Rearrangeable Nonblocking Conditions for Four Elastic Optical Data Center Networks

: Four variants of elastic optical data center network (DCN) architectures based on optical circuit switching were proposed in an earlier study. The necessary and su ﬃ cient values of frequency slot units (FSUs) per ﬁber required for these four DCNs in the sense of there being strictly nonblocking (SNB) were derived, but no results in the sense of being rearrangeable nonblocking (RNB) were presented. In reality, only limited bandwidths are available, and reducing the value of FSUs per ﬁber has become a critical task to realize nonblocking optical DCN architectures in practice. In this paper, we derive the su ﬃ cient value of FSUs per ﬁber required for the four DCNs to be RNB by two multigraph approaches. Our results show that the proposed RNB conditions in terms of FSUs per ﬁber for a certain two of the four DCNs reduce their SNB results down to at least half for most cases, and even down to one-third.

connections, and a network is called rearrangeable nonblocking (RNB) if a new connection can be accommodated by rearranging some existing connections [14]. An RNB network is also defined as one where any set (or frame) of connections can be routed simultaneously. The necessary and sufficient number of FSUs per fiber required for these four DCNs in the sense of their being SNB were given in [13], but no results in the sense of being RNB were proposed.
The four DCNs usually require a great number of FSUs per fiber to be SNB, especially when m max is growing. However, the resource of FSUs per fiber in practical systems is limited since for the EON switches, and the C-band has only around 350 available FSUs (1530-1565 nm). Reducing the value of FSUs per fiber is a challenging task, and this issue has been studied in various research on EONs [8][9][10][11]. In order to reduce the value of FSUs per fiber to realize nonblocking optical DCN architectures in practice, we studied the four DCNs in the sense of being RNB in this paper, and derived the sufficient number of FSUs per fiber by adopting two multigraph approaches. Our results show that two of the proposed RNB conditions reduced the SNB results significantly.
The rest of the paper is organized as follows: In Section 2, we give a brief review of the four DCN architectures and introduce the notations used in the paper. In Section 3, we prove the RNB conditions for the DCN1 and DCN3 networks. In Section 4, we prove the RNB conditions for the DCN2 and DCN4 networks. Section 5 concludes the paper.

Preliminaries and Notations
In this section, we will review the four elastic optical DCN architectures proposed in [13] and introduce the notations used in this paper. The four elastic optical DCN architectures require bandwidth-variable, waveband-selective switches (BV-WSSs) [15,16], bandwidth-variable space switches (BV-SSs), passive combiners (PCs) and ToR switches. Both BV-WSSs and BV-SSs, the latter of which consist of BV-WSSs and PCs, can switch wavebands with flexible bandwidths without spectrum conversion capabilities. Each ToR switch consists of q bandwidth-variable transponders (BVTs), which are divided into two parts: the transmission part, denoted by BVT-T, and the receiving part, denoted by BVT-R. The part of each ToR switch consisting of q BVT-Ts (or BVT-Rs) and a PC (or BV-WSS) is denoted by ToR-T (or ToR-R) (see Figure 1). A BVT-T can use any m consecutive FSUs of its output; i.e., the frequency of its output is arbitrarily tunable. In addition, a BVT-T is connected to a BVT-R in a strict one-to-one manner, and thus a BVT-T does not simultaneously send connections to two or more BVT-Rs. All connections generated from the same ToR switch occupy different FSUs, so that all of them can be sent through one fiber connecting the ToR-T (or ToR-R) to the OCS network.
A network is called strictly nonblocking (SNB) if a connection will never be blocked by existing connections, and a network is called rearrangeable nonblocking (RNB) if a new connection can be accommodated by rearranging some existing connections [14]. An RNB network is also defined as one where any set (or frame) of connections can be routed simultaneously. The necessary and sufficient number of FSUs per fiber required for these four DCNs in the sense of their being SNB were given in [13], but no results in the sense of being RNB were proposed.
The four DCNs usually require a great number of FSUs per fiber to be SNB, especially when mmax is growing. However, the resource of FSUs per fiber in practical systems is limited since for the EON switches, and the C-band has only around 350 available FSUs (1530-1565 nm). Reducing the value of FSUs per fiber is a challenging task, and this issue has been studied in various research on EONs [8][9][10][11]. In order to reduce the value of FSUs per fiber to realize nonblocking optical DCN architectures in practice, we studied the four DCNs in the sense of being RNB in this paper, and derived the sufficient number of FSUs per fiber by adopting two multigraph approaches. Our results show that two of the proposed RNB conditions reduced the SNB results significantly.
The rest of the paper is organized as follows: In Section 2, we give a brief review of the four DCN architectures and introduce the notations used in the paper. In Section 3, we prove the RNB conditions for the DCN1 and DCN3 networks. In Section 4, we prove the RNB conditions for the DCN2 and DCN4 networks. Section 5 concludes the paper.

Preliminaries and Notations
In this section, we will review the four elastic optical DCN architectures proposed in [13] and introduce the notations used in this paper. The four elastic optical DCN architectures require bandwidth-variable, waveband-selective switches (BV-WSSs) [15,16], bandwidth-variable space switches (BV-SSs), passive combiners (PCs) and ToR switches. Both BV-WSSs and BV-SSs, the latter of which consist of BV-WSSs and PCs, can switch wavebands with flexible bandwidths without spectrum conversion capabilities. Each ToR switch consists of q bandwidth-variable transponders (BVTs), which are divided into two parts: the transmission part, denoted by BVT-T, and the receiving part, denoted by BVT-R. The part of each ToR switch consisting of q BVT-Ts (or BVT-Rs) and a PC (or BV-WSS) is denoted by ToR-T (or ToR-R) (see Figure 1). A BVT-T can use any m consecutive FSUs of its output; i.e., the frequency of its output is arbitrarily tunable. In addition, a BVT-T is connected to a BVT-R in a strict one-to-one manner, and thus a BVT-T does not simultaneously send connections to two or more BVT-Rs. All connections generated from the same ToR switch occupy different FSUs, so that all of them can be sent through one fiber connecting the ToR-T (or ToR-R) to the OCS network. The DCN1 architecture, denoted by DCN1(r, q, k), is given in Figure 2a. A DCN1(r, q, k) network contains one r × r BV-SS and r ToR switches, each of which consists of q BVT-Ts (or BVT-Rs) and is attached to an input (or output) fiber with k FSUs of the BV-SS. The DCN2 architecture, denoted by DCN2(s, r, q, k), is a variant of DCN1(r, q, k) and is given in Figure 2b. A DCN2(s, r, q, k) network contains one r × r BV-SS and r groups of s ToR switches, which are combined by one PC into (or directed from one BV-WSS to) one input (or output) fiber connecting to the BV-SS. We use ToR-T (or ToR-R) u-i to denote the ith ToR-T (or ToR-R) in group u, where 1 ≤ u ≤ r and 1 ≤ i ≤ s. The DCN3 architecture is denoted by DCN3(r, q, k, p), and it contains p r × r BV-SSs and r ToR switches ( Figure  3a). The output (or input) fiber of ToR-T u (or ToR-R v) is connected to one BV-WSS (or PC) which connects to the uth input (or vth output) of each BV-SS. Finally, the DCN4 architecture is denoted by DCN4(s, r, q, k, p), and it is obtained from a DCN2(s, r, q, k) network by adopting p BV-SSs to connect ToR-Ts and ToR-Rs (Figure 3b). The DCN1 architecture, denoted by DCN1(r, q, k), is given in Figure 2a. A DCN1(r, q, k) network contains one r × r BV-SS and r ToR switches, each of which consists of q BVT-Ts (or BVT-Rs) and is attached to an input (or output) fiber with k FSUs of the BV-SS. The DCN2 architecture, denoted by DCN2(s, r, q, k), is a variant of DCN1(r, q, k) and is given in Figure 2b. A DCN2(s, r, q, k) network contains one r × r BV-SS and r groups of s ToR switches, which are combined by one PC into (or directed from one BV-WSS to) one input (or output) fiber connecting to the BV-SS. We use ToR-T (or ToR-R) u-i to denote the ith ToR-T (or ToR-R) in group u, where 1 ≤ u ≤ r and 1 ≤ i ≤ s. The DCN3 architecture is denoted by DCN3(r, q, k, p), and it contains p r × r BV-SSs and r ToR switches (Figure 3a). The output (or input) fiber of ToR-T u (or ToR-R v) is connected to one BV-WSS (or PC) which connects to the uth input (or vth output) of each BV-SS. Finally, the DCN4 architecture is denoted by DCN4(s, r, q, k, p), and it is obtained from a DCN2(s, r, q, k) network by adopting p BV-SSs to connect ToR-Ts and ToR-Rs (Figure 3b).   The four DCN architectures serve m-slot connections with m ≤ mmax. To guarantee that each fiber occupying k FSUs is sufficient to carry connections served by all BVT-Ts, the value of k is assumed to be k ≥ qmmax (or k ≥ sqmmax) for the DCN1 and DCN3 (or DCN2 and DCN4) architectures. An m-slot connection from a BVT-T in ToR-T u (or ToR-

BV-WSS
FSUs in each fiber are numbered from 1 to k. To set up a connection (u, v, m) (or (u-i, v-j, m)), the same sets of m adjacent FSUs must be found in both the fiber connecting ToR-T u (or ToR-T u-i) with one BV-SS and the fiber connecting this BV-SS with ToR-R v (or ToR-R v-j). If those sets do not exist, the connection is blocked. The necessary and sufficient values of k for DCN1 to DCN4 in the sense of being SNB were given in [13]. We quote the SNB results for the DCN1 and DCN2 networks in Lemmas 1 and 2 for further comparison in Section 3 and 4. consists of q BVT-Ts (or BVT-Rs), as given in Figure 1.  The four DCN architectures serve m-slot connections with m ≤ mmax. To guarantee that each fiber occupying k FSUs is sufficient to carry connections served by all BVT-Ts, the value of k is assumed to be k ≥ qmmax (or k ≥ sqmmax) for the DCN1 and DCN3 (or DCN2 and DCN4) architectures. An m-slot connection from a BVT-T in ToR-T u (or ToR-

BV-WSS
FSUs in each fiber are numbered from 1 to k. To set up a connection (u, v, m) (or (u-i, v-j, m)), the same sets of m adjacent FSUs must be found in both the fiber connecting ToR-T u (or ToR-T u-i) with one BV-SS and the fiber connecting this BV-SS with ToR-R v (or ToR-R v-j). If those sets do not exist, the connection is blocked. The necessary and sufficient values of k for DCN1 to DCN4 in the sense of being SNB were given in [13]. We quote the SNB results for the DCN1 and DCN2 networks in Lemmas 1 and 2 for further comparison in Section 3 and 4.  The four DCN architectures serve m-slot connections with m ≤ m max . To guarantee that each fiber occupying k FSUs is sufficient to carry connections served by all BVT-Ts, the value of k is assumed to be k ≥ qm max (or k ≥ sqm max ) for the DCN1 and DCN3 (or DCN2 and DCN4) architectures. An m-slot connection from a BVT-T in ToR-T u (or ToR-T u-i) to a BVT-R in ToR-R v (or ToR-R v-j) in a DCN1 or DCN3 (or a DCN2 or DCN4) is denoted by (u, v, m) (or (u-i, v-j, m)), where 1 ≤ u, v ≤ r and 1 ≤ i, j ≤ s.
FSUs in each fiber are numbered from 1 to k. To set up a connection (u, v, m) (or (u-i, v-j, m)), the same sets of m adjacent FSUs must be found in both the fiber connecting ToR-T u (or ToR-T u-i) with one BV-SS and the fiber connecting this BV-SS with ToR-R v (or ToR-R v-j). If those sets do not exist, the connection is blocked. The necessary and sufficient values of k for DCN1 to DCN4 in the sense of being SNB were given in [13]. We quote the SNB results for the DCN1 and DCN2 networks in Lemmas 1 and 2 for further comparison in Sections 3 and 4.

RNB DCN1 and DCN3 Networks
In this section, we first consider the RNB DCN1 network and then the RNB DCN3 network. In order to derive the sufficient value of k for a DCN1(r, q, k) network in the sense of being RNB, we propose a multigraph approach and a routing algorithm in the following.

Multigraph Approach and Routing Algorithm
Given a DCN1(r, q, k) network and a frame F of connections, we propose Multigraph Approach A, given below, to convert the DCN1(r, q, k) network for frame F into a multigraph G F .

Multigraph Approach A:
Let each left vertex u (or right vertex v) in multigraph G F be ToR-T u (or ToR-R v) of the DCN1(r, q, k) network. In multigraph G F , there is an edge connecting vertexes u and v if there is an m-slot connection from a BVT-T in ToR-T u and it is destined to a BVT-R in ToR-R v, i.e., (u, v, m) (see Figure 4a). Note that we call G F a multigraph [17] because multiple connections between ToR-T u and ToR-R v are allowed, and thus there could be more than one edge connecting vertexes u and v in G F .

RNB DCN1 and DCN3 Networks
In this section, we first consider the RNB DCN1 network and then the RNB DCN3 network. In order to derive the sufficient value of k for a DCN1(r, q, k) network in the sense of being RNB, we propose a multigraph approach and a routing algorithm in the following.

Multigraph Approach and Routing Algorithm
Given a DCN1(r, q, k) network and a frame F of connections, we propose Multigraph Approach A, given below, to convert the DCN1(r, q, k) network for frame F into a multigraph GF.

Multigraph Approach A:
Let each left vertex u (or right vertex v) in multigraph GF be ToR-T u (or ToR-R v) of the DCN1(r, q, k) network. In multigraph GF, there is an edge connecting vertexes u and v if there is an m-slot connection from a BVT-T in ToR-T u and it is destined to a BVT-R in ToR-R v, i.e., (u, v, m) (see Figure  4a). Note that we call GF a multigraph [17] because multiple connections between ToR-T u and ToR-R v are allowed, and thus there could be more than one edge connecting vertexes u and v in GF.  In Property 1, we show that GF is q-edge-colorable. Property 1. Given a DCN1(r, q, k) network and a frame F of connections, let GF be the corresponding multigraph constructed by Multigraph Approach A. Multigraph GF is q-edge-colorable.
Proof. Let Δ(GF) be the maximum degree of GF. Since each ToR switch consists of q BVT-Ts and q BVT-Rs, at most q m-slot connections can be generated from a ToR-T (or destined to a ToR-R). Thus, we have Δ(GF) ≤ q. From the construction of GF, we can see that GF is a bipartite multigraph. In addition, GF is q-edge-colorable according to graph theory [17] if GF is a bipartite multigraph with Δ(GF) ≤ q. □ In a DCN1(r, q, k) network, we use Iu (or Ov) to denote the fiber connecting ToR-T u (or ToR-R v) and the BV-SS, where k ≥ qmmax and 1 ≤ u, v ≤ r. In addition, we partition each fiber with k FSUs into q parts, each of which consists of mmax consecutive FSUs. Each part is called a window, and these q windows, denoted by Wl for 1 ≤ l ≤ q, are numbered from 1 from left to right. We use |Wl| to represent the size of window Wl, and also use Iu,l (or Ov,l) to represent the lth window in fiber Iu (or Ov) for 1 ≤ u, v ≤ r and 1 ≤ l ≤ q.
Recall that GF is q-edge-colorable (Property 1). Let colors 1, 2,…, q be adopted to edge color GF. We route each (u, v, m) for the RNB condition [14] using Routing Algorithm A given below.

Routing Algorithm A:
Connection (u, v, m) is routed in windows Iu,c and Ov,c if color c is assigned to the corresponding edge of (u, v, m) in GF (see Figure 4b). In Property 1, we show that G F is q-edge-colorable. Property 1. Given a DCN1(r, q, k) network and a frame F of connections, let G F be the corresponding multigraph constructed by Multigraph Approach A. Multigraph G F is q-edge-colorable.
Proof. Let ∆(G F ) be the maximum degree of G F . Since each ToR switch consists of q BVT-Ts and q BVT-Rs, at most q m-slot connections can be generated from a ToR-T (or destined to a ToR-R). Thus, we have ∆(G F ) ≤ q. From the construction of G F , we can see that G F is a bipartite multigraph. In addition, G F is q-edge-colorable according to graph theory [17] if G F is a bipartite multigraph with ∆(G F ) ≤ q.
In a DCN1(r, q, k) network, we use I u (or O v ) to denote the fiber connecting ToR-T u (or ToR-R v) and the BV-SS, where k ≥ qm max and 1 ≤ u, v ≤ r. In addition, we partition each fiber with k FSUs into q parts, each of which consists of m max consecutive FSUs. Each part is called a window, and these q windows, denoted by W l for 1 ≤ l ≤ q, are numbered from 1 from left to right. We use |W l | to represent the size of window W l , and also use I u,l (or O v,l ) to represent the lth window in fiber I u (or O v ) for 1 ≤ u, v ≤ r and 1 ≤ l ≤ q.
Recall that G F is q-edge-colorable (Property 1). Let colors 1, 2, . . . , q be adopted to edge color G F . We route each (u, v, m) for the RNB condition [14] using Routing Algorithm A given below.

Routing Algorithm A:
Connection (u, v, m) is routed in windows I u,c and O v,c if color c is assigned to the corresponding edge of (u, v, m) in G F (see Figure 4b).

RNB Sufficient Conditions
A sufficient value of k for a DCN1(r, q, k) network in the sense of being RNB is derived in Property 2.
Proof. This property holds if Routing Algorithm A is feasible, and Routing Algorithm A is feasible if each m-slot connection can be carried by the corresponding windows. Since m ≤ m max , each m-slot connection can be carried by any window W l if |W l | = m max for 1 ≤ l ≤ q, which implies that each fiber has k = q·m max FSUs. Therefore, when Routing Algorithm A is applied, a DCN1(r, q, k) network with k ≥ q·m max is RNB.
Comparing Equation (3) with Equation (1), we have k RNB /k SNB ≤ 1/2 for m max ≥ 2 and q ≥ 3. Property 2 implies that k RNB reduces the SNB DCN1 result given in [13], namely, k SNB , down to at least half for most cases. In addition, numerical results are given in Table 1 which show that k RNB can reduce k SNB down to as low as one third, for example, the cases with m max ≥ 6 and q = 4, and the cases with m max ≥ 4 and q ≥ 8. Table 1. Numerical results of k required for being an SNB or RNB DCN1(r, q, k) network for m-slot connections with q = 4, 8, 10 and 1 ≤ m ≤ m max , where k SNB and k RNB are given in Equations (1)  The sufficient condition for being an RNB DCN1(r, q, k) network (Property 2) is also the necessary condition if only one connection rate m max is considered (Property 3).

Property 3.
Suppose only one connection rate, m max , is considered. Then, the DCN1(r, q, k) network is RNB if and only if k ≥ k RNB = q·m max .
Proof. The sufficient condition of this property is true since it is a special case with one connection rate of Property 2. In addition, the necessary condition holds when q connections (u, u, m max ) for 1 ≤ u ≤ r are generated from each ToR-T u.
From the architectures of the DCN1 and DCN3 networks (see Figures 2a and 3a), we can see that a DCN1(r, q, k) network for k ≥ q·m max functions the same as a DCN3(r, q, k, p) network with p = 1. Thus, we derive Property 4 immediately.

Property 4.
A DCN3(r, q, k, p) network for m-slot connections with 1 ≤ m ≤ m max and k ≥ q·m max is RNB if p ≥ 1.
Proof. This property is true for two reasons: i) a DCN3(r, q, k, p) network with p = 1 functions as well as a DCN1(r, q, k) network, and ii) a DCN1(r, q, k) network for k ≥ q·m max is RNB (Property 2).
For a DCN1(r, q, k) (or DCN3(r, q, k, p)) network, recall that the resource of FSUs per fiber in practical systems is limited, namely, k ≤ 350. This implies that to have an RNB DCN1(r, q, k) (or DCN3(r, q, k, p)) network for m-slot connections with 1 ≤ m ≤ m max in the real word, we also need q·m max ≤ 350 due to Property 2 (or Property 4).

RNB DCN2 and DCN4 Networks
Similar to Section 3, we first consider the RNB DCN2 network and then the RNB DCN4 network. For the DCN2 network (Figure 2b), I u-i (or O v-j ) is used to represent the fiber connecting ToR-T u-i (or ToR-R v-j) and the uth PC (or vth BV-WSS), and I' u (or O' v ) is used to represent the fiber connecting the uth PC (or vth BV-WSS) and the BV-SS for 1 ≤ u, v ≤ r and 1 ≤ i, j ≤ s. Next, we will propose Multigraph Approach B and Routing Algorithm B for the DCN2 network in the sense of being RNB by modifying Multigraph Approach A and Routing Algorithm A, respectively.

Multigraph Approach B:
Given a DCN2(s, r, q, k) network and a frame F of connections, multigraph G' F is constructed in the following way. Let each left vertex u (or right vertex v) in G' F be the uth (or vth) group of s ToR-Ts u-i (or ToR-Rs v-j) for 1 ≤ i, j ≤ s. An edge is added between two vertexes u and v in G' F if there is an m-slot connection from the uth ToR-T group destined to the vth ToR-R group (see Figure 5a).
Appl. Sci. 2020, 10, x FOR PEER REVIEW 6 of 8 DCN3(r, q, k, p)) network for m-slot connections with 1 ≤ m ≤ mmax in the real word, we also need q·mmax ≤ 350 due to Property 2 (or Property 4).

RNB DCN2 and DCN4 Networks
Similar to Section 3, we first consider the RNB DCN2 network and then the RNB DCN4 network. For the DCN2 network (Figure 2b), Iu-i (or Ov-j) is used to represent the fiber connecting ToR-T u-i (or ToR-R v-j) and the uth PC (or vth BV-WSS), and I'u (or O'v) is used to represent the fiber connecting the uth PC (or vth BV-WSS) and the BV-SS for 1 ≤ u, v ≤ r and 1 ≤ i, j ≤ s. Next, we will propose Multigraph Approach B and Routing Algorithm B for the DCN2 network in the sense of being RNB by modifying Multigraph Approach A and Routing Algorithm A, respectively.

Multigraph Approach B:
Given a DCN2(s, r, q, k) network and a frame F of connections, multigraph G'F is constructed in the following way. Let each left vertex u (or right vertex v) in G'F be the uth (or vth) group of s ToR-Ts u-i (or ToR-Rs v-j) for 1 ≤ i, j ≤ s. An edge is added between two vertexes u and v in G'F if there is an m-slot connection from the uth ToR-T group destined to the vth ToR-R group (see Figure 5a). Since each group of ToR switches can generate at most sq m-slot connections, we derive that Δ(G'F) ≤ sq, and thus G'F is sq-edge-colorable [17]. Let colors 1, 2,…, sq be used to edge-color G'F. We adopt Routing Algorithm B, shown below, to route each (u-i, v-j, m) in association with the edgecoloring of G'F for the RNB condition. Since each group of ToR switches can generate at most sq m-slot connections, we derive that ∆(G' F ) ≤ sq, and thus G' F is sq-edge-colorable [17]. Let colors 1, 2, . . . , sq be used to edge-color G' F . We adopt Routing Algorithm B, shown below, to route each (u-i, v-j, m) in association with the edge-coloring of G' F for the RNB condition.

Routing Algorithm B:
Connection (u-i, v-j, m) is routed in windows I u-i,c , I' u,c , O' v,c and O v-j,c if color c is assigned to the corresponding edge of (u-i, v-j, m) in G' F (see Figure 5b).

Property 5.
A DCN2(s, r, q, k) network for m-slot connections with 1 ≤ m ≤ m max is RNB if Proof. The proof is similar to that of Property 2.

Property 6.
Suppose only one connection rate, m max , is considered. Then a DCN2(s, r, q, k) network is RNB if and only if k ≥ k' RNB = sq·m max .
Proof. The proof is similar to that of Property 3.

Property 7.
A DCN4(s, r, q, k, p) network for m-slot connections with 1 ≤ m ≤ m max and k ≥ sq·m max is RNB if p ≥ 1.
Proof. From the topology of the DCN4 architecture (see Figure 3b), we can see that a DCN4(s, r, q, k, p) network with p = 1 and k ≥ sq·m max functions as well as a DCN2(s, r, q, k) network. According to Property 5, the property holds immediately.
Comparing Equation (2) with Equation (4), we have k' RNB /k SNB ≤ 1/2 for m max ≥ 2 and sq ≥ 3. Property 5 implies that k' RNB reduces the SNB DCN2 result given in [13], namely, k' SNB , down to at least half for most cases, and even down to one third. In addition, numerical results are given in Table 2, which shows that k RNB can reduce k SNB down to as low as one third, for example, all the cases with m max ≥ 4, s = 3 and q ≥ 4. Again, due to the limited resource of FSUs per fiber in practical systems, to have an RNB DCN2(s, r, q, k) (or DCN4(s, r, q, k, p)) network for m-slot connections with 1 ≤ m ≤ m max in the real word, we need sq·m max ≤ 350 due to Property 5 (or Property 7). Table 2. Numerical results of k required for being an SNB or RNB DCN2(s, r, q, k) network for m-slot connections with s = 3, q = 4, 8, 10 and 1 ≤ m ≤ m max , where k' SNB and k' RNB are given in Equations (2) and (4)

Conclusions
Four variants of elastic optical DCN architectures, called DCN1, DCN2, DCN3 and DCN4, were proposed in [13]. The four DCNs in the sense of being SNB usually require a large number of FSUs per fiber. To reduce the value of FSUs, we considered the four DCNs in the sense of their being RNB in this paper. We proposed two multigraph approaches to firstly prove the sufficient number of FSUs per fiber for these four DCNs in the sense of there being RNB. Our results show that the proposed RNB conditions in term of FSUs per fiber for the DCN1 and DCN2 networks reduce their SNB results down to at least half in most scenarios, and even down to one third. In addition, we show that the sufficient condition for an RNB DCN3 (or DCN 4) network is exactly the same as that derived for an RNB DCN1 (or DCN 2) network. The proposed multigraph approaches can be applied to all Clos-like architectures for studying RNB conditions.