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

Journal of Systems Architecture 48 (2003) 175–191 www.elsevier.com/locate/sysarc

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

*

School of Computer, Information and Communication Engineering, Sangji University, Wonju, Kangwon 220-702, South Korea

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. Ó 2002 Elsevier Science B.V. All rights reserved. Keywords: Banyan network; Multicasting network; Nonblocking switching network; Photonic crosstalk; SE-disjoint routing

1. Introduction 1.1. Motivation 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

*

Tel.: +82-33-730-0484; fax: +82-33-745-2433. E-mail address: [email protected] (Y. Tscha).

elements (SEs) [9,18,23–28]. 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 [7,18,26–28]. SEs based on DCs are connected with fiber links to construct a large-scale photonic switching network.

1383-7621/03/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. doi:10.1016/S1383-7621(02)00135-2

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Banyan, Shuffle, and Baseline networks and theirs reverse versions, referred to as Banyan-type networks 1 are typical backbones of the photonic switching networks, as extensively used in parallel processing and electronic switching systems [10,16–20,24,28]. An attractive feature for highspeed 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., Oðlog2 N Þ for N
N switching) is also desirable for photonic switching technologies, compared with a crossbar network of OðN Þ. The photonic Banyan-type networks promise virtually unlimited bandwidth. However, they possess their own problems, for instance, pathdependent signal loss and crosstalk. Studies show that the crosstalk in these networks is more severe than the loss [14,15,18,23–26]. 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. 1.2. Problem and literature review A typical system-level approach to the zerocrosstalk 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 (firstorder) 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 nonblocking 3 switching, also has been an active topic by many researchers for its application to switching systems [6–10,13–15,25,26]. 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 SEdisjoint one-to-one connections in the photonic Banyan network and discussed the scheduling lengths in terms of average number of routing rounds.

3

1

Throughout this paper we refer to Banyan, Shuffle, Baseline networks and their topological equivalents [17] as Banyan-type networks in stead of multistage interconnection networks (MINs). Some of the important works early done on the networks can be found, for instance, in [2,19–23,29–31]. 2 Even if only one signal passes through a photonic SE, a small portion of it may leave at the other unintended output. Hence, this stray signal may arrive at the input of the next stage SE and will result in the second-order crosstalk [18,26]. Nevertheless, this is much smaller than the first-order crosstalk. In this paper, we will consider only the first-order crosstalk as in [9,11,14,15,23–25].

By definition in [3] there are three kinds of a nonblocking network. In the rearrangeable nonblocking network, some of existing connections may be reconfigured for it to be nonblocking. The wide-sense nonblocking is achieved by a clever algorithm that carefully selects the connection paths in order to avoid all the forthcoming blocking states of the network without disturbing existing connections. The strictly (strictsense) nonblocking network has no such constraints. Every strictly nonblocking network is also wide-sense nonblocking and the wide-sense nonblocking network is also rearrangeable nonblocking. But not vice versa. On the other hand, the terms switching, routing and scheduling may be used interchangeably in this paper, as in [9].

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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 SEdisjoint 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

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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.

1.3. Approach and paper organization 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 Banyantype networks to be zero-crosstalk under the multicast connections. Unfortunately, the optimal scheduling is NPcomplete 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

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picts a 16
16 network. There are four stages, 1, 2, 3, 4 (¼ log2 16), respectively, from left to right. Sixteen inputs (outputs) are numbered 0 through 15, respectively, from top to bottom. We assume every photonic SE is capable of multicasting: lower and upper broadcasts, as well as unicasting: straight and crossswitching [7,8,13,25]. The unique feature of the Banyan-type networks is that the path between each input–output pair is uniquely determined by the corresponding bit of the binary representation of the destination output [17,19]. If i (1 6 i 6 n)-th bit is 0 then, the active input link of the SE at stage i is connected to the upper output link (UP operation); otherwise to the lower output link (DOWN operation). Consider, for instance, the routing to destination output 5 in Fig. 1. Since the corresponding binary representation of the output is 0101, the sequence of decisions made by the SE at each stage from left to right is given as UP, DOWN, UP, and DOWN, respectively. Thus, any path to the des-

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 SEdisjoint 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.

2. Preliminaries 2.1. Banyan-type networks and crosstalk Without loss of generality, we use the N
N (N ¼ 2n ) reverse Baseline network [17] as the representative of Banyan-type networks. Fig. 1 de-

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Fig. 1. A 16
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tination can be easily and uniquely established by such bit-hunting. Denote by hw; Si a connection from an input w to a non-empty set S of destination outputs, where w 2 f0; 1; . . . ; N 1g, S  f0; 1; . . . ; N 1g. Then, hw; Si is one-to-one (i.e., unicast) if and only if jSj ¼ 1 (say, h12; f12gi) and one-to-many (i.e., multicast) otherwise (say, h0; f0; 3gi), where jSj denotes the cardinality of S. Photonic crosstalk occurs when two or more connections go through some SE in common at the same time. In Fig. 1, for example, h0; f0; 3gi and h3; f4; 7gi share the SE at stage 2. Hence, the output signals of the SE would be distorted and be the source of bit-error if they are established simultaneously. Moreover, h0; f0; 3gi and h1; f1; 5gi collide at several links, respectively, between stage 1 and 2, 2 and 3, 3 and 4. Such link-blocking is another source of the crosstalk. If two connections intersect at a link between stage j and j þ 1, 1 6 j 6 n 1, evidently, they use commonly the SEs at both stages. Thus, the SE-disjoint routing will prevent both types of the crosstalk. Consider the upper four connections h0f0; 3gi, h1; f1; 5gi, h2; f2; 6gi, and h3; f4; 7gi in Fig. 1 in which they all go through the SE at stage 2 in common. In order for the zero-crosstalk each connection must be exclusively entered into the network hence, four routing rounds are inevitable. By doing so the link-blocking also will not show up throughout the network.

2.2. Crosstalking condition We first consider a simple topological property of Banyan-type networks. It is due to the buddy property [2] of the networks. Observation 1. In the N
N (N ¼ 2n ) Banyantype network, an SE at stage j, 1 6 j 6 n, is reachable to 2j inputs and 2n jþ1 outputs, respectively. That is, there exists a back-to-back fully binary tree in which the root is an SE at stage j and the number of leaves, i.e., inputs (respectively, outputs), in the left (right) subtree is 2j ð2n jþ1 Þ such that 2j  2n jþ1 ¼ 2N (for instance, see thick lines and squares in Fig. 2).

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Let Cði; jÞ denote the number of bits in common prefix between n-bit binary representations of i and j. For instance, Cð1001; 1011Þ ¼ 2 and Cð1000; 0100Þ ¼ 0. Given any hw; Ri and hx; Si, w 6¼ x, R \ S ¼ ;, it follows that 0 6 Cðw; xÞ, Cðr; sÞ 6 n 1 and 0 6 Cðw; xÞ þ Cðr; sÞ 6 2ðn 1Þ for some r 2 R and s 2 S. With this notation a condition that connections will pass through an SE in common is as follows. Lemma 1. In the N
N (N ¼ 2n ) photonic reverse Baseline network, hw; Ri and hx; Si use a common SE (i.e., make crosstalk) if and only if Cðw; xÞ þ Cðr; sÞ P n 1 for some r 2 R and s 2 S. Proof Necessity: Assume hw; Ri and hx; Si pass through commonly some SE at stage j (1 6 j 6 n). Clearly, the common SE is reachable to both inputs w and x, and also to some outputs r 2 R and s 2 S. We note that inputs (outputs) of the reverse Baseline network are consecutively numbered 0 through N 1, respectively, from top to bottom. Considering Observation 1, the numerical difference between the inputs w and x is given as 1 6 jw xj 6 2j 1. 4 Given the n-bit binary representations of w and x, we have Cðw; xÞ P n j. Similarly, we have 1 6 jr sj 6 2n jþ1 1. Therefore, Cðr; sÞ P n ðn j þ 1Þ ¼ j 1 and finally, Cðw; xÞ þ Cðr; sÞ P n 1. Sufficiency: Assume that Cðw; xÞ þ Cðr; sÞ P n 1 for some r 2 R and s 2 S. Let Cðw; xÞ P t for some t, 0 6 t 6 n 1 then, 1 6 jw xj 6 2n t 1. Since Cðr; sÞ P n ðt þ 1Þ, we have 1 6 jr sj 6 2tþ1 1. This implies that there exists a back-toback fully binary tree in which the root is the SE at stage n t and the number of inputs (respectively, outputs) in the left (right) subtree is 2n t ð2tþ1 Þ, resulting 2n t  2tþ1 ¼ 2N (Observation 1). First n ðt þ 1Þ bits between the n-bit binary representations of r and s must be the same. Cleary, the

4 For any set S, jSj denotes its cardinality. Notation ji jj is used to denote the absolute numerical difference between two integers i and j. Distinction between these two notations comes without confusion.

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Fig. 2. Input intersection windows IWi SE and output intersection windows OWi SE in the N
N (N ¼ 2n ) Banyan-type networks: (a) n is even (n ¼ 4), (b) n is odd (n ¼ 5).

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routing of two connections hw; Ri and hx; Si will result in intersection at stage n t.  A direct consequence from Lemma 1 is as follows: hw; Ri and hx; Si intersect at stage j (1 6 j 6 n), for the first time, if and only if Cðw; xÞ ¼ n j and Cðr; sÞ P j 1 for some r 2 R and s 2 S. For h12; f12gi and h14; f8; 10; 14gi shown in Fig. 1, it is seen that Cð12; 14Þ ¼ Cð1100; 1110Þ ¼ 2 for input 12 and 14, and so does for output 12 and 14. Thus, h12; f12gi and h14; f8; 10; 14gi intersect at stage 2 for the first time. Letting Cðw; xÞ P ðn 1Þ=2 and Cðr; sÞ P ðn 1Þ=2 in Lemma 1, it is seen that the number of connections that pass through an SE in common is at most 2bðnþ1Þ=2c at the center stage ðn þ 1Þ=2 for odd n (respectively, n=2 or ðn=2Þ þ 1 for even n), where bxc denotes the greatest integer less than or equal to x. This leads to the following definition. Given the input (output) set f0; 1; . . . ; N 1g of the reverse Baseline network, let IWfSE (respectively, OWfSE ) be fdSE  f ; dSE  f þ 1; . . . ; dSE  f þ ðdSE 1Þg, where f 2 f0; 1; . . . ; ðN =dSE Þ 1g, dSE ¼ 2bðnþ1Þ=2c . Each IWfSE (respectively, OWfSE ) is called input (output) intersection window. Hence, dSE is said to be intersection window size, i.e., jIWfSE j ¼ jOWfSE j ¼ dSE . Fig. 2 shows input and output intersection windows in the N
N Banyantype switching networks where N ¼ 16 and 32, respectively. With this definition, connections intersecting at some SE are characterized by Observation 2. Observation 2. Assume that hw; Ri and hx; Si intersect at some SE in the Banyan-type network, w 6¼ x, R \ S ¼ ;. It is true that either (1) w, x 2 IWfSE for some f 2 f0; 1; . . . ; ðN =dSE Þ 1g (i.e., Cðw; xÞ P bn=2cÞ or (2) R \ OWgSE and S \ OWgSE for some g 2 f0; 1; . . . ; ðN =dSE Þ 1g (i.e., Cðr; sÞ P bn=2c for some r 2 R and s 2 S).

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dows allows us to easily analyze the worst case crosstalk in the multicast connections. The next section will cover these topics in detail.

3. Multicast and worst case crosstalk Given hw; Si, let S be partitioned into a set of subsets fSfSE g such that, for each SfSE , SfSE ¼ S \ OWfSE and SfSE 6¼ ;, where f 2 f0; 1; . . . ; ðN =dSE Þ 1g. Each hw; SfSE i is said to be subconnection of hw; Si. For the output set {8,10,14} of h14; f8; 10; 14gi in Fig. 1, it is seen that f8; 10g  OW2SE and f14g  OW3SE . Hence, the connection has two unique subconnections h14; f8; 10gi and h14; f14gi. By definition every one-to-one connection is subconnection itself. For any hw; Si and set of its subconnections fhw; SfSE ig, it follows that SfSE  OWfSE , 1 6 jSfSE j 6 dSE and that S ¼ [SfSE , \SfSE ¼ ;, 1 6 jfhw; SfSE igj 6 N =dSE . An SSG GSE ¼ ðVSE ; ESE Þ is defined as follows. VSE is a set of vertices corresponding to the subconnections, and ESE is a set of edges such that an edge is drawn between two vertices if they are corresponding to the crosstalking subconnections. Fig. 3 depicts GSE from the connections shown in

<3,{4,7}> <2,{2}>

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The upper bound on the number of routing rounds in the SE-disjoint multicasting can be derived in terms of the window size dSE . More importantly, splitting each multicast connection into several subconnections based on intersection win-

<14,{8,10}>

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Fig. 3. GSE from connections shown in Fig. 1.

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Fig. 1. There are no direct edges among the vertices that correspond to the subconnections with an identical input. Because these subconnections are defined from a multicast connection, they never makes the crosstalk even if they use some SE in common (for instance, see the vertices corresponding to h2; f2gi and h2; f6gi). Definition 1. In GSE , physical degree of a vertex v, denoted by pdðvÞ, is the number of vertices adjacent to v. Logical degree of v, ldðvÞ, is defined to be the number of vertices that are adjacent to v and correspond to subconnections with different inputs. Physical (respectively, logical) degree of GSE , pdðGSE Þ ðldðGSE ÞÞ, is the maximal physical (logical) degree of a vertex in it. It is seen in Fig. 3 that pdðh3; f4; 7giÞ ¼ 5, ldðh3; f4; 7giÞ ¼ 3, pdðGSE Þ ¼ 5, and ldðGSE Þ ¼ 3. The upper bounds on pdðGSE Þ and ldðGSE ) are respectively as follows. Lemma 2. It holds for any GSE that pdðGSE Þ 6 0:5ndSE for even n and pdðGSE Þ 6 0:25ðn þ 1ÞdSE for odd n, respectively, where dSE ¼ 2bðnþ1Þ=2c . Proof. We first consider odd n. Suppose an arbitrary subconnection hw; SgSE i such that w 2 IWfSE , SgSE  OWgSE for some f , g 2 f0; 1; . . . ; ðN =dSE Þ 1g. Bearing the condition given in Lemma 1, we will count all subconnections that may intersect hw; SgSE i at stage 1 through n (see Fig. 2 for convenience). Noting jOWgSE j ¼ dSE , it follows that hw; SgSE i and dSE 1 other subconnections can maximally intersect for any stage j such that j P ðn þ 1Þ=2, provided that jSgSE j ¼ 1. Next we consider the intersections at stage 1; 2; . . . ; ðn 1Þ=2, respectively. An SE at stage 1 is reachable from 2 inputs and 2n outputs (i.e., N =dSE output intersection windows) thus, hw; SgSE i may intersect ðN =dSE Þ 1 other subconnections (note that constant ‘‘1’’ accounts for the subconnection hw; SgSE iÞ. At stage 2 an SE is reachable from 22 inputs and 2n 1 outputs (i.e., ðN =2Þð1=dSE Þ output intersection windows), respectively. Excluding the subconnections considered at stage 1, it follows that ½ðN =22 1 Þðð1=dSE Þ 1Þ22 1 additional subconnections can maximally intersect hw; SgSE i at

stage 2. Following the similar argument, it is seen that ½ðN =23 1 Þðð1=dSE Þ 1Þ23 1 additional subconnections at stage 3;...;½ðN =2ððn 1Þ=2Þ 1 Þð1=dSE Þ 1 2ððn 1Þ=2Þ 1 additional subconnections at stage ðn 1Þ=2. Therefore, the maximal number of subconnections that will intersect hw;SgSE i (i.e., the maximal physical degree of the vertex corresponding hw;SgSE i) is given by ðdSE 1Þþ Pðn 1Þ=2 to j 1 ½ðN =2 Þðð1= dSE Þ 1Þ2j 1 ¼ 0:25ðn þ 1Þ
j¼1 dSE . Similarly, Pn=2 it is given that pdðGSE Þ6 ðdSE 1Þ þ j¼1 ½ðN =2j 1 Þðð1=dSE Þ 1Þ2j 1 ¼ 0:5n dSE for even n.  Lemma 3. For any GSE , it is given that ldðGSE Þ 6 2dSE 2 when n is even and ldðGSE Þ 6 1:5dSE 2 when n is odd, respectively. Proof. By definition of logical degree of a vertex, the term ½ðN =2j 1 Þðð1=dSE Þ 1Þ of each formula given in the proof part of Lemma 2 becomes Pðn 1Þ=21. This leads to that ldðGSE Þ 6 ðdSE 1Þ þ j¼1 2j 1 ¼ 1:5dSE 2 for odd n, and ldðGSE Þ6ðdSE 1Þþ P n=2 j 1 ¼ 2dSE 2 for even n, respectively.  j¼1 2 Note that pdðGSE Þ ¼ OðldðGSE Þ  log2 N Þ holds for sufficient large N . We also know that Lemma 3 is true for one-to-one connections in the Banyantype networks because every one-to-one connection is subconnection itself thus, the equality pdðGSE Þ ¼ ldðGSE Þ always holds. Furthermore this is true for any multicast connections fhw; Sig in which the output set S of each hw; Si is such that S  OWfSE for some f 2 f0; 1; . . . ; ðN =dSE Þ 1g. The fanout capability of each multicast is restricted to the stage dðn þ 1Þ=2e through n, as in [13], where dxe denotes the least integer greater than or equal to x. Any connection with such output constraint will have only one subconnection regardless of whether it is one-to-one or oneto-many. Observation 3. It holds that pdðGSE Þ ð¼ ldðGSE ÞÞ 6 1:5dSE 2 (respectively, 2dSE 2) when n is odd (even), where GSE is given from a set of connections fhw; Sig in which S of each hw; Si is such that S  OWfSE for some f 2 f0; 1; . . . ; ðN =dSE Þ 1g.

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4. Upper bound on scheduling length In this section we consider GSE -coloring problem. The properly coloring 5 [4] of GSE is equivalent to assigning the SE-sharing subconnections into different routing rounds so that at most one subconnection holds each SE at any given time. With this abstraction the number of the colors used for coloring GSE becomes the number of routing rounds (i.e., scheduling length) needed for the SEdisjoint routing of a set of given multicast connections. 4.1. Optimal bound It is well-known that the problem finding the chromatic number of an arbitrary graph is NPcomplete [5]. A self-evident result immediately follows. Finding the chromatic number of any GSE is NP-complete. This implies that an optimal GSE coloring algorithm has a time complexity which is exponential to the number of (sub)connections to be set up. Hence, we need a coloring algorithm that has a polynomial time complexity and gives a good approximation to the optimal. By definition of GSE , the subconnections intersecting at an SE correspond to a clique 6 [4] in GSE . For instance, in the lower connected component of GSE in Fig. 3, the vertices h12; f12gi, h13; f13gi, h14; f14gi, and h15; f15gi build up the clique of size four. A self-evident result on the maximal clique in GSE is as follows. Corollary 1. The size of the maximal clique in GSE is dSE hence, the upper bound on the chromatic number for GSE is at least dSE .

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upper bound on the scheduling length is at least dSE because a clique of size k needs k distinct colors for it to be properly colored. 4.2. Approximation bound We now give an approximation algorithm that properly colors GSE with ldðGSE Þ þ 1 colors. This of course does not guarantee the bound given by Corollary 1 nevertheless, the result based on Lemma 3 will promise somewhat reasonable bound on the scheduling length because coloring GSE with pdðGSE Þ þ 1 colors would be too high (see Lemma 2). Definition 2. Given GSE , denote by brotherðj; wÞ a set of vertices which are adjacent to a vertex w and correspond to the subconnections with an input j in common. For instance, in Fig. 3, brother ð1;h0;f0;3giÞ ¼ fh1;f1gi;h1;f5gig, and jfbrotherðj;h0;f0;3giÞgj ¼ jffh1; f1gi; h1; f5gig; fh2; f2gi; h2; f6gig; fh3; f4; 7giggj ¼ 3ð¼ ldðh0; f0; 3giÞÞ. In order for GSE to be properly colored with ldðGSE Þ þ 1 colors, all vertices in each brotherðj; wÞ must be colored with the same color so that w and fbrotherðj; wÞg are colored with jfbrotherðj; wÞgj þ 1 colors, at most. For the lower connected component shown in Fig. 3, four colors will suffice for it to be properly colored, provided that h14; f14gi and h14; f8; 10gi are colored with the same color. Suppose v1 , v2 2 brotherðj; wÞ for some j and w. Assume that v1 is colored but v2 is under coloring (see Fig. 4). The properly coloring of v2 is done as follows. If the color of v1 has not been used for any vertex adjacent to v2 then, it is used for v2 .

No matter how to design an efficient optimal coloring algorithm, it is evident that the optimal vertex under coloring

brother(j,w)={v1,v2,...,vq}

5

For a graph G, we say that G is properly colored if no adjacent vertices are colored with the same color. The chromatic number of G is the smallest number of colors for which G can be properly colored. 6 A clique of a graph G is a completely connected subgraph of G. The size of a clique is the number of vertices in it. The maximal clique is a clique whose size is not less than any others.

v1

v2

vq

x

w path under coloring

Fig. 4. Coloring the brother vertices.

y

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Otherwise, choose some color used in GSE but, neither for the vertices adjacent to v1 nor to v2 , then, color v2 and re-color v1 with it, respectively. If this impossible, a new color is chosen, and v2 and v1 are colored and re-colored respectively with it. In Fig. 4, assume vertices in brotherðj; wÞ are colored in the order v1 ; v2 ; . . . ; vq . Then, in the worst case, each vk , 1 6 k 6 q, will be colored one time and re-colored (q k) times, q 6 N =dSE . A detailed description of our algorithm, SC, is given in Fig. 5. Note that the path taken by the algorithm builds up the depth-first spanning tree [1] for each connected component of GSE . The vertex under coloring is always chosen by the depth-first search, with respect to the vertex colored most recently. Let us consider Fig. 6, a subgraph G0SE of the upper connected component shown in Fig. 3, for pedagogical purpose. Suppose the coloring has done in the order h3; f4; 7gi, h2; f2gi, h1; f1gi, and h0; f0; 3gi. Then, G0SE looks like Fig. 6 (a). We are now to color h1; f5gi adjacent to h0; f0; 3gi. Noting brotherð1; h0; f0; 3giÞ ¼ fh1; f1gi, h1; f5gig and ColorsðAdjðh1; f5giÞÞ \ Colorsðbrotherð1; h0; f0; 3giÞÞ ¼ f1; 3g \ f1g ¼ f1g 6¼ ;, color ‘‘1’’ cannot be used for h1; f5gi (violation of the properly coloring). Hence, by line 5 through 6 and line 9 in Fig. 5, it follows that UsedColor0 -Colors (Adj(brother ð1; h0; f0; 3giÞÞ ¼ f1; 2; 3g f1; 2; 3g ¼ ;. Thus, new color ‘‘4’’ is chosen due to line 10 in Fig. 5, then h1; f1gi is re-colored and h1; f5gi is colored with it, respectively. Finally, since Colors(Adjðh2; f6giÞÞ \ Colorsðbrother
ð2; h3; f4; 7giÞÞ ¼ f1; 4g \ f2g ¼ ;, color ‘‘2’’ is used for h2; f6gi (line 3 through 4 in Fig. 5). Therefore, all vertices are properly colored such that vertices in each brother(j; w) are colored with the same color (Fig. 6(b)). The total number of colors used becomes 4 (¼ ldðGÞ þ 1). Letting colors (i.e., integers) correspond to routing rounds, it is seen that the SE-disjoint multicasting of these connections can be done in four rounds. However, three rounds are optimal in this example. Theorem 1. Using algorithm SC in Fig. 5, any GSE is properly colorable with ldðGSE Þ þ 1 colors. Proof. See Appendix A.



Let us consider the connections shown in Fig. 1. From its corresponding GSE in Fig. 3 it follows that ldðGSE Þ ¼ 3 and four colors suffice for the properly coloring of it by Theorem 1. Coloring by the algorithm in Fig. 5 will make the twelve subconnections partitioned into four rounds such that subconnections all assigned to the same round never share any SE when they are set up simultaneously in the network. A possible scenario of four routing rounds, for instance, is as follows: Round 1 ¼ fh3; f4; 7gi; h12; f12gi; h10; f9; 11gig; Round 2 ¼ fh2; f2gi; h2; f6gi; h13; f13gig; Round 3 ¼ fh1; f1gi; h1; f5gi; h14; f14gi; h14; f8; 10gig; Round 4 ¼ fh0; f0; 3gi; h15; f15gig. This guarantees the zero-crosstalk of the network. 4.3. Evaluation of algorithm It is seen that the time to compute ColorsðAdjðbrotherðj; wÞÞÞ (line 6 of VC(v) in Fig. 5) for a vertex v dominates the time complexity of VC(v) and it requires OðN1 þ dðN1 þ N2 ÞÞ at most for GSE ¼ ðVSE ; ESE Þ represented by an adjacency list [1], where jVSE j ¼ N1 , jESE j ¼ N2 , d ¼ pdðwÞ. The computation will be repeated for N1 vertices. Hence, the time complexity of SC in Fig. 5 is given as OðN12 þ dN1 ðN1 þ N2 ÞÞ. Since N1 ¼ OðN Þ and d ¼ OðndSE Þ (Lemma 2), the complexity becomes OðN22:5 log2 N ) for N2 ¼ OðN Þ and OðN 3:5 log2 N Þ for N2 ¼ OðN 2 Þ, respectively. Now, we consider the effectiveness of our approximation algorithm. Given an NP-complete problem P, let OPT ðPÞ and APRðPÞ be optimal and approximation solutions respectively found by their corresponding algorithms. It is said that the approximation is good provided that ½APRðPÞ=OPT ðPÞ 6 c for some constant c [5]. By Corollary 1, the upper bound on the optimal length by OPT ðPÞ is at least dSE , that is, OPT ðPÞ P dSE . With Theorem 2 (Section 5), it follows that APRðPÞ 6 2dSE 1 for even n and APRðPÞ 6 1:5dSE 1 for odd n, respectively. Our approximations appear to be good because ½APRðPÞ=OPT ðPÞ ¼ ½ð2dSE 1Þ=dSE  < 2 for even

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Fig. 5. The algorithm SC.

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Theorem 2. The maximum scheduling length given by SC in Fig. 5 for SE-disjoint multicasting in the Banyan-type network is 2dSE 1 for even n and 1.5dSE 1 for odd n, respectively, where dSE ¼ 2bðnþ1Þ=2c .

1 <3,{4,7}>

2

<2,{2}>

<2,{6}>

1

<1,{1}>

<1,{5}> <0,{0,3}>

(a)

3

Proof. Direct consequence from Lemma 3 and Theorem 1. 

vertex under coloring

1 <3,{4,7}>

2

<2,{2}>

<2,{6}>

2

4

<1,{1}>

<1,{5}>

4

re-colored with 4 (b)

<0,{0,3}>

3

Fig. 6. Application example of the algorithm SC.

n and ½APRðPÞ=OPT ðPÞ ¼ ½ð1:5dSE 1Þ=dSE  < 1:5 for odd n, with the equality OPT ðPÞ ¼ dSE . 7

5. Results and comparison 5.1. Under the crosstalk-free constraint Given our algorithm, it is a direct matter to show the upper bound on the scheduling length for the SE-disjoint multicasting in the Banyan-type switching networks.

7

Note that performance ratio is viable for GSE only, where the size of the maximal clique (Corollary 1) and the maximal logical degree (Lemma 3) are on hand in advance. For any graph of N vertices, the best-known ratio on vertex-coloring is OðN = log2 N Þ (see p. 133 in [5]).

Corollary 2. Let PSE be the scheduling length required for SE-disjoint connection setup in the DCbased Banyan-type switching network. Assume PSE is 2dSE 1 for even n and PSE is 1.5dSE 1 for odd n, respectively. Then, the network is crosstalk-free and furthermore, (1) rearrangeable nonblocking for multicast connections, (2) strictly nonblocking for one-to-one connections, and (3) wide-sense nonblocking for a set of multicast connections fhw; Sig such that S  OWgSE , g 2 f0; 1; . . . ; ðN =dSE Þ 1g, for each hw; Si. Proof. Recall that the SE-disjoint routing of connections guarantees the nonblocking (link-disjoint) network. The proofs are self-evident by definition of each nonblockingness (footnote 3) and Theorem 2 and Observation 3.  Corollary 3. The photonic Banyan-type network is crosstalk-free and strictly nonblocking for multicast connections if its SE-disjoint scheduling length is 0.5ðndSE Þ þ 1 for even n and 0:25ððn þ 1ÞdSE Þ þ 1 for odd n, respectively. Proof. Straightforward by Lemma 2.  It is interesting to note that Corollary 2 is not only new but also general in the sense that the bound in Theorem 2 coincides with that is required for strictly nonblocking one-to-one connection networks under crosstalk-free constraint [14,15,23, 24]. Compared with the strictly nonblocking multicasting network (Corollary 3), the wide-sense nonblocking network given by a simple restriction that keeps the output(s) of each multicast to the subset of an output intersection window greatly reduces the scheduling length. Routing is also much simpler than the rearrangeable nonblocking

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networks. For instance, the interval-splitting algorithm [6] is directly applicable. It should be noticed that the scheduling length 2 is the lower bound of the SE-disjoint multicasting because N (sub)connections are established in the N
N Banyan-type network that consists of log2 N stages and N =2 SEs at each stage. In reality, the subconnection-by-subconnection routing would be carefully considered whether it is admittable or not in the switching network under consideration since a multicast connection is set up per subconnection-basis and hence, its all subconnections may not be established at the same time. Note that the bound given by each corollary above also corresponds to the number of network planes (respectively, wavelengths) that is required for making space-division (wavelength-division) Banyan-type networks nonblocking and crosstalkfree under the multicast connections.

5.2. Under the nonblocking constraint So far we have discussed on the SE-disjoint multicasting for the crosstalk-free network. Another issue many researchers have been interested in is the link-disjoint (i.e., nonblocking). Consider the N
N (N ¼ 2n ) Banyan-type multicasting network, as Fig. 1. For the simplicity, we say that a link is at stage i if it is between stage i and i þ 1, 1 6 i 6 n 1. Cleary, link-blocking may happen at stage 1 through n 1. The link at stage i is reachable to 2i inputs and 2n i outputs, respectively. It is seen that at most, 2bn=2c connections can collide at the stage n=2 (respectively, ðn 1Þ=2 or ðn þ 1Þ=2) for even (odd) n. A necessary and sufficient condition for any two connections to intersect at some link in the Banyan-type network (similar to Lemma 1) is easily derived as follows. In the N
N (N ¼ 2n ) reverse Baseline network, connections hw; Ri and hx; Si (w 6¼ x; R \ S ¼ ;) collide at some link if and only if Cðw; xÞ þ Cðr; sÞ P n for some r 2 R and s 2 S, where 1 6 Cðw; xÞ, Cðr; sÞ 6 n 1 and Cði; jÞ denotes the number of bits in common prefix between the n-bit binary representations of i and j. Letting dL ¼ 2bn=2c and replacing with it all occurrence of dSE in

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definitions given so far, the upper bound on the link-disjoint scheduling is as follows. Corollary 4. The maximum scheduling length given by SC in Fig. 5 for link-disjoint multicasting in the Banyan-type network is 1.5dL 1 for even n and 2dL 1 for odd n, respectively, where dL ¼ 2bn=2c . It is easy to identify the bound is also within double of the optimal upper bound. The network satisfying Corollary 4 is of course rearrangeable nonblocking for multicast connections. With analogy to Corollary 2, it is found that the network is also strictly nonblocking for one-to-one connection, as in [7,8,13–15,24], and wide-sense nonblocking for multicast connections in which the output(s) of each multicast is restricted to the subset of an output intersection window defined by it size dL [13]. Comparison with related results is summarized in Table 1. As for the SE-disjoint multicasting our result can be regarded as the first trial on the Banyan-type network. 8 Concerning the link-disjoint multicasting, there is a set of new results, extended our early works [13], on the wide-sense nonblocking parallel Banyan network which are capable of multicasting without any fanout constraint [25]. More recently, two works have been given on the elimination of the first-order crosstalk in the vertically and horizontally replicated and

8

With the proliferation of a large-scale ATM or optical switching technology, a lot of researches has been being active on overcoming the blocking nature of the Banyan-type switching network for application to high-speed broadband switching systems (see [10,26,27] and references in there). Most of them are based on the Batcher–Banyan network [29] or its variants (say, [6,31]), where for instance, sorting network is used for resolving the link conflicts between connections and Banyan for routing to the final outputs. Example of an early made commercial switch based on this concept is ‘‘starlite’’ [30]. Nowadays small-scale switches (say, 16
16 or 32
32) or routers are based on high-speed bus or crossbar switching technologies [26]. The study on multicast switching in the Banyan-type network, without adding to it any extra network such as copy or sorting or distribution network, has been witnessed recently [7–9,11–15,23–25], even some works done early for unicasting [7,8,17,19–22].

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Table 1 Comparison with related results Scheduling length ensuring

Connection type Multicast (this paper)

SE-disjoint Scheduling Rearrangeable 2dSE 1 for even n nonblocking and 1.5dSE 1 for odd n crosstalk-free

One-to-one [14,15,24]

One-to-one [9,11]

–

Average scheduling length

Wide-sense nonblocking and crosstalk-free

2dSE 1 for even n 1.5dSE 1 for odd n (S  OWfSE for any hw; Si)

1:5dSE 1 for even n dSE 1 for odd n (allowing one crosstalk)

–

Strictly nonblocking and crosstalkfree

0.5ndSE þ 1 for even n 0.25 ðn þ 1ÞdSE þ 1 for odd n

2dSE 1 for even n 1:5dSE 1 for odd n

–

Multicast (this paper)

Multicast [13]

One-to-one [7,8]

Multicast [25]

–

dL (also in [23])

–

1.5dL 1 for even n

–

0.25n dL 1 for even n

Link-disjoint scheduling Rearrangeable 1:5dL 1 for even n 2dL 1 for odd n nonblocking and crosstalk-free Wide-sense non– blocking and crosstalk-free

Strictly nonblocking and crosstalkfree

–

2dL 1 for odd n (S  OWfL for any hw; Si) 0.25 ndL 1 for even n 0.5 ðn 1ÞdL 1 for odd n

expanded parallel Banyan networks [23,24]. Their works, however, are limited to one-to-one connections.

6. Conclusions The abundant bandwidth of a large-scale photonic switch will not come without the reduction of crosstalk at SEs. In this paper, we have studied the SE-disjoint routing for the DC-based photonic Banyan-type networks to be zero-crosstalk under the multicast connections. So as not to route multiple connection through SEs in common, connections are grouped into several routing rounds so that connections all assigned to each round do not share any SE or link and they can pass through the network without causing the (first-order) crosstalk.

0.5ðn 1ÞdL 1 for odd n(also various results) 1:5dL 1 for even n 2dL 1 for odd n

N =2

Finding an optimal number of routing rounds (i.e., scheduling length) is NP-complete hence, we have proposed an approximation algorithm whose upper bound is less than double of the optimal upper bound. The bound we have shown is 2dSE 1 for even n and 1.5dSE 1 for odd n, respectively, where dSE ¼ 2bðnþ1Þ=2c . Interestingly, this allows Banyan-type multicasting networks to be rearrangeable nonblocking and crosstalk-free. The same result is also sufficient for making the networks crosstalk-free and strictly (respectively, wide-sense) nonblocking under one-to-one connections (some restricted multicast connections). We have also given some results on nonblocking (link-disjoint) routing of multicast connections in the Banyan-type network. We expect several notions and definitions introduced in this paper, such as input/output intersection window, subconnection, SSG, and

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virtual degree, will be helpful for studying multicast switches. Extension of our theory to modified Banyan-type networks, vertically replicated and horizontally expanded versions [23,24] but allowing the multicast connections, is left for further research. We look forward to developing another algorithm or heuristic that guarantees more tight bound on the scheduling length with an improved time complexity.

Appendix A. Proof of Theorem 1 Proof. We first assume SSG GSE ¼ ðVSE ; ESE Þ has one connected component. Let v be the vertex 0 0 under coloring. Let G0SE ¼ ðVSE ; ESE Þ be a subgraph of GSE that consists of colored vertices and edges among them (not including v). Denote by 00 00 G00SE ¼ ðVSE ; ESE Þ the subgraph obtained by add0 ing v to GSE , i.e., G00SE ¼ G0SE [ fvg. Recall that the vertex-coloring path taken by algorithm SC in Fig. 5 is the depth-first spanning tree of GSE . The proof proceeds by mathematical induction on jVSE j, the number of vertices in G0SE . For 0 jVSE j P 1, let w in G0SE be a vertex adjacent to 0 v, i.e., w 2 AdjðvÞ. Letting m ¼ jVSE j, the proof is as follows (the proofs for m ¼ 0 and 1 are trivial). (1) Case 1 (m ¼ 2): Let x be another vertex in G0SE , except w. Then, it follows that x 2 AdjðwÞ and v 2 AdjðwÞ. Assume w and x are colored with color ‘‘1’’ and color ‘‘2’’, respectively (note that colors correspond to positive integers in the algorithm SC of Fig. 5). Supposing v 2 brotherðj; wÞ, whether x 2 brotherðj; wÞ or not is checked by line 1 of Fig. 5. If so line 2 and otherwise line 13 are followed, respectively. In the former case, it is seen that ldðG0SE Þ ¼ ldðG00SE Þ ¼ 1 and ColorsðAdjðvÞÞ \ Colorsðbrotherðj;wÞÞ ¼ ColorsðwÞ \ ColorsðxÞ ¼ f1g \ f2g ¼ ;. The color of x, ‘‘2’’, is used for v and thus, x and v are colored with the same color (line 3 through 4). This does not violate the properly coloring. Hence, the total number of colors used is 2 (¼ ldðG00SE Þ þ 1 ¼ 1 þ 1). The execution of line 13 implies x 62 brotherðj;wÞ. If x is adjacent to v (that is, x 2 AdjðvÞ) it fol-

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lows that ldðG00SE Þð¼ 2Þ > ldðG0SE Þð¼ 1Þ and UsedColors-ColorsðAdjðvÞÞ ¼ f1;2g f1;2g ¼ ;, thus line 18 through 20 are considered. It is given that h ¼ minfC ColorsðAdjðvÞÞg ¼ minff1;2;3;...;pg f1;2gg ¼ f3g (assume p P 3). Color ‘‘3’’ is chosen for v hence, three (¼ ldðG00SE Þ þ 1 ¼ 2 þ 1) colors are used up so far, without violating the properly coloring of G00SE . If x 62 AdjðvÞ, then it means that ldðG00SE Þð¼ 2Þ > ldðG0SE Þð¼ 1Þ but UsedColorsColorsðAdjðvÞÞ ¼ f1;2g f1g ¼ f2g 6¼ ; (line 13 through 14). Hence, v is colored with color ‘‘2’’ (line 15 through 16). The number of used colors is 2 ð< ldðG00SE Þ þ 1Þ in this case. Therefore, it follows that ldðG00SE Þ62 and three colors suffice for all cases. (2) Case 2 (m ¼ k (>2)): By the induction hypothesis, let G0SE be properly colored with ldðG0SE Þ þ 1 colors. Letting ldðG0SE Þ ¼ t 0 ( P 1), it follows that jColorsðVSE Þj ¼ t þ 1. 0 0 For any y 2 VSE of GSE , the assumption implies that y and fbrotherðj; y)} must be properly colored with ldðyÞ þ 1 colors, at most. Since ldðyÞ 6 ldðG0SE Þ, y is adjacent to the vertices which are all properly colored with t different colors, at most (please note this means that ldðyÞ 6 t, not pdðyÞ 6 t). Denote by f1; 2; 3; . . . ; t þ 1g the number of colors used for G0SE , that is, let UsedColors ¼ f1; 2; 3; . . . ; t þ 1g. We are now to prove the case for m ¼ k þ 1 as follows. (3) Case 3 (m ¼ k þ 1): Let v 2 brotherðj; wÞ such that brotherðj; wÞ ¼ fv1 ; v2 ; . . . ; vj ; vg. Without loss of generality, we assume the set of colored vertices in brotherðj; wÞ is fv1 ; v2 ; . . . ; vj g (note jColorsðfv1 ; v2 ; . . . ; vj gÞj ¼ 1 if fv1 ; v2 ; . . . ; vj g 6¼ ;. Depending upon line 1, two cases are as follows, respectively. (3.1) Line 2 is executed: Noting v 2 AdjðwÞ and fv1 ; v2 ; . . . ; vj g ¼ 6 ;, we get ldðG00SE Þ ¼ 0 ldðGSE Þð¼ tÞ. If ColorsðAdjðvÞÞ \ Colors ðbrotherðj; wÞÞ ¼ ; then, line 3 through 4 and otherwise line 5 are considered, respectively. In the former case, the color used for fv1 ; v2 ; . . . ; vj g is also used for v, and G00SE is still properly colored with t þ 1 colors. In the latter case, the color for v is given as h ¼ UsedColors–ColorsðAdjðvÞÞ.

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If h ¼ 6 ; then (line 7), v must be adjacent to the vertices that are colored with t different colors, at most. Since ldðG00SE Þ ¼ ldðG0SE Þ (¼ t), there exists at least one color not used for AdjðvÞ (note that 0 jColorsðVSE Þj ¼ t þ 1). Hence, with such h, v is colored and v1 ; v2 ; . . . ; vj are recolored, respectively. G00SE remains properly colored and the number of used colors is still t þ 1 (¼ ldðG00SE Þ þ 1). Otherwise (i.e., h ¼ ;) (line 9), it is seen that ldðG00SE Þ > ldðG0SE Þ. This means ldðG00SE ÞP t þ 1 because G0SE is a properly colored graph with t þ 1 colors (by the induction hypothesis) and v is adjacent to vertices which are colored with t þ 1 colors. Thus, a new color not used so far is required for v to be properly colored. By line 10 through 12, a new color is given as h ¼ C UsedColors ¼ f1; 2; . . . ; pg f1; 2; . . . ; t þ 1g ¼ ft þ 2g (assume p P t þ 2) is chosen. And it is used for both v and its brothers fv1 ; v2 ; . . . ; vj g, respectively. G00SE is still properly colored. The maximal number of colors used is t þ 2 ( 6 ldðG00SE Þ þ 1). (3.2) Line 13 is executed: This case implies either brotherðj;wÞ ¼ fvg or Colorsðfv1 ; v2 ;...;vj gÞ ¼ ;. Hence, the same procedures applied to 3.1) above are followed such that brotherðj;wÞ in line 6 through 12 is replaced with the singleton v. And the re-coloring for fv1 ;v2 ;...;vj g in brotherðj;wÞ is void. It is seen that G00SE is properly colored with ldðG00SE Þþ1 colors, at most. (4) So far we have shown that G00SE is always properly colored with ldðG00SE Þ þ 1 colors, at most, for all cases. Repeating such procedure (line 22) and letting G00SE ¼ GSE , GSE remains properly colored with ldðGSE Þ þ 1 colors. Let GSE have r (> 1) connected components G1SE , G2SE ; . . . ; GrSE . Evidently, there exists some GkSE such that ldðGkSE Þ ¼ ldðGSE Þ, k 2 f1; 2; . . . ; rg. Each GiSE , 1 6 i 6 r, can be colored independently and properly with ldðGkSE Þ þ 1 colors, at most. This completes the proof of the theorem. 

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Yeonghwan Tscha was born in Pochon, Korea, in 1960. He received the B.Sc. degree from Inha University in 1983, the M.Sc. degree from Korea Advanced Institute of Science and Technology (KAIST) in 1985, and the Ph.D. degree from Inha University in 1993, respectively, all in Computer Science.In 1985, he joined the Electronics and Telecommunications Research Institute (ETRI) and was engaged in R & D in the area of ISDN, CCS No.7, and Protocol Engineering. During 1986–1987, he was a Guest Scientist at NIST (formerly, NBS), USA, and was participated in the project on automated protocol methodology. He left ETRI in 1990 and completed his Ph.D. degree in 1993. Since 1994 he has been an Associate Professor with the School of Computer, Information, and Communication Engineering, Sangji University, Wonju, Korea.His research includes communication protocols, network architectures, applied graph theory, and algorithms, with current interests in layerless switching/ routing and protocols for mobile ad hoc networks and pervasive computing.Dr. Tscha enjoys lure-fishing (especially, Siniperca Scherzeri) and riverside trekking/backpacking.