EQUIVALENCE OF THE 1-RATE MODEL TO THE CLASSICAL
MODEL ON STRICTLY NONBLOCKING SWITCHING NETWORKS∗
W. R. CHEN†, F. K. HWANG†, AND XUDING ZHU‡
Abstract. In the 1-rate(f) network, each link can carry up to f messages for some integer f.
The classicalmodelis the specialcase when f = 1. We show that a network is strictly nonblocking under the 1-rate(f) model if and only if it is strictly nonblocking under the classical model.
Key words. switching network, 1-rate network, multirate network, graph coloring, flow, strictly
nonblocking
AMS subject classifications. 68M10, 15C15, 90B18 DOI. 10.1137/S0895480102414806
1. Introduction. A switching network consists of a set of nodes and a set of
(directed) links. Typically, an outlink of a node is the inlink of another node, and vice versa. There are two special types of nodes: the inputs and the outputs. Each input (output) is a node which has no inlink (outlink) and exactly one outlink (inlink).
We view a network as a directed graph G = (V, E), where each vertex is a node and each edge is a link. The inputs and outputs are subsets I, O of V . To emphasize the special roles of the inputs and outputs, we denote a network as G = (V, E, I, O). A network is called acyclic if the directed graph G is acyclic; i.e., G contains no directed cycles.
Let G = (V, E, I, O) and f be a positive integer. The 1-rate(f) network, denoted b y (G, f), is a network G together with the capacity constraint that each edge can carry up to f messages. If f = 1, then the 1-rate network (G, 1) is the classical model. In other words, a classical model is a network in which each edge can carry at most one message. In this paper, we consider only 1-rate networks.
A traffic of (G, f) is a sequence of input-output pairs (i, j), where i ∈ I and
j ∈ O. There are two types of traffics: requests and cancellations. A request is a pair
(i, j) such that neither of i, j has appeared in more than f − 1 previous uncancelled requests. Namely, the pair requests a connection in the network. A cancellation is a previous request whose connection in the network is to be removed. A request (i, j) is
routed if a directed i-j-path is chosen, without exceeding the capacity of the edges. So
a request (i, j) can be routed in the network (which has already routed many previous requests) if and only if there exists a directed i-j-path, each of whose edges has not been used more than f − 1 times.
A state S of (G, f) is a collection of (not necessarily distinct) directed paths of
G joining vertices of I to vertices of O such that each edge e is contained in at most f directed paths. Given a state S, let S(e) denote the number of directed paths ∗Received by the editors September 19, 2002; accepted for publication (in revised form) July 28, 2003; published electronically January 22, 2004.
http://www.siam.org/journals/sidma/17-3/41480.html
†Department of Applied Mathematics, National Chiao Tung University, Hsinchu 300, Taiwan (fhwang@math.nctu.edu.tw). The research of these authors was partially supported by ROC National Science Councilgrant NSC 91-2115-M-009-010 and the NationalChiao Tung University Lee-MTI Center.
‡Department of Applied Mathematics, National Sun Yat-sen University, Kaoshiung 80424, Taiwan (zhu@math.nsysu.edu.tw). This author’s research was partially supported by ROC National Science Councilgrant NSC 91-2115-M-110-003.
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containing e. Then 0 ≤ S(e) ≤ f. A state is blocking if there exist a vertex i ∈ I and
j ∈ O such that both i and j are contained in fewer than f directed paths in S, and
every directed i-j-path of G contains an edge e with S(e) = f. We say that (G, f) is strictly nonblocking if there is no blocking state.
The classical model is, of course, the dominating model in the study of switching networks. Recently, the multirate network has received increasing attention due to the popular attempt to integrate multimedia service into one network. Since the theory of the classical model is well established, it is profitable to ask how much of it can be extended to the multirate model. The 1-rate model is the simplest multirate model but also has its own application. It is used in the digital symmetrical matrices in time-space switching [7, 10]. The principle of providing more links between two nodes, known as statistical line grouping in [8], was promoted as a major technique to cut down network blocking. On the other hand, strict nonblockingness is one of the most fundamental properties of a switching network. Therefore, asking whether one model implies the other on this property can serve as a natural start to explore the relation between the classical model and the multirate model. In this paper we prove that if G = (V, E, I, O) is an acyclic network, then the strict nonblockingness of a 1-rate network (G, f) is equivalent to that of the classical model (G, 1).
2. Strictly nonblocking for (G, f) implies the same for (G, 1). We first
prove the implication in one direction.
Theorem 1. If (G, f) is strictly nonblocking for some positive integer f, then (G, 1) is strictly nonblocking.
Proof. It suffices to prove that if (G, 1) has a blocking state, then (G, f) has a
blocking state. Suppose S is a blocking state of (G, 1). Let S be the collection of
directed paths of G which is obtained by duplicating f times each directed path of
S. Then S is a state of (G, f) and for each edge e of G, S(e) = f × S(e). As S is a
blocking state of (G, 1), there is an input i ∈ I and an output j ∈ O such that none of i, j is contained in any directed path of S, and any directed i-j-path of G contains an edge e with S(e) = 1. Then both of i and j are contained in no directed paths of
S, and every directed i-j-path of G contains an edge e with S(e) = f. Therefore S
is a blocking state of (G, f).
In the remainder of this paper, we shall prove the other direction; i.e., if for some integer f ≥ 1, (G, f) has a blocking state, then (G, 1) has a blocking state. Let S be a blocking state of (G, f). Then there exist i ∈ I and j ∈ O such that both i, j are contained in at most f − 1 directed path of S, and any directed i-j-path contains an edge e with S(e) = f. We need to construct a blocking state S for (G, 1). One may
attempt to partition the directed paths in S into f classes such that (i) directed paths which share an edge belong to different classes;
(ii) there exists a class C not containing any directed path with end vertex i or j. If such a partition exists, then it is easy to verify that the class C is a blocking state of (G, 1). However, such a partition may not exist. Consider the following network: Figure 1 shows an example of (G, 2), where G is a simple digraph (a pair of double links indicates a link carrying two paths). The collection of directed paths
S = {P1, P2, P3, P4} in Figure 1 is a blocking state for (G, 2), where input i and
output j each has generated one path, and hence a new request (i, j) is legitimate. However, it is impossible to partition the paths into two classes in such a way that directed paths sharing an edge belong to different classes, because every two directed paths share an edge. Thus to construct the blocking state S for (G, 1), we need to
use directed paths not contained in the collection S.
4
P
1
P2
P3
P
i
j
Fig. 1. An example.3. Strictly nonblocking for (G, 1) implies the same for (G, 2). In this
section, we consider the case f = 2.
Theorem 2. Suppose G is acyclic. If (G, 1) is nonblocking, then (G, 2) is
non-blocking.
Proof. Let S be a blocking state for (G, 2). Thus there exist i ∈ I and j ∈ O
such that both i, j are contained in at most one directed path of S, and any directed
i-j-path contains an edge e with S(e) = 2.
We shall construct a blocking state for (G, 1). For each vertex v of G, denote by
E+(v) the outlinks of v and by E−(v) the inlinks of v. Let E(v) = E+(v) ∪ E−(v).
Let s+(v) = e∈E+(v) S(e) = P ∈S |P ∩ E+(v)|, s−(v) = e∈E−(v) S(e) = P ∈S |P ∩ E−(v)|, and s(v) = s+(v) + s−(v) = P ∈S |P ∩ E(v)|.
Since each directed path P ∈ S connects a vertex of I to a vertex of O, we conclude that for each vertex v ∈ I ∪ O, |P ∩ E+(v)| = |P ∩ E−(v)|. Hence s+(v) = s−(v)
and s(v) = 2s+(v). Let E
1 = {e ∈ E : S(e) = 1}, and let E2 = {e ∈ E : S(e) = 2}.
Then s(v) = |E1∩ E(v)| + 2|E2∩ E(v)|. If v ∈ (I ∪ O), then s(v) is even, and hence
|E1∩ E(v)| is even. Let G1= (V, E1) be the subgraph of G induced by the edge set
E1. As each vertex of V − (I ∪ O) has even degree in G1, we can decompose G1into
an edge-disjoint union of (not necessarily directed) cycles and paths, say
E1= (P1∪ P2∪ · · · ∪ Pl) ∪ (C1∪ C2∪ · · · ∪ Cm),
where each path Pk connects two vertices of I ∪ O. We color the edges of each Pkand
Clby two colors, a and b, as described below.
Given an undirected cycle (or a path), there are two choices for the positive
direction of the cycle (or path). If the cycle is drawn on the plane, then either the
clockwise direction or the counterclockwise direction can be chosen as the positive
direction. For a path with end vertices i and j, one can traverse the path from i
to j or from j to i. Once a positive direction is chosen, then those directed edges
that agree with the positive direction of the cycle (or path) are called forward edges, and those directed edges that oppose the positive direction are called backward edges. Arbitrarily choose a positive direction of Cl (or Pk) and color the forward edges of
Cl (or Pk) by color a and backward edges by color b, except that if there exist an
edge incident to i and an edge incident to j, then they should both be colored a. Observe that if these two edges are contained in a same path, then it is easy to see that they are in the same direction. Therefore whether the two edges are contained in a same path or contained in two distinct paths, by appropriately choosing the positive directions of the paths, they are both forward edges. So the required coloring exists. Let Ea ⊂ E1 be the edges of color a and Eb ⊂ E1 be the edges of color b. Let
B1 = Ea∪ E2 and B2 = Eb ∪ E2. Suppose v ∈ (I ∪ O). Let ia(v) (respectively,
oa(v)) be the number of inlinks (respectively, outlinks) of v of color a, and let ib(v)
(respectively, ob(v)) be the number of inlinks (respectively, outlinks) of v of color b.
If Pk (or Cl) contains v, then either Pk(or Cl) contains two inlinks or two outlinks
of v which are colored by distinct colors or it contains one outlink and one inlink of
v which are colored by the same color. Therefore
ia(v) + ob(v) = oa(v) + ib(v).
Let i2(v) = |E2∩ E−(v)| and o2(v) = |E2∩ E+(v)|. Then
s−(v) = i a(v) + ib(v) + 2i2(v) and s+(v) = o a(v) + ob(v) + 2o2(v). As s+(v) = s−(v), we conclude that i a(v) + i2(v) = oa(v) + o2(v) and ib(v) + i2(v) = ob(v) + o2(v).
Let H1 be the directed subgraph of G induced by the edge set Ea∪ E2 and H2
be the directed subgraph of G induced by the edge set Eb∪ E2. Then for each vertex
v ∈ (I ∪ O), the number of inlinks of v in H1 is ia(v) + i2(v), and the number of
outlinks of v in H1 is oa(v) + o2(v). So the number of inlinks of v is equal to the
number of outlinks of v. As G is acyclic, H1 is acyclic. Therefore H1, and similarly
H2, can be decomposed into directed paths joining vertices of I to vertices of O. For
k = 1, 2, denote by Sk the collection of directed paths which form a decomposition
of Hk. For each edge e of G, 0 ≤ Sk(e) ≤ 1 and S(e) = S1(e) + S2(e). Moreover,
both i and j are not contained in any directed paths of S2. Any directed i-j-path of
G contains an edge e with S(e) = 2, and hence S2(e) = 1. Therefore S2 is a blocking
state of (G, 1).
4. Strictly nonblocking for (G, 1) implies the same for (G, f). In this
section, we prove that the strict nonblocking of the classical model implies the strict nonblocking of the 1-rate(f) model for any f ≥ 1. Our proof needs a result concerning integer flows of graphs.
Let G be a directed graph. An integer flow of G is a mapping φ : E → Z which assigns to each edge e ∈ E an integer φ(e) such that for each vertex v of G,
e∈E+(v)
φ(e) =
e∈E−(v)
φ(e).
An integer flow φ is called a nonnegative k-flow if for each edge e, 0 ≤ φ(e) ≤ k − 1. Lemma 1 is due to Little, Tutte, and Younger [9].
Lemma 1. For each nonnegative k-flow f of G, there exist k − 1 nonnegative 2-flows φt(t = 1, 2, . . . , k − 1) such that φ =k−1t=1φt.
Lemma 2. Suppose G is acyclic. If S is a state of (G, f), then there are f states
S1, S2, . . . , Sf of (G, 1) such that for each edge e of G, S(e) =fi=1Si(e).
Proof. Let S be a state of (G, f). Let G be the directed graph obtained from G
by identifying all the inputs and outputs, i.e., identifying all the vertices of I ∪ O into a single vertex v∗. We view S as a weight assignment to the edges of G. It is easy to
see that for each vertex v of G,
e∈E+(v)
S(e) =
e∈E−(v)
S(e),
and for each edge of G,
0 ≤ S(e) ≤ f.
Therefore S is a nonnegative (f + 1)-flow of G. By Lemma 1, G has f nonnegative
2-flows St (t = 1, 2, . . . , f) such that S = k−1t=1St. Each nonnegative 2-flow St
corresponds to a collection of edge disjoint directed cycles of G. As G is acyclic,
each directed cycle C contains the vertex v∗. In other words, each directed cycle C
corresponds to a directed path of G joining a vertex of I to a vertex of O. Thus each
Stis indeed a state of (G, 1).
Theorem 3. If (G, 1) is strictly nonblocking, then (G, f) is strictly nonblocking
for any f ≥ 1.
Proof. Assume (G, f) is not strictly nonblocking and S is a blocking state of
(G, f). Then there exist i ∈ I and j ∈ O such that both i and j are contained in fewer than f directed paths in S, and every directed i-j-path of G contains an edge e with S(e) = f. By Lemma 2, there exist f states, S1, S2, . . . , Sf, of (G, 1) such that
for every edge e,
S(e) =
f
k=1
Sk(e).
As both i and j are contained in fewer than f directed paths in S, there exists 1 ≤ a, b ≤ f such that i is not contained in any path of Sa, and j is not contained in
any path of Sb. If a = b, then Sa is a blocking state of (S, 1). Assume a = b. Then
Sa∪ Sb is a blocking state of (G, 2). By Theorem 2, (G, 1) has a blocking state.
Corollary 1. Suppose G = (V, E, O, I) is an acyclic network. Then for any
positive integers f, f, (G, f) is strictly nonblocking if and only if (G, f) is strictly
nonblocking.
Proof. The strictly nonblocking of (G, f) is equivalent to the strictly nonblocking
of (G, 1) for any integer f. Hence strictly nonblocking of (G, f) is equivalent to strictly nonblocking of (G, f).
5. Some concluding remarks. Some other implications between the classical
model and the multirate model are available from the literature. These involve some other notions of nonblockingness. A network is wide-sense nonblocking if every request can be routed, provided all routing follows a given algorithm. A network is
rearrange-ably nonblocking if all requests can be routed if they are given at once (instead of the
usual “sequential” model).
Let C(n1, r1, m, n2, r2) denote the 3-stage Clos network whose nodes are
parti-tioned into three stages (parts):
The first stage consists of r1 nodes each with n1 inlinks and m outlinks; the
second stage consists of m nodes each with r1 inlinks and r2 outlinks; and the third
stage consists of r2 nodes each with m inlinks and n2outlinks such that there exists
a link from each stage-i node to each stage-(i + 1) node but no other links between two nodes.
Clos [4] proved the following lemma.
Lemma 3. C(n1, r1, m, n2, r2) is strictly nonblocking under the classical model if
and only if
m ≥ min{n1+ n2− 1, n1r1, n2r2}.
Hwang and Yeh, as reported in [6], proved a similar result under a model slightly more general than the 1-rate(f) model; suppose each input has capacity f0, each
output has capacity f
0, each link between stage 1 and stage 2 has capacity f1, and
each link between stage 2 and stage 3 has capacity f2.
Lemma 4. C(n1, r1, m, n2, r2; f0, f0, f1, f2) is strictly nonblocking if and only if
m ≥ min{n1f1, n2r2f2} − 1 f0 + min{n1r1f1, n2r2} − 1 f0 + 1. By setting f0= f0 = f1= f2= f, we obtain the following corollary.
Corollary 2. C(n1, r1, m, n2, r2) is strictly nonblocking under the 1-rate(f)
model if and only if
m ≥ min{n1+ n2− 1, n1r1, n2r2}.
Note that the conditions in Lemmas 3 and Corollary 2 are the same. Hence we obtain the following theorem.
Theorem 4. For C(n1, r1, m, n2, r2), strictly nonblocking under the classical
model implies the same for the 1-rate(f) model, and vice versa.
Benes [1] proved the following lemma.
Lemma 5. C(n, 2, m, n, 2) is wide-sense nonblocking under the classical model if
and only if m ≥ 3n 2 .
On the other hand, Fishburn et al. [5] proved the following lemma.
Lemma 6. C(n, 2, m, n, 2) is wide-sense nonblocking under the 1-rate(f) model
if and only if m ≥ 3n 2.
By comparing Lemmas 5 and 6, we obtain the following theorem.
Theorem 5. For C(n, 2, m, n, 2), wide-sense nonblocking under the classical
model does not imply the same for the 1-rate model.
Finally, Chung and Ross [3] proved the following lemma.
Lemma 7. Rearrangeably nonblocking under the classical model implies the same
for the 1-rate(f) model.
For the other direction, only special cases have been proved. Slepian (see [1]) proved the following result (he ignored the terms n1r1 and n2r2, which reflect the
boundary effects).
Lemma 8. C(n1, r1, m, n2, r2) is rearrangeably nonblocking under the classical
model if and only if m ≥ max{min{n1, n2r2}, min{n1r1, n2}}.
On the other hand, Hwang and Yeh, as reported in [6], proved the following lemma.
Lemma 9. C(n1, r1, m, n2, r2; f0, f0, f1, f2) is rearrangeably nonblocking if and only if m ≥ max min{n1f1, n2r2f2} f0 , min{n1r1f1, n2f2} f 0 .
By setting f0= f0 = f1= f2, we obtain the following corollary.
Corollary 3. C(n1, r1, m, n2, r2) is rearrangeably nonblocking under the
1-rate(f) model if and only if m ≥ max{min{n1, n2r2}, min{n1r1, n2}}.
By comparing Lemma 8 and Corollary 3, we obtain the following theorem. Theorem 6. For the 3-stage Clos network, rearrangeably nonblocking under the 1-rate(f) model implies the same for the classical model.
Note that all these results deal with the very special 3-stage Clos networks. Chung and Ross, and the authors, are the only exceptions to attack the much harder general networks.
To summarize, we have
classical 1-rate(f) remark
strict =⇒ proved
⇐= proved
wide-sense =⇒ not true
⇐= possible
rearrangeable =⇒ proved
⇐= possible
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