Equivalent Permutation Capabilities Between Time-Division Optical Omega Networks and Non-Optical Extra-Stage Omega Networks

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1 518 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 9, NO. 4, AUGUST 2001 Equivalent Permutation Capabilities Between Time-Division Optical Omega Networks and Non-Optical Extra-Stage Omega Networks Xiaojun Shen, Associate Member, IEEE, Fan Yang, and Yi Pan, Senior Member, IEEE Abstract Because signals carried by two waveguides entering a common switch element would generate crosstalk, a regular multistage interconnection network (MIN) cannot be directly used as an optical switch between inputs and outputs in an optical network. A simple solution is to use a 2 2 cube-type MIN to provide the connections, which needs a much larger hardware cost. A recent research proposed another solution, called the time-domain approach, that divides the optical inputs into several groups such that crosstalk-free connections can be provided by an regular MIN in several time slots, one for each group. Researchers studied this approach on Omega networks and defined the class set to be the set of -permutations realizable in two time slots on an Omega network. They proved that the size of is larger than the size of class, where consists of all -permutations admissible to a regular (nonoptical) Omega network. This paper first presents an optimal ( log ) time algorithm for identifying whether a given permutation belongs to class or not. Using this algorithm, this paper then proves an interesting result that the class is identical to the class +1which represents the set of -permutations admissible to a nonoptical one-extra stage Omega network. Index Terms Conflict graph, crosstalk-free connection, dilated MIN, Omega network, optical switch, time-domain approach. I. INTRODUCTION DESIGNING a cost-efficient switching device is a key task to support switching function for all-optical networks in the future. Multistage interconnection networks (MINs) have been used in telecommunication networks for many years and also proposed to be used as optical switching topology by many researchers [1] [8]. Among these MINs, researchers have paid most attention to the well known multistage cube-type networks (MCTNs), including the indirect cube network, Omega network, baseline network, etc. [9], [10]. An ( ) MCTN consists of columns (called stages) of 2 2 switching elements (SEs). The stages are numbered 0 to, from Manuscript received October 14, 1999; approved by IEEE/ACM TRANSACTIONS ON NETWORKING Editor J. Chao. This work was supported in part by the National Science Foundation under Grant C-CR This paper was presented in part at the 18th IEEE International Performance, Computing, and Communications Conference (IPCCC), Phoenix/Scottsdale, AZ, February X. Shen is with the Computer Science Telecommunication Program, University of Missouri-Kansas City, Kansas City, MO USA ( xshen@cstp.umkc.edu). F. Yang is with the IBM Corporation 414/6-4, Poughkeepsie, NY USA ( yangfan@us.ibm.com). Y. Pan is with the Department of Computer Science, Georgia State University, Atlanta, GA USA ( pan@cs.gsu.edu). Publisher Item Identifier S (01) Fig. 1. An example of Omega network and an admissible 8-permutation. left to right. Each stage consists of SEs, and the output ports from one stage are connected to the input ports of the next stage in certain pattern. For example, the inter-stage connection used by an Omega network is known as shuffle-exchange. Suppose we use -bit binary numbers from 0to to label the addresses of input (or output) ports, from top to bottom. The shuffle exchange connects output port from stage to the input port of stage,. The source inputs are also shuffled before entering the stage 0. Fig. 1 shows an 8 8 Omega network. For convenience, the SEs in each stage are also labeled with an -bit binary number, top to bottom, from 0 to, asshown in Fig. 1. Used asanonoptical switch, an Omega network can support multiple connections between inputs and outputs. An -permutation is called admissible if conflict-free paths can be established, one for each (input/output) pair defined by the permutation. Fig. 1 also illustrates that the following permutation is admissible to the Omega network It is known that there are permutations admissible to the Omega network [11]. We use to represent the set /01$ IEEE

2 SHEN et al.: TIME-DIVISION OPTICAL OMEGA NETWORKS AND NON-OPTICAL EXTRA-STAGE OMEGA NETWORKS 519 Fig. 2. A dilated Omega network and four crosstalk-free paths. Fig. 3. An extra-stage Omega network provides two disjoint paths between 100 and 111. of all admissible permutations to the Omega network. Since all MCTNs are proven topologically equivalent [9], [10], we select the Omega network for the purpose of study. Using MINs for optical networks is a natural and feasible extension of their applications. However, one significant problem an optical switch introduces is the crosstalk which is caused by undesired coupling between signals carried by two waveguides [1] when they meet at a common SE. Crosstalk reduces the signal-to-noise ratio and limits the network size. One way to deal with the problem is to use a regular MIN to provide the connection, with only half of the input and output ports being used. This MIN is called a dilated MIN, in which the crosstalk can be eliminated by ensuring that only one input of every switch is active at any time. This method is called network dilation or space-domain approach [1] [5]. Fig. 2 shows a dilated 4 4 Omega network, in which only one of the two input (or output) ports of each switch at the first and last stages is allowed to use. In a dilated network, a connection between an input and output is established by choosing an appropriate path in the network so that no switch in the network will have both input ports active at the same time. Fig. 2 shows four crosstalk-free (CF) paths for four (input/output) pairs. Obviously, an dilated Omega has the same hardware cost as that of a regular Omega. It is shown in [2] that an dilated Omega network used as an optical switch has the same permutation capability as that of an regular Omega network used as a nonoptical switch, but the hardware cost is more than doubled. Qiao et al. proposed a time-domain approach [6] that extends the method used in the reconfiguration with time-division multiplexing (RTDM) to avoid crosstalk. Specifically, RTDM partitions the set of required connections into several subsets, each of which is a CF mapping for an undilated network, meaning that at most one path goes through a switch element in each subset. By establishing the connections for each of these CF mappings in a separate time slot, crosstalk is avoided without the need for a dilated network. Note that a CF mapping admissible to an undilated network can contain, at most, connections. Qiao et al. [1] used to denote the set of -permutations realizable with two CF mappings on an Omega network, and proved that the size of is larger than the size of class.an interesting question left unanswered is how to characterize the set. In other words, how do we know if a given permutation belongs to? Moreover, compared with, how much larger is the set? This paper first presents an optimal time algorithm for the membership problem. Using this algorithm, this paper then proves an interesting result that, where denotes the set of admissible -permutations to the extrastage Omega network. An extra-stage Omega network is simply obtained by adding one more stage of shuffle in front of an Omega network [12] [15]. An extra-stage Omega network provides two paths between an input and an output and allows much more permutations to be admitted. Gazit and Malek [11] presented an algorithm to compute the size of. Their formula shows that is much larger than. Therefore, the results of this paper implies that is also much larger than. Fig. 3 shows an example of one extra-stage Omega network. This paper is organized as follows. Section II introduces previous results and related terms. Section III presents the membership algorithm; Section IV proves ; and Section V concludes this paper. II. PRELIMINARIES ction, we introduce the window method and other related notions that will be used in the following sections. For a blocking network, determining the admissibility of a given permutation is a fundamental problem. The window method [16], [17] has been used to deal with this problem for Omega and extra-stage Omega networks. Later, this method will be modified in this paper and used for the class. Given an -permutation, a transition sequence [16], [17] for a pair is the -bit binary representation of followed by the -bit binary representation of. Given a permutation, if we list all transition sequences, one for each pair, we obtain an matrix called transition matrix. We define a window to be a collection of consecutive columns in. Specifically, window. It is shown [16] that a permutation is admissible if and only if the rows in any window (i.e., the -bit numbers) are distinct. As an example, Fig. 4 shows all windows of a given 8-permutation which are admissible to the 8 8 Omega network of Fig. 1, where the 8 disjoint paths are shown. Note that the definition of windows here is slightly different from the definition used in [16], [17]. A key point behind this window method is that the -bit number of row in window, denoted by, is exactly the

3 520 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 9, NO. 4, AUGUST 2001 Fig. 4. An illustration of windows. Fig. 5. An illustration of optical windows. output port address at stage passed by path. Therefore, if all numbers are distinct in, the paths will go through different exit ports at stage, without conflicting. Moreover, the first bits of row in is the label of the SE at stage which provides connection for path. This observation can be easily verified by tracing the path from source to destination. In this sefirst, the source address is shuffled to the input port at stage 0. This input port is at the switch with label. Its two output port addresses are and. Depending on or 1, we can set the switch such that the path goes through the output port. Then, this address is shuffled to the input port at stage 1. Similarly, we can set the corresponding switch such that this path goes through the port at stage 1. Repeating this argument, we conclude that this path goes through the exit port at stage. Finally, this path reaches the output port at stage. Note that this path is unique. At stage, we must set the switch labeled such that the path takes output address. Thus, the above observation is true. III. AN MEMBERSHIP ALGORITHM FOR CLASS Since class consists of permutations that can be realized in two time slots, the membership problem here is equivalent to the following problem: is it possible to divide the pairs of a given permutation into two CF groups such that they can go through the network in two different time slots? For this problem, the window method cannot directly solve this problem, because two conflict-free paths in a regular Omega network become conflicting paths if they go through a common SE in an optical network. For example, path (000, 101) and (100, 011) in Fig. 1 are conflicting optical paths. For optical MIN, only one path is allowed to cross an SE to avoid crosstalk. Therefore, we need to modify the definition of windows. The key observation to the solution is that the path Fig. 6. The conflict graph of the permutation in Fig. 5 is two-colorable. for pair is unique and the label of the SE used by the path in stage is exactly equal to the first bits of row in. Therefore, given a transition matrix, we define an optical window to be a collection of consecutive columns. Specifically, the optical window. Fig. 5 shows an example of the optical windows. It is evident that the -bit number of row in, denoted by, is the label of the SE used by path at stage. Therefore, a group of pairs is a CF mapping if and only if their corresponding rows in are distinct in every optical window. In order to solve the membership problem, we introduce the notion of conflict graph. Definition: The conflict graph of an -permutation is the graph, where, and there is an optical window such that. It is clear that vertices and in the conflict graph are adjacent if and only if they share a common SE at some stage. Fig. 6 shows an example of the conflict graph. Theorem 1: A permutation belongs to class if, and only if, its conflict graph is two-colorable. Proof: A graph is two-colorable if each vertex of can be assigned a color, black or white, such that the vertices of the same color are not adjacent. If, then we can partition the pairs into two CF mappings, and. We color a vertex in black if, and white otherwise. Let, be

4 SHEN et al.: TIME-DIVISION OPTICAL OMEGA NETWORKS AND NON-OPTICAL EXTRA-STAGE OMEGA NETWORKS 521 Fig. 7. Routing paths for the two CF mappings. (a) Group A. (b) Group B. any two black vertices. Since path and path in do not share a common SE, their rows must be distinct in every optical window. From the definition of a conflict graph, we know that there is no edge between and. Similarly, there is no edge between two white vertices. Therefore, the conflict graph is two-colorable. On the other hand, if the graph is two-colorable, we can put in group A if is colored black and in group B otherwise. Now, any two paths, and, in group A do not conflict (share a common SE) because there is no edge between and in graph, and hence, pairs in form a CF mapping. Similarly, pairs in form another CF mapping. Therefore, belongs to. Fig. 6 shows that the conflict graph of the permutation of Fig. 5 is two-colorable and therefore the permutation belongs to. The two CF mappings (groups) are Group A Group B Fig. 7 illustrates how the two CF groups are routed through the Omega network in two time slots. The following admissibility algorithm shows how to construct the conflict graph in time. After the conflict graph is constructed, a simple linear algorithm checks the two-colorability. Algorithm Two-CF-Mappings ( : permutation); Step 1. Create vertices,,,,,. Step 2. Construct edge set as follows: for to do //check window one by one// (i) initialize array with, such that the -bit number of row of,. // can be obtained from by deleting the first column and adding a new column from matrix.// (ii) initialize array (empty) ; (empty) ; (iii) for to do ; // is a -bit number// if then //The number occurs first time in row.// else if then // occurs second time in row.// else return ( is not in ) //If occurs more than two times, the conflict //graph will not be two-colorable.// (iv) for to do if and then ; ; add edge in graph. Step 3. Use depth-first search to color with black and white. If successful, return ( is in ) and output: Group A is black Group B is white else return ( is not in ); End. The time complexity can be analyzed as follows: Step 1) needs time; Step 2) needs time because there are at most inner loops for each statement in Step 2, and there are at most outer loops; Step 3) needs time because there are at most edges. Note that any path can conflict with at most one of other paths in each stage. If it conflicts with more than one path, the algorithm will exit at Step 2 (iii), and the graph is not two-colorable. Therefore, the time complexity of the algorithm is. It is optimal because there are switches to be set individually. A deeper analysis on the lower bound of this problem can be found in [17]. IV. A PROOF OF Before we show the proof of, we need to introduce a method to identify any permutation that belongs to (see [17] for detail). Actually, the method used to identify an permutation follows the method used for permutation. As we mentioned before, an extra-stage Omega network provides two different paths between any source and its destination. Given a source and a destination, the two paths connecting and can

5 522 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 9, NO. 4, AUGUST 2001 Fig. 8. Windows of the permutation of Fig. 4 used for an extra-stage 828 Omega network. -bit se- be conveniently represented by the following quence, which is also called a transition sequence: The value of can be 0 or 1, which specifies one of the two paths. For example, in Fig. 8, the sequence and represent two paths from source 000 to destination 101. When the -bit is fixed, the path is chosen. Similar to Omega network, the -bit number also is the exit port address at stage taken by this path. Given an -permutation, if we list all transition sequences, two for each pair, then we obtain a transition matrix, in which row is the transition sequence of with, and row is the transition sequence of with. We also define a window of as consecutive columns. The middle column is called column because it contains the value of for each path. Window is defined as. As an example, Fig. 8 shows all windows for the 8-permutation of Fig. 4 for an extra-stage Omega network. In [17], a conflict graph is used to identify permutations admissible to, where contains vertices, one for each row in and if and only if in some window. Theorem 2 [17]: A permutation is admissible to network if and only if its conflict graph is twocolorable (see [17] for detail). Now, let us prove. Theorem 3: The set and the set are identical. Proof: Given a permutation, let and be the transition matrix and conflict graph used for determining or not. Let and be the transition matrix and conflict graph for determining or not. From Theorem 1 and 2, we need only to show that is two colorable if and only if is two colorable. First, let us study the relationship between and. Comparing Figs. 4 and 8, we immediately see that can be expanded to by doubling each row and inserting column. On the other hand, can be obtained from by deleting column and combining every two adjacent rows into a single row. Second, let us investigate the structure of. Because row and are identical except -bit values, we use to represent row and to represent row. Then,, where,. Because row and row are distinct in any window except,wehave where row and row are identical in some window ; row and row are identical in some window ;. Obviously, if, and only if,. Therefore, graphs, and, are two subgraphs of and are isomorphic to each other. The graph is obtained from and by connecting each vertex and its corresponding vertex. Fig. 9 shows the structure of for the permutation of Fig. 4. Obviously, if is two-colorable, then can be two-colored by coloring in with the opposite color from in. Therefore, is two colorable if, and only if, or is two-colorable.

6 SHEN et al.: TIME-DIVISION OPTICAL OMEGA NETWORKS AND NON-OPTICAL EXTRA-STAGE OMEGA NETWORKS 523 Fig. 9. The graph of G is obtained from G and G. Third, we show that the conflict graph used for class is isomorphic to. Let, row and row in are identical in some optical window. From the expansion of to, we know window of can be obtained from of by duplicating every row and inserting column.if,, then in some window of,row and row are identical. Since both of them have,row and row will also be identical if their -bits are deleted. Therefore,. On the other hand, if,row and row are identical in some optical window. They will remain identical if both are inserted with an -bit, which implies that. Therefore, is isomorphic to in. Thus is two-colorable if, and only if, is two-colorable. V. CONCLUSION In this paper, we have presented an membership algorithm for the class of permutations, where is the set of permutations which are realizable with two CF-mappings by an Omega network. Based on this algorithm, we have proven that the class of is identical to the class of, which is the set of permutations admissible to an extra-stage Omega network. Previously, Qiao et al. [6] proved that the size of is larger than the size of class, but. The permutation in Fig. 4 is a good example which belongs to, but not (or ). An interesting problem for future work is to find the relationship between the time-division optical -extra-stage Omega network and the nonoptical extra-stage Omega network, where. We can similarly define classes, and,. We conjecture that. REFERENCES [1] C. Qiao, R. G. Melhem, D. M. Chiarulli, and S. P. Levitan, A time domain approach for avoiding crosstalk in optical blocking multistage interconnection networks, IEEE J. Lightwave Technol., vol. 12, pp , Oct [2] K. Padmanabhan and A. Netravali, Dilated network for photonic switching, IEEE Trans. Commun., vol. COM-35, pp , Dec [3] T. O. Murphy, C. T. Kemmerer, and D. T. Moser, A Ti : LiNbO dilated Benes photonic switch module, in Proc. Topical Meeting on Photonic Switching, Salt Lake City, UT, Mar. 1991, Postdeadline Paper PD3. [4] R. A. Thompson, The dilated slipped Banyan switching network architecture for use in an all-optical local area network, J. Lightwave Technol., vol. 9, pp , Dec [5] J. E. Watson et al., A low-voltage 82 8 Ti : LiNbO switch with a dilated Benes architecture, J. Lightwave Technol., vol. 8, pp , May [6] C. Qiao, A high speed interconnection paradigm for multiprocessors and its applications to optical interconnection networks, Ph.D. dissertation, Dept. Comput. Sci., Univ. Pittsburgh, Pittsburgh, PA, [7] Y. Pan, C. Qiao, and Y. Yang, Optical multistage interconnection networks: New challenges and approaches, IEEE Commun. Mag., vol. 37, pp , Feb [8] Y. Yang, J. Wang, and Y. Pan, Permutation capability of optical multistage interconnection networks, J. Parallel Distrib. Comput., vol. 60, no. 1, pp , Jan [9] C. Wu and T. Feng, On a class of multistage interconnection networks, IEEE Trans. Comput., vol. C-29, pp , Aug [10] D. K. Pradhan and K. L. Kodandapani, A uniform representation of single and multistage interconnection networks used in SIMD machines, IEEE Trans. Comput., vol. 29, pp , Sept [11] I. Gazit and M. Malek, On the number of permutations performable by extra-stage multistage interconnection networks, IEEE Trans. Comput., vol. 38, pp , [12] C. T. Lea and D. J. Shyy, Tradeoff of horizontal decomposition versus vertical stacking in rearrangeable nonblocking networks, IEEE Trans. Commun., vol. 39, pp , June [13] D. J. Shyy and C. T. Lea, Lg (N; m; p) strictly nonblocking networks, IEEE Trans. Commun., vol. 39, pp , Oct [14] G. B. Adams III and H. J. Siegel, The extra stage cube: A fault-tolerant interconnection network for supersystems, IEEE Trans. Comput., vol. C-31, pp , May [15] C. L. Wu, T. Y. Feng, and M. C. Lin, Star: A local network system for real-time management of imagery data, IEEE Trans. Comput., vol. C-31, pp , Oct [16] X. Shen, M. Xu, and X. Wang, An optimal algorithm for permutation admissibility to multistage interconnection networks, IEEE Trans. Comput., vol. 44, pp , Apr [17] X. Shen, An optimal O(NlgN) algorithm for permutation admissibility to extra-stage cube-type networks, IEEE Trans. Comput., vol. 44, pp , Sept

7 524 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 9, NO. 4, AUGUST 2001 Xiaojun Shen (A 92) received the B.S. degree from Tsinghua University, Beijing, China, in 1968, the M.S. degree from Nanjing University of Science and Technology, Nanjing, China, in 1982, and the Ph.D. degree in computer science from the University of Illinois at Urbana-Champaign in He has been with the faculty of University of Missouri Kansas City since 1989, where he is currently an Associate Professor in Department of Computer Networking. His research interests include discrete mathematics, algorithms, computer networking, and parallel processing, with focus on interconnection networks. Fan Yang received the M.S. degree from the University of Missouri Kansas City. He is a Software Engineer with IBM Corporation, Poughkeepsie, NY, where he works in the RS/6000 SP CSS Switch Software Development Team. His research focus has involved switch network management. Yi Pan (SM 96) received the B.Eng. degree in computer engineering from Tsinghua University, Beijing, China, in 1982, and the Ph.D. degree in computer science from the University of Pittsburgh, Pittsburgh, PA, in Currently, he is an Associate Professor in the Department of Computer Science, Georgia State University, Atlanta. Previously, he was with the Department of Computer Science, University of Dayton, Dayton, OH. His research interests include parallel algorithms and architectures, optical communication and computing, wireless networks, high-performance data mining, distributed computing, task scheduling, and networking. He has authored more than 110 research papers, including over 46 papers in international journals, including numerous IEEE publications. Dr. Pan is currently an Associate Editor of the IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, Area Editor-In-Chief of the Journal of Information, Editor of the Journal of Parallel and Distributed Computing Practices, Associate Editor of the International Journal of Parallel and Distributed Systems and Networks, and serves on the editorial board of The Journal of Supercomputing. He has also served as a Guest Editor of special issues for several journals, and as General Chair, Program Chair, Vice Program Chair, Publicity Chair, Session Chair, and as a member for Steering, Advisory, and Program Committees for numerous international conferences and workshops. He received the Outstanding Scholarship Award of the College of Arts and Sciences at the University of Dayton (1999), a Fellowship from the Japanese Society for the Promotion of Science (1998), an AFOSR Summer Faculty Fellowship (1997), NSF Research Opportunity Awards (1994 and 1996), and the Best Paper Award from PDPTA (1996). He is an IEEE Computer Society Distinguished Visitor and a member of the IEEE Computer Society. He is listed in Men of Achievement and Marquis Who s Who in the Midwest.