# Rearrangeable Nonblocking Conditions for Four Elastic Optical Data Center Networks
## 2 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 \(r\times r\) 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 \(r\times r\) 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 \(i\)th ToR-T (or ToR-R) in group \(u\), where \(1\leq u\leq r\) and \(1\leq i\leq s\). The DCN3 architecture is denoted by DCN3(\(r\), \(q\), \(k\), \(p\)), and it contains \(p\times r\) 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 \(u\)th input (or \(v\)th output) of each BV-SS. Finally, the DCN4 architecture is denoted by DCN4(\(s\), \(r\)  
Figure 1: A ToR switch consisting of \(q\) BVTs. \(q,k,p\)), 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 2(b)). The four DCN architectures serve \(m\)-slot connections with \(m\leq m_{\max}\). 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 \(k\geq qm_{\max}\) (or \(k\geq sqm_{\max}\)) 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 \(1\leq u\), \(v\leq r\) and \(1\leq i\), \(j\leq s\). 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. **Lemma 1.