by Lin, Du, Hu and Xue [9] under the symmetric model.
Corollary 3.0.7. C(n, 2n − 1, r) is rearrangeable for the (0, 1] recursive half channel model and point-to-point assignments.
Kim and Du observe a case:
Theorem 3.0.8. C(n, m, r) is rearrangeable for broadcast assignments with weights chosen from {ω1, · · · , ωh}, where 1 ≥ ω1 > · · · > ωh > 0 and ωj divides ωj−1 for 2 ≤ j ≤ h if
m > (n − 1) min
1≤k≤min(n−1,r)(k + r1/k).
4 Numerical Comparison and Conclusion
In this section, we compare some necessary conditions of C(n, m, r) Clos networks which are WSNB for [b, B] broadcast assignments. Let us denote Kim-Du [8] as KD, Yang [13] as Y, Kabaci´nski-Danilewicz [7] as KaD and Chan-Chan-Yeung [2] as CY.
n = 10 n = 20 n = 40
KD Y and KaD KD Y and KaD KD Y and KaD
r b 0.1 0.4 0.6 b 0.1 0.4 0.6 b 0.1 0.4 0.6
4 112 360 72 36 227 760 152 76 457 1560 312 156 12 147 477 96 48 299 1005 201 101 603 2063 413 207 20 159 515 103 52 323 1086 218 109 652 2229 446 223 28 168 544 109 55 342 1147 230 115 689 2355 471 236
Table 1: B = 1
Note that, for B = 1, Yang [13]’s result is the same with Kabaci´nski-Danilewicz [7]’s. And Kim-Du [8]’s result is better than Yang [13] and Kabaci´nski-Danilewicz [7]’s result only when b <= 1/4 or 1/4 < b ≤ 1/3 and n > 16.
The following two tables show some comparisons between the results un-der B < 1 ( except Yang’s result which constrains B = 1).
n 10 20 40
CY KD KaD CY KD KaD CY KD KaD
r for all b ≤ 0.25 for all b ≤ 0.25 for all b ≤ 0.25
4 146 112 149 306 227 309 626 457 629
12 193 147 196 404 299 408 828 603 831 20 208 159 212 437 323 441 894 652 898 28 220 168 224 461 342 465 944 689 948
Table 2: B = 0.75
Finally, we compare the results of Chen-Chen-Yeung and Kim-Du.
n 10 30 100 is better than Chen-Chen-Yeung’s for B ≥ 2332. Table 2 shows this property.
For B < 2332, hCY(n, B) < hKD(n, B) for all n < h(B) and hKD(n, B) <
hCY(n, B) for n large. Hence Kim-Du’s result is better than Chen-Chen-Yeung’s for B < 2332 and n large. Table 3 shows this property, where h(0.4) = 40. But notice that, for some B, h(B) is very large.
Appendix
Let Mj denote the vector (Mj(1), · · · , Mj(r2)), where Mj(k) is the sum of weights loaded on link between middle switch j and output switch k. In Figure 2, for example, M1 = (0.4, 0, 0), M2 = (0.6, 0.4, 0), M3 = (0, 0.2, 0.6) and M4 = (0, 0.4, 0) .
Proof of Theorem 2.3.12. Let C = (x, O(x), ω) be a new f -cast request.
We have |O(x)| ≤ f and ω ≤ p+11 . Under the k-limited algorithm let m0 be
the number of middle switches blocking the new request from the x’s input
Consider the m00× |O(x)| destination matrix M by discarding at most m0 rows whose corresponding middle switches are blocking the new request from the input switch B where those |O(x)| columns are corresponding to output switches in O(x). Suppose that any k middle switches cannot satisfy this new multicast request. Let t1(j) be the number of elements in the j-th row whose values are greater than 1 − ω and t1 = min1≤j≤m00t1(j). We obtain, a part of the request to f − t1 output switches by using the middle switch corresponding to the s-th row and delete those f − t1 columns from M for finding the next middle switch to route the remaining destinations. Gen-erally, assume there are only ti−1 output switches which are needed to be routed by using m00× ti−1 destination matrix M(i−1) for i < k. Let ti(j) be the number of elements in the j-th row whose values are greater than 1 − ω and ti be the minimum of ti(j) for all j. Then,
where ti 6= 0 for i < k. Otherwise, it is a contradiction to the assumption that any k middle switches can not satisfy the new multicast request. When i = k, each row vector has at least one element whose value is greater than 1 − ω. Therefore,
Since a geometric mean is not less than the minimum of a sequence, the minimum m00 can be obtained from (2), (3) and (4) as re-quest can be routed through this network.
Proof of Theorem 2.4.2. Under the algorithm R(l), assume we partition middle switches M into MS and MLwhose sizes are mS and mL, respectively.
The algorithm forces a small call (ω ≤ q+11 ) to use only MS but allows a large call (ω < q+11 ) to use not only ML but also MS. Let C = (x, O(x), ω) be a request compatible to the existing configuration. First assume ω ≤ q+11 . From Theorem 2.3.12, setting mS ≥ s > (βn1−b)(q+1)q k + (βn2−b)(q+1)q f1/k, C can be routed. Next assume ω > q+11 . Let MS0 be a subset of MS blocking C from x’s input switch B and ML0 be a subset of ML blocking the request from B and their sizes are m0S and m0L. Since q = bB1c, we have each link from B to ML0 carrying exactly q calls. Because of the compatibility, the maximum total weights going to the middle stage out of the input switch is at most (βn1− ω)k. Therefore,
m0L q
q + 1+ m0S(1 − B) ≤ m0L q
q + 1 + m0S(1 − ω) ≤ (βn1− ω)k. (6)
Let MS00 ⊆ MS\ MS0 and ML00 ⊆ ML\ ML0 be the subset of MS and ML which are available for C, respectively. Their sizes are denoted as m00S and m00L. To find out the maximum number of blocking links to output switches in O(x), let us consider (m00S + m00L) × |O(x)| destination matrix M. Suppose that any k middle switches from MS00∪ ML00 can not satisfy this new multicast request. We will use the same notation for ti(j) and ti as Theorem 2.3.12 but ti = minj∈MS00∪ML00ti(j).
We apply a method similar to Theorem 2.3.12 to contruct the destination matrix and obtain the minimum number of middle switches as
m00L q
q f1/k suffices to route all small requests. For this m∗S, if
the request can be routed.
Therefore, m = mL + mS ≥ m∗S + m∗L > s + l middle switch modules suffice.
Proof of Theorem 3.0.6. Assume ω1, · · · , ωh are those h rates with ω1 >
ω2 > · · · > ωi−1 > 1/2 ≥ ωi > ωi+1 > · · · > ωh and ωj divides ωj−1 for i + 1 ≤ j ≤ h. We will prove this theorem by induction on h. For h = 1, each link can carry no more than one call due to ω1 > 1/2 so this three-stage Clos network is nonblocking and rearrangeable if m > min1≤k≤min(n2−1,f )(n1− 1)k + (n2 − 1)f1/k (see Corollary 2.3.1). Assume that this Clos network is rearrangeable for h = h0 − 1. Consider i integers u1, u2, · · · , ui−1 and v such that ωj+ujωh0 ≤ 1 < ωj+(uj+1)ωh0 for j ≤ i−1 and vωh0 ≤ 1 < (v+1)ωh0. If a link blocks a new connected request C = (x, O(x), ωh0), then the blocking link is carrying either one ωj-call for some j ≤ i − 1 and some ωk-calls, h ≥ k ≥ i, with total weight ujωh0 (Uj-blocking), or no such ωj-call but all ωk-calls with total weight vωh0 (V -blocking). Let us assume that m0 middle switches are blocking this new ωh0-call from input stage with m0 > (n1− 1)k.
Because all connected requests were able to duplicate their messages at most k times at the input switch, at least n1 input ports should have carry full weights that are ωj+ujωh0 or vωh0. This is a contradiction to our assumption for the compatible new ωh0-call. Hence, m0 ≤ (n1− 1)k.
Consider m00 middle switches by discarding those m0 middle switches blocking this new connection request from input stage. Suppose that no k middle switches among m00 can route the new ωh0-call. Because each output switch in O(x) has at most (n2−1) output ports which are either Uj-blocking for some j ≤ i − 1 or V -blocking for the new call, the total number of block-ing links between middle stage and output switches in O(x) is no more than (n2−1)|O(x)| ≤ (n2−1)f . By using the similar approach as Theorem 2.3.12, we can obtain,
m00≤ (n2− 1)f /t1, (11)
m00 ≤ (n2− 1)ti−1/ti for i < k, (12) m00 ≤ (n2− 1)tk−1 for i = k. (13) A minimum of a sequence is not larger than its geometric mean so that, from
(11), (12) and (13), we can obtain,
m00≤
·
(n2− 1)f
t1 · (n2− 1)t1
t2 · · · (n2− 1)tk−2
tk−1 · (n2− 1)tk−1
¸1/k
= (n2−1)f1/k. We showed that the nonblocking multicast Clos network for the switching network is also rearrangeable for multirate-multicast communications for the recursive half channel model.
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