A Saturaton Bnary Neural Network for Crossbar Swtchng Problem Cu Zhang 1, L-Qng Zhao 2, and Rong-Long Wang 2 1 Department of Autocontrol, Laonng Insttute of Scence and Technology, Benx, Chna bxlkyzhangcu@163.com 2 Graduate School of Engneerng, Unversty of Fuku, Bunkyo 3-9-1, Fuku-sh, Japan nkzlq@hotmal.com, wang@u-fuku.ac.jp Abstract. A saturaton bnary neural network s proposed to solve the crossbar swtchng problem. In the proposed algorthm, neurons are updated accordng to dfferent formula, then neurons enter nto saturatng state, and as a result, t makes the neural network escape from a local mnmum stagnaton. The proposed algorthm has been tested on a large number of nstances and compared wth other algorthms. The expermental results show that the proposed algorthm s superor to ts compettors. Keywords: Saturaton bnary neuron model, Combnatoral optmzaton problems, Crossbar swtchng problem. 1 Introducton Bnary Hopfeld networks (Hopfeld networks wth two-state threshold neurons) (HNNs) have been appled to large amount of dffcult combnatoral optmzaton problems [1]-[3] because of ts advantages. The advantages nclude massve parallelsm, convenent hardware mplementaton of the neural network archtecture, and a common approach for solvng varous optmzaton problems [4]. One of the combnatoral optmzaton problems s the real-tme control of a crossbar swtch (crossbar swtch problem (CSP)). The Hopfeld neural network archtecture has been appled to CSP used for swtchng hgh-speed packets at maxmum throughput [5]- [7]. However the work by Wlson and Pawley showed that the Hopfeld neural networks often faled to converge to vald solutons. When t converged, the obtaned soluton was often far from the optmal soluton. Snce ther report varous modfcatons have been proposed to mprove the convergence of the Hopfeld neural networks. In ths paper we propose a saturaton bnary neuron model and use t to construct a Hopfeld-type neural network for effcently solvng the crossbar swtchng problem. In the proposed saturaton bnary neuron model, once the neuron s n exctatory state, then ts nput potental s n postve saturaton where the nput potental can only be reduced but cannot be ncreased, and once the neuron s n nhbtory state, then ts H. Deng et al. (Eds.): AICI 2011, Part III, LNAI 7004, pp. 254 261, 2011. Sprnger-Verlag Berln Hedelberg 2011
A Saturaton Bnary Neural Network for Crossbar Swtchng Problem 255 nput potental s n negatve saturaton where the nput potental can only be ncreased but cannot be reduced. Usng the saturaton bnary neuron model, a saturaton bnary neural network s constructed to solve the crossbar swtchng problem. The smulaton results show that the saturaton bnary neural network can fnd better solutons than the orgnal Hopfeld neural network. 2 Saturaton Bnary Neural Network In the proposed algorthm, a novel neuron updatng rule s proposed. To avod the neural network enterng a local mnmum stagnaton, the neuron network s updated accordng to the state of the neuron. Accordng to dfferent states, the updatng rule s dfferent. In ths secton, the proposed saturaton bnary neural network s ntroduced. In bnary neuron model, the updatng method of nput potental U s especally mportant. In conventonal neuron model [8] [9], the nput potental U s updated from the Eq. 1 or Eq.2 The updatng process of the neuron s no matter wth the state of the neuron. Where method: t) / U ( t 1) + (1) U ( t 1) U + + (2) ( derves from the energy functon E based on the gradent descent E( V1, V2 V,..., V ) The Hopfeld-type bnary neural network s usually constructed usng the above neuron model. In order to mprove the global convergence qualty and shorten the convergence tme, we propose a new neuron model called saturaton bnary neuron model (SBNM) whch conssts of the followng mportant deas. (1). Once the neuron s n exctatory state (the output V 1), then the nput potental s assumed to be n postve saturaton. In the postve saturaton, the nput potental U can only be reduced but cannot be ncreased. For the case of V 1: f / < 0 U ( t 1) U + n (3) + (4)
256 C. Zhang, L.-Q. Zhao, and R.-L. Wang else U ( t 1) U + (5) (2). Once the neuron s n nhbtory state (the output V 0), then the nput potental s assumed to be n negatve saturaton. Then the nput potental can only be ncreased but cannot be reduced. For the case of V 0 f / > 0 else U ( t 1) U + + (6) U ( t 1) U + (7) Note that the nput/output functon n the McCulloch-Ptts neuron [8] or hysteress McCulloch-Ptts neuron [9] can be used to update the output V. Usng the above neuron model, we construct a Hopfeld-type neural network called saturaton bnary neuron network (SBNN). Comparng to the McCulloch-Ptts neuron [8] and hysteress McCulloch-Ptts neuron [9], the neuron updatng rule s mproved n the proposed SBNN. The followng procedure descrbes the synchronous parallel algorthm usng the SBNN to solve combnatoral optmzaton problems. Note that N s the number of neuron, targ_cost s the target total cost set by a user as an expected total cost and t_lmt s maxmum number of teraton step allowed by user. 1. Set t0 and set targ_cost, t_lmt, and other constants. 2. The ntal value of U for 1,,N s randomzed. 3. Evaluate the current output V (t) for 1,,N. 4. Check the network, f targ_cost s reached, then termnate ths procedure. 5. Increment t by 1. f t> t_lmt, then termnate ths procedure. 6. For 1,,N a. Compute Eq. 3 to obtan b. Update U (t+1), usng the proposed saturaton bnary neuron model( Eq. 4-Eq. 7). 7. Go to the step 3. 3 Crossbar Swtchng Problem In the communcaton systems, crossbar packet swtches route from the nput to output where a message packet s transmtted from the source to the destnaton. The randomly ncomng packets must be controlled to elmnate conflct at the crossbar swtch where the conflct s that two or more packets may smultaneously access to a sngle output [10]. The goal of crossbar swtchng problem s to maxmze the throughput of packets through a crossbar swtch. In packet-swtched
A Saturaton Bnary Neural Network for Crossbar Swtchng Problem 257 telecommuncaton networks, swtches are located at nodes, routng randomly arrvng packets so that they may be transmtted from the source to the destnaton, the basc swtch s the n n crossbar swtch. It conssts of a grd of n nput lnes by n output lnes wth a swtch at each of the cross. Thus, the crossbar swtch can route a packet arrvng at any nput lne to any output lne. Multple packets, however, can arrve smultaneously at dfferent nput lnes destned for the same output lne and cannot be routed to the same output lne at the same tme wthout collson. In such a case, one packet s sent, and others must be blocked and queued at each nput lne for the next transmsson tme perod. These are based on the asynchronous transfer mode (ATM) protocol. Thus, at most one packet can be transmtted from each nput lne and to each output lne. Ths s a physcal constrant of crossbar swtches [11]. Queue Managers Neural Network b c [0110] Input Lnes d [0001] Crossbar Cross-pont Control [0000] d a [1001] Input Request matrx R 0110 0001 0000 1001 a b c d Optmal Confguraton Matrces C1 0100 0000 0001 C2 0010 0001 0000 1000 1000 Fg. 1. Schematc archtecture of crossbar control wth an example of nput request matrx and ts optmal confguraton matrces
258 C. Zhang, L.-Q. Zhao, and R.-L. Wang To show the request for packet transmsson, n n crossbar swtches can be represented by an n n bnary request matrx R(r j ). Rows and columns of the matrx R are assocated wth nputs and outputs, respectvely, of the crossbar swtch. A matrx element r j 1 ndcates that there s a request for swtchng at least one packet from nput lne to output lne j of the swtch; otherwse r j 0. If we consder the crossbar swtch for pont-to-pont connectons, then at most one cross-pont may be closed on any row or column of the swtch durng packed transmsson. The state of the swtch can be represented by an n n bnary confguraton matrx C(c j ), where c j 1 ndcates that nput lne s connected to output lne j by the closed cross-pont (j). c j 0 ndcates that cross-pont (j) s open. For proper operaton of the swtch, there should be at most one closed cross-pont n each row and each column. The throughput of the swtch s optmal when the matrx C, whch s a subset of the matrx R (.e., c j r j for every (,j)), contans at most a 1 n each row/column. In other words, we call the throughput maxmum f C has a maxmum overlap wth R, and r c j j j s the maxmum [12]. Thus, crossbar packet swtchng s bascally a combnatoral optmzaton problem, whch fnds the confguraton matrx C havng a maxmum overlap wth R. Example of optmal matrces s shown n Fg. 1 for a 4 4 crossbar. Thus the crossbar swtchng problem can be mathematcally transformed nto the followng optmzaton problem: Constrant conon: Maxmze: n n c j j r 1 1 j n n 2 n n 2 ( 1) + ( 1) 0 1 c j 1 j j 1 c (9) 1 j Equaton (8) s used to maxmze the throughput of packets through a crossbar swtch, and equaton (9) demonstrated the physcal constrants of the crossbar swtch. (8) 4 Solvng CSP Usng SBNN The crossbar swtchng problem has been ntroduced above. The crossbar swtch s controlled by a neural network that has one neuron n correspondence to each swtch cross-pont. Row request vectors from all the nlets are suppled to the neural network, whch used them to compute an optmal confguraton matrx for the swtch. The resultng row confguraton vectors are then returned to the correspondng queue manager, whle the crossbar swtch cross-ponts selected by the computed confguraton matrx are closed. Each queue manager presents to ts nput lne a sngle packet destned to the output lne selected by the row vector returned by the neural network, whch thus gets routed through the closed cross-pont to ts proper output lne. The queue manager also updates t row request vector by cleanng the selected column bt, provded that no packets reman queued for that output.
A Saturaton Bnary Neural Network for Crossbar Swtchng Problem 259 The crossbar swtch problem can be solved by constructng an approprate energy functon and mnmzng the energy functon to zero (E0) usng an n n twodmensonal Hopfeld neural network. The objectve energy functon of the crossbar swtchng s gven by: n n n n A B ckrk 1 + c r 2 1 k 1 2 j 1 k 1 2 kj kj 2 1 E (10) Where A and B are coeffcents. We can get the total nput (μ j ) of neuron by usng the partal dervaton term of the energy functon. Then the weghts and threshold of the neural network are derved as follows: w j, kl Ar kl r kj δ k Br kl r δ l lj δ δ k lj 1, ( k ) 0, ( k ) 1, ( l j ) 0, ( l j ) (11) 5 Smulaton Result Snce our archtecture s vald for any Hopfeld neural networks as llustrated n the prevous secton, we used the archtecture for some randomly generated problems and a large number of real crossbar swtch problems. We now present smulaton results of the archtecture when appled to real crossbar swtchng. The parameters A and B were set to A1.0, B2.0. The weghts and external nput currents were all the same as the orgnal Hopfeld neural network. In smulatons, 100 smulaton runs wth dfferent randomly generated ntal states were performed on each of these nstances. To evaluate our results, we compared the results of the orgnal Hopfeld neural network [9] wth our results. Informaton of the crossbar swtch as well as all the results s shown n Table I. In Table I, the column labeled optmal s the global convergence tmes among 100 smulatons and the column labeled steps s the average number of teraton steps requred for the convergence n the 100 smulatons. The smulaton results show that the proposed saturaton bnary neuron network wth could almost fnd optmum soluton to most crossbar swtch problems wthn short computaton tmes whle the orgnal Hopfeld neural network could hardly fnd any optmum soluton to the crossbar swtchng, especally for large sze problems.
260 C. Zhang, L.-Q. Zhao, and R.-L. Wang Table 1. Smulaton result Crossbar Swtches Hopfeld Network Proposed Algorthm optmal steps optmal steps 4x4 100 3 100 2 6x6 100 4 100 3 8x8 100 5 100 5 10x10 100 14 100 8 20x20 100 41 100 24 30x30 100 91 100 36 50x50 100 120 100 79 80x80 86 440 100 184 100x100 43 479 88 296 200x200 9 745 47 437 300x300 -- -- 14 653 6 Concluson We have proposed a saturaton bnary neuron model and use t to construct a Hopfeld-type neural network called saturaton bnary neural network (SBNN). In the proposed algorthm, neurons are updated accordng to dfferent formula, then neurons enter nto saturatng state, and as a result, t makes the neural network escape from a local mnmum stagnaton. The SBNN s used to solve the crossbar swtchng problem. The smulaton results show that SBNN s capable of fndng better solutons than other method. Also, t can be seen that the SBNN s problem ndependent and can be used to solve other combnatoral optmzaton problems. References 1. Hopfeld, J.J.: Neurons and physcal systems wth emergent collectve computatonal abltes. Proc. Natl. Acad. Sc. USA 79, 2554 2558 (1982) 2. Smth, K., Palanswam, M., Krshnamoorthy, M.: Neural technques for combnatoral optmzaton wth applcatons. IEEE Trans. Neural Networks 9(6), 1301 1318 (1998)
A Saturaton Bnary Neural Network for Crossbar Swtchng Problem 261 3. Hopfeld, J.J., Tank, D.W.: Neural computaton of decsons n optmzaton problems. Bol. Cybern. 52, 141 152 (1985) 4. Zeng, X.C., Martnez, T.: A newrelaxaton procedure n the Hopfeld neural networks for solvng optmzaton problems. Neuron Processng Lett. 10, 211 222 (1999) 5. Marrakch, A., Troudet, T.: A neural net arbtrator for large crossbar packet swtches. IEEE Trans. Crcuts Syst. 36(1), 1039 1041 (1989) 6. Troudet, T.P., Walters, S.M.: Neural-network archtecture for crossbar swtch control. IEEE Trans. Crcuts Syst. 38(1), 42 56 (1991) 7. Xa, G., Tang, Z., L, Y., Wang, J.: A bnary Hopfeld neural network wth hysteress for large crossbar packet-swtches. Neurocomputng 67, 417 425 (2005) 8. McCulloch, W.S., Ptts, W.H.: A logcal calculus of deas mmanent n nervous actvty. Bull. Math. Bophys 5, 115 133 (1943) 9. Takefuj, Y., Lee, K.C.: An artfcal hysteress bnary neuron: A model suppressng the oscllatory behavors of neural dynamcs. Bol. Cybern. 64, 353 356 (1991) 10. Matsuda, S.: Theoretcal Lmtatons of a Hopfeld Network for Crossbar Swtchng. IEEE Transactons on Neural Networks 12(3) (May 2001) 11. Ntnaware, V.N., Lmaye, S.S.: Folded archtecture of scheduler for area optmzaton n an on-chp swtch fabrc. Internatonal Journal of Hybrd Informaton Technology 4(1) (January 2011) 12. L, Y., Tang, Z., Xa, G., Wang, R.: A Postvely Self-Feedbacked Hopfeld Neural Network Archtecture for Crossbar Swtchng. IEEE Transactons on Crcuts and Systems-I: Regular Papers 52(1) (January 2005)