Scheduling length for switching element disjoint multicasting in Banyan-type switching networks

Abstract

Due to the stringent bit-error requirement in fiber optics, preventing the forthcoming crosstalk at switching elements (SEs) is very crucial in the large-scale photonic switches. In this paper, we consider the SE-disjoint multicasting for a photonic Banyan-type switching network. This ensures that at most, one connection holds each SE in a given time thus, neither photonic crosstalk nor link blocking will arise in the switching network implemented with optical devices such as directional couplers, splitters and combiners. Routing a set of connections under such constraint usually takes several routing rounds hence, it is desirable to keep the number of rounds (i.e., scheduling length) to a minimum. Unfortunately, finding the optimal length is NP-complete in general, we propose an algorithm that seeks an approximation solution less than double of the optimal upper bound. The bound permits the Banyan-type multicasting network to be rearrangeable nonblocking as well as crosstalk-free. It is given that the same bound guarantees wide-sense nonblocking for multicast connections, provided that the multicasting capability of the network is restricted to the second half of the whole stages, and strictly nonblocking for one-to-one connections, yet allowing to be free from the crosstalk. We also consider other crosstalk-free multicasting and the link-disjoint multicasting for the Banyan network. The theory developed in this paper gives a unified foundation on designing nonblocking and crosstalk-free photonic Banyan-type networks under the multicast connections.

Introduction

With the advent of optical wide-band technologies, it has been shown that large-scale photonic switching networks can be constructed using LiNbO₃ directional couplers (DCs) as switching elements (SEs) [9], [18], [23], [24], [25], [26], [27], [28]. A 2×2 electro-optical DC, implemented by fabricating two waveguides close to each other, consists of two optical inputs, two optical outputs, and one control input. The voltage of the control input puts the DC in either one of the two states: the straight and the cross. A photonic SE capable of broadcasting an optical input signal to two outputs also can be designed by combining DCs with other optical devices like splitters or combiners [7], [18], [26], [27], [28]. SEs based on DCs are connected with fiber links to construct a large-scale photonic switching network.

Banyan, Shuffle, and Baseline networks and theirs reverse versions, referred to as Banyan-type networks¹ are typical backbones of the photonic switching networks, as extensively used in parallel processing and electronic switching systems [10], [16], [17], [18], [19], [20], [24], [28]. An attractive feature for high-speed switching is that the routing path between any pair of input–output in the networks is uniquely established by the corresponding bits of the binary representation of the destination output. The small number of SEs between an input–output pair (i.e., O(log₂N) for N×N switching) is also desirable for photonic switching technologies, compared with a crossbar network of O(N).

The photonic Banyan-type networks promise virtually unlimited bandwidth. However, they possess their own problems, for instance, path-dependent signal loss and crosstalk. Studies show that the crosstalk in these networks is more severe than the loss [14], [15], [18], [23], [24], [25], [26]. The photonic crosstalk occurs when two or more connections pass through a DC-based SE in common, and it becomes the noise. Due to the stringent bit-error requirement in fiber optics, preventing the forthcoming crosstalk at the DC-based SEs is very important in the high-speed photonic switches.

A typical system-level approach to the zero-crosstalk in the switching network is to ensure that at most one input of each SE will be used at any given time, i.e., SE-disjoint routing,² so as not to route multiple connections through the same SEs simultaneously [9], [11]. This implies that a set of connections must be grouped into several routing rounds so that connections all assigned to the same round will never share any SE or link and can pass through the network at once without any (first-order) crosstalk. Thus the SE-disjoint scheduling usually needs several routing rounds, the key factor pertaining to it is to keep the number of rounds (referred to as scheduling length) as few as possible. Depending upon switching technologies considered, the scheduling length can also be interpreted in various criteria, for instance, as the number of time slots in the time domain approach [18], as the number of copies of the switching network in the space domain approach [14], [23], and as the number of wavelengths in the wavelength domain approach [12], [24], but all for high-speed switching.

A companion issue in the Banyan-type networks, called nonblocking³ switching, also has been an active topic by many researchers for its application to switching systems [6], [7], [8], [9], [10], [13], [14], [15], [25], [26]. Nonblocking implies the link-disjoint switching of connections hence, its principles and results are applicable to the SE-disjoint routing. In general, the rearrangeable nonblocking is suitable for synchronous switching and comes with less SEs under a complex routing algorithm while the wide-sense or strict-sense nonblocking is useful for an asynchronous system under a simple control but at the cost of high hardware complexity.

Qiao and Zhou [11] and Pan et al. [9] studied various scheduling algorithms for establishing SE-disjoint one-to-one connections in the photonic Banyan network and discussed the scheduling lengths in terms of average number of routing rounds.

Vaez and Lea [14], [15] presented upper bounds of the scheduling lengths which are necessary and sufficient to make Banyan networks, respectively, wide-sense nonblocking and strictly nonblocking under various crosstalk constraints. However, the results are applicable to the multi-Banyan networks and multi-Benes networks when they are loaded with one-to-one connections only.

Pankaj [12] proposed asymptotic upper bounds on the number of wavelengths which are needed to support multicasting capability in the photonic multi-Benes networks, respectively, nonblocking in the wide-sense and in the strict-sense.

In [13], we studied the parallel Banyan networks, respectively, nonblocking in the strict-sense and the wide-sense for the multicast connections. The works are based on the necessary condition for connections to be intersected in the networks therein, the optimality of the nonblocking conditions may not be justified.

Kabacinski and Danilewicz [25] extended our results and proposed a class of wide-sense nonblocking networks which are capable of multicasting without any fanout constraint.

More recently, a couple of works has been given on the elimination of the first-order crosstalk in the vertically and horizontally replicated and expanded parallel Banyan networks [23], [24]. The networks considered, however, were limited to one-to-one connections.

In summary, there are many results on the SE-disjoint routing or nonblocking switching for the Banyan-type one-to-one connection network (too many to cite here!) but relatively few on the multicast. Up to now, we are not aware of archival literature on the SE-disjoint routing or related issues for the Banyan-type mutlicasting networks, as also recently pointed out in [13], [25] (see also footnote 8 in Section 5).

To the best of our knowledge, the SE-disjoint multicasting preventing the (first-order) crosstalk in the DC-based photonic Banyan-type networks is still open issue. The purpose of this paper is to address the subject. The implementation issue or survey on photonic switches is beyond the scope of this paper (see [27], [28]). Our study deserves to receive attention, due to the generality of the multicast connection, the abstraction of the scheduling length, the extensive application of the Banyan-type networks, and the need for large-scale multicast switches supporting multiparty communications over the wide area network.

The overall approach taken to the problem is as follows. We first characterize the necessary and sufficient condition for any connections to pass through an SE in common in the Banyan-type switching networks. The crosstalk relationship among the multicast connections is highly complex, compared with that among the one-to-one connections, hence, we introduce the notion of subconnection by which each multicast connection is uniquely splitted into multiple subconnections. SE-sharing graph (SSG) is proposed to represent the SE-intersection (i.e., crosstalk) relationship among the (sub)connections. The scheduling problem reduces to SSG-coloring (SC) problem and, considering the worst case crosstalk, the maximum number of distinct colors that is needed to properly color any SSG is found. Let colors correspond to distinct routing rounds then, we finally get the upper bound on the number of routing rounds (i.e., scheduling length) for the photonic Banyan-type networks to be zero-crosstalk under the multicast connections.

Unfortunately, the optimal scheduling is NP-complete in general, we present an algorithm that gives an approximation solution in a polynomial time. Given the algorithm, the scheduling length is always within double of the optimal upper bound. It allows the Banyan-type network to be rearrangeable nonblocking and crosstalk-free for a set of multicast connections. The same bound makes the network, respectively, strictly nonblocking for one-to-one connections and wide-sense nonblocking for multicast connections under that the multicasting capability of the network is restricted to the second half of the whole stages. Other crosstalk-free and/or nonblocking conditions for the Banyan-type multicasting network are also discussed.

The rest of this paper is as follows. Preliminaries on the Banyan-type networks are given in Section 2. We discuss the crosstalk problem in the networks. Section 3 introduces some definitions that will be used throughout the paper and presents a graph that represents the crosstalk relationship among the multicast connections. Some definitions are due to previous our works [13]. An approximation algorithm for scheduling SE-disjoint multicasting and the upper bound on the scheduling length are given in Section 4. Correctness and evaluation of the algorithm are explored. Section 5 is devoted to the result of this paper and comparison with other works. Conclusions are drawn in Section 6.