1 Scalable Non-Blocking Networks with Fixed Size Routers Tor Skeie, Olav Lysne and Geir Horn Keywords — Nonblocking networks, Clos-networks Abstract — The generation of non-blocking Clos-networks from custom routing chips is studied. In particular we ap- proach the question of how to construct non-blocking net- works with P ports using as few routers as possible. Un- like previous attempts we study this problem for fixed size routers, and for any P , not only those values arising from straightforward recursion of the Clos scheme. We describe a practical way to circumvent the shortcomings of the de- veloped theory for non-blocking networks when the size of the router cannot be manipulated. I. Introduction In a nonblocking circuit switching network any connec- tion between an idle input port and an idle output port can be realized without interfering with any of the other existing connections. The original application area for such networks was telephone switching, but due to their low blocking probability nonblocking networks have also been used for multiprocessor interconnection [1], and they have been advocated for ATM switching [2]. In 1950 Shannon gave a theoretical lower bound for the number of crosspoints needed in any rearrangeable or non- blocking network with N -inputs and N -outputs [3]. Shan- non did, however, not show how such networks should be constructed, and to this end we have seen a large body of research. In 1953 Clos published his seminal paper which presented a construction of strict sense nonblocking net- works that subsided previous results significantly in the need of crosspoints [4]. Since then we have seen a series of refinements and analyzes, e.g. [5], [6], [7], [8], [9], [10]. A construction for rearrangeable non-blocking networks that realizes the lower bound given by Shannon was first pre- sented by Margulis [11]. After the ground-breaking work of Clos his result has been extended to different traffic requirements like mul- tirate traffic sources [12], [13]. Likewise there is a body of results that address the issue of path-choosing for new connections, see e.g. [14] and [15]. This paper concentrates on complexity of strict sense non-blocking networks, but our focus is different from pre- vious work in five respects: This work is supported by Esprit under the projects OMI- MACRAM ´ E and OMI-ARCHES Tor Skeie, who will present the paper, is a research fellow at the Department of Informatics, University of Oslo, Box 1080, Blindern, N-0316 OSLO,Norway, E-mail: torsk@ifi.uio.no, Phone: +47 22 06 73 43, Fax: +47 22 85 24 01 Olav Lysne is a researcher at the Department of Informatics, Uni- versity of Oslo, Box 1080, Blindern, N-0316 OSLO,Norway, E-mail: olavly@ifi.uio.no,Phone: +47 22 85 24 28, Fax: +47 22 85 24 01 Geir Horn is a research scientist at SINTEF, Box 124 Blindern, N- 0314 Oslo, NORWAY, E-mail: Geir.Horn@si.sintef.no, Phone: +47 22 06 73 00, Fax: +47 22 06 73 50 • Most of the previous work assumes that the networks can be built using crossbars of arbitrary size. In modern digital switching systems, however, the elements of the net- works are custom integrated circuits of standard size called routers, thus we focus on the construction of nonblocking networks from fixed size routers. • Previous results only tell you how to build networks of size P for only very few values of P when your router is of fixed size. We aim at answering the question of how to build low complexity nonblocking networks of size P with routers of size p for any P and p. • Instead of using the number of crosspoints as complexity measure we focus on the number of routers that is needed in order to build a network. • Previous work on complexity for Clos-networks present results that are valid as the size of the networks grow to- wards infinity. This paper focuses on network sizes in the range that they are actually built today. • Most of the work addressing the non-blocking issue, has communication schemes where all the input ports are lo- cated in one end-stage and the output ports are found in the other. However, in the communication environment we are investigating the ports can serve both as input and out- put channels, also concurrently, using bidirectional links. Therefore we do not distinguish between input- and out- put ports. II. Construction steps The general idea behind Clos networks is to organize the routers in three stages, such that between the stages each pair of routers is connected via a link. Strict non- blockingness of such networks may be guaranteed by en- suring that there are sufficiently many midstage routers – if a connection is requested between two ports, there is al- ways a mid stage router that can be reached from both of the ports using idle links. If m is the number of mid- stage routers, and p e is the number of external ports on each end-stage router, then the requirement for nonblock- ingness is m ≥ 2p e − 1. In the full paper we prove that the bidirectional communication scheme has no implication on the number of midstage elements needed for the network to be strictly non-blocking. Assume now that the size of the router is 8. This means that each mid stage router can only connect to 8 end stage routers, thus the largest Clos network that can be gener- ated straightforwardly is one with 24 ports using 13 routers, see figure 1. In order to build a nonblocking network with arbitrary many ports, we make use of 5 different construction steps: Recursion: In the above example we built a Clos network of with 24 ports from a router with 8 ports. We may now