Neural Network Based Algorthm for Mult-Constraned Shortest Path Problem Jyang Dong 1,2, Junyng Zhang 2, and Zhong Chen 1 1 Department of Physcs, Fujan Engneerng Research Center for Sold-State Lghtng, Xamen Unversty, Xamen 361005, P.R. Chna 2 Natonal Key Laboratory for Radar Sgnal Processng, Xdan Unversty, X an 710071, P.R. Chna jydong@xmu.edu.cn Abstract. Mult-Constraned Shortest Path (MCSP) selecton s a fundamental problem n communcaton networks. Snce the MCSP problem s NP-hard, there have been many efforts to develop effcent approxmaton algorthms and heurstcs. In ths paper, a new algorthm s proposed based on vectoral Autowave-Competed Neural Network whch has the characterstcs of parallelsm and smplcty. A nonlnear cost functon s defned to measure the autowaves (.e., paths). The M-paths lmted scheme, whch allows no more than M autowaves can survve each tme n each neuron, s adopted to reduce the computatonal and space complexty. And the proportonal selecton scheme s also adopted so that the dscarded autowaves can revve wth certan probablty wth respect to ther cost functons. Those treatments ensure n theory that the proposed algorthm can fnd an approxmate optmal path subject to multple constrants wth arbtrary accuracy n polynomal-tme. Comparng experment results showed the effcency of the proposed algorthm. 1 Introducton Provdng Qualty-of-Servce (QoS) guarantees n packet networks gves rse to several challengng ssues. One of them s how to determne a feasble path that satsfes a set of constrants whle mantanng hgh utlzaton of network resources. The latter objectve mples the need to mpose an addtonal optmalty requrement on the feasblty problem. Ths can be done through a prmary cost functon accordng to whch the selected feasble path s optmzaton [1,2]. In general, multconstraned path selecton, wth or wthout optmzaton, s an NP-complete problem that cannot be exactly solved n polynomal tme [3]. Heurstcs and approxmaton algorthms wth polynomal- and pseudo-polynomal-tme complextes are often used to deal wth ths problem. One common heurstc approach s to fnd the k-shortest paths wth respect to a cost functon defned based on the lnk weghts and the gven constrants, hopng that one of these paths s feasble [4]. Increasng k mproves the performance of ths approach, but the computatonal cost becomes excessve that cannot be used for onlne network operaton. Another approach s to explot the dependences among the constrants, and to solve the path selecton problem assumng specfc schedulng schemes at network routers [5]. Specfcally n QoS D. Lu et al. (Eds.): ISNN 2007, Part I, LNCS 4491, pp. 776 785, 2007. Sprnger-Verlag Berln Hedelberg 2007
Neural Network Based Algorthm for Mult-Constraned Shortest Path Problem 777 routng, f weghted far queung servce dscplne s beng used and the constrants are bandwdth, queung delay, delay-jtter, and loss, then the problem can be reduced to standard shortest path problem by expressng all the constrants n terms of bandwdth. As we can see, ths approach only deals wth specal cases of the problem. Jaffe proposed a new soluton called mult-label routng for the MCSP problem [6]. It s smple, easy to mplement. And most mportantly, t can fnd the approxmate optmal path wth multple constrants. Unfortunately, the computatonal complexty s also exponentally ncreased wth the network scale for the reasons of too many labels to be reserved and too many vectors to be compared. Some modfcatons are suggested for the mult-label method to reduce the computatonal and space complexty [7], e.g., lmtng the total labels of each node, and the algorthmc loops are reduced to about 1/3 of the orgnal mult-label algorthm. However, a large number of labels are needed for the mproved mult-label routng algorthm to ensure the approxmate optmal paths can be found. Therefore, the computatonal and space complexty are not reduced essentally. In ths paper, the Autowave-Competed Neural Network (ACNN) [8,9] s vectorzed and appled successfully to the MCSP problem. The M-paths lmted scheme s adopted to reduce the computatonal and space complexty. All the autowaves propagatng to a neuron have to compete wth the paths reserved on the neuron s threshold, and only M autowaves (.e., paths) can survve the competton. The wnners wll be reserved for the next competton by replacng the old paths on the neuron s threshold, whle the losers wll be dscarded. At the same tme, the wnners wll propagate forward to ts adjacent neuron. A nonlnear cost functon s defned to measure the paths. However, dfferng from the tradtonal MCSP algorthms whch focus on fndng the paths wth mnmum cost functon, the proposed algorthm uses the cost functon to buld on a proportonal path selecton scheme. The dscarded paths can revve wth certan probablty. So the optmal path can also be found n theory by the proposed algorthm even wth small M. Furthermore, the vectoral ACNN based algorthm s parallel n archtecture and n runnng mode. The rest of paper s organzed as follows. Secton 2 gves the defnton of MCSP problems. Secton 3 ntroduces the scalar ACNN for tradtonal shortest path problem (no constrant). The vectoral ACNN, the nonlnear cost functon and the new algorthm of MCSP s defned n secton 4. Smulaton results are presented n secton 5 and some concludng remarks are drawn n secton 6. 2 Problem Defnton Consder a network that s represented by a drected graph G = (V,E), where V s the set of nodes whch represent swtches, routers, and hosts and E s the set of edges whch represent communcaton lnks. Each edge (,j) E s assocated wth a prmary cost parameter c(,j) and K QoS weghts, ω (, j k ), k = 1,2,, K, all parameters are nonnegatve. Gven K constrants, Ck, k = 1,2,, K. The MCSP problem s to fnd a path P from a source node s to the destnaton node d such that [1]:
778 J. Dong, J. Zhang, and Z. Chen () ω ( P) = w (, j) C for k = 1,2,...,K k k k (, j) P () cp ( ) = c (, j) s mnmzed over all feasble paths (, j) P satsfyng () For the smplcty of expresson and computaton, the cost parameter can be regarded as one of the constrants on the lnk, e.g., let ω 0 (, j) = c(, j). Then the MCSP problem s to fnd a path satsfed all K+1 constrants and wth the mnmum cost from node s to node d. To solve the MCSP problem, one can fnd frstly all the paths satsfed the constrants, then chooses the one wth mnmum cost from those paths [6]. Ths paper also treats the cost parameter as a constrant of the lnk. Each lnk n the network s assocated wth multple parameters whch can be roughly classfed nto addtve and non-addtve [10]. For the addtve parameters (e.g., delay, jtter, admnstratve weght), the cost of an end-to-end path s gven by the sum of the ndvdual lnk values along that path. In contrast, the cost of a path wth respect to a non-addtve parameter, such as bandwdth, s determned by the value of that constrant at the bottleneck lnk. It s known that constrants assocated wth non-addtve parameters can be easly deal wth a preprocessng step by prunng all lnks that do not satsfy these constrants [5]. Hence, n ths paper we wll manly focus on addtve QoS parameters and assume that the optmalty of a path s evaluated based on an addtve parameter. (1) 3 ACNN for Shortest Path Problem The Shortest Path (SP) problem s well-documented. Many practcal algorthms have been developed for shortest path problem. Recently, we proposed a neural network model called autowave-competed neural network (ACNN) for the SP problem [8,9]. The ACNN neuron s conssted of three parts,.e., the mnmum selector, the autowave generator and the threshold updater, see Fg.1. w 1 w 2 w j θ u y Fg. 1. Neuron model of ACNN
Neural Network Based Algorthm for Mult-Constraned Shortest Path Problem 779 The ACNN neuron can be descrbed wth the followng equatons, Z ( t) = { j w and y ( t 1) > 0} (2) j j 0 Z( t) = φ u () t = mn ( yj( t 1) + wj) otherwse j Z () t (3) u() t u() t < θ( t 1) y( t) = f[ u( t), θ( t 1)] = (4) 0 otherwse θ ( t 1) y( t) = 0 θ( t) = h[ y( t), θ( t 1)] = (5) y () t otherwse where s the ndex of neuron, t s the tme (or says the teratons). u () t, θ () t and y () t are the nternal actvty, the threshold and the output of neuron at tme t respectvely. w s the connecton weght from neuron to neuron j. j z () t s the set of neurons whch fred at tme t and s reachable to neuron. When appled to the SP problem, an ACNN somorphc to the weghted graph G should be constructed,.e., each node of G s correspondng to a unque neuron of the network, and w s assocated wth the weght of the edge j (, j ) n G, see Fg. 2. All neurons are ntalzed wth nfnte threshold and zero-nternal-actvty. Fre the source neuron to run the network, and the frng would nspre some autowaves (a) (b) Fg. 2. ACNN topology for SP problem. (a) A weghted dgraph. The crcles wth numbers nner are the vertexes, and the numbers on edges are the costs assocated wth the correspondng edges. (b) The ACNN model for the SP problem of the graph shown n (a). The crcles wth numbers nner are the neurons, and the squares wth nner are the summators on the correspondng lnks. propagatng through the whole network. When passed through the neuron, the travelng dstance of an autowave would be recorded on the threshold θ () t f t s the shortest. All neurons decrease ther thresholds progressvely untl the network stops.
780 J. Dong, J. Zhang, and Z. Chen When the network stops, the threshold θ s correspondng to the dstance of the shortest path from the source neuron to the neuron. The ACNN based shortest path algorthm s parallel, non-parameter, and flexble, whch can easly be modfed to sut for the other problems concerned wth shortest path [9]. Furthermore, t s sutable for large-scale network. Reader can refer to the reference [8] for more detals about ACNN. 4 Vectoral ACNN for MCSP Problem 4.1 Vectoral ACNN In order to solve the MCSP problem, a vectoral ACNN must be establshed, n whch the connecton weght, the threshold, the nternal actvty and the output of the neurons should be vectored. (1) The vector-formed connecton weght from neuron to the reachable neuron j should be w j = [ ω (, j), ω (, j),, ω (, )] 1 2 K j, where ω (, j k ), k = 1,, K s the k th QoS parameters on the lnk (, j ). (2) A path ( s j ) can be wrtten as a vector-formed P = ( p, d ), where d = ( d1, d2,, d K ) and p = (, s,, j, ) are the total weghts and the node sequence of the path respectvely, and dk = ωk(, j), k = 1,, K. (, j) P (3) If an autowave wth travelng path P = ( s ) reaches neuron, the m nternal actvty of neuron would be u = P= ( p, d ) f d satsfes the QoS constrants, where m s a temporal label for the autowave. (4) When passed through neuron, the autowave whose travelng path s P may be m recorded on the threshold θ = P= ( p, d), where m s the ndex for path P. (5) Neuron outputs smultaneously all paths n ts threshold to ts neghborhood..e., y = { P= ( p, d ) P θ }. Furthermore, a cost functon must be defned for a path P so that we can make a comparson between two dfferent paths. It s documented that the performance of an MCSP algorthm s closely concerned wth the cost functon. A varety of cost functons have been proposed and dscussed. For example, n [6] the author proposed for a two-constrant problem a cost functon f ( P) = αω1( P) + βω2( P ), then one can search fast for a feasble path usng Djkstra s shortest path algorthm by mnmzng the cost functon f ( P ). However, no performance guarantee for path P s gven wth respect to ndvdual constrants. In [11] the authors proposed an algorthm that dynamcally adjusts the values of α and β wthn a logarthmc number of calls to Djkstra s shortest path algorthm. However the problem s stll unsolved. Some researchers [1] have recently proposed a nonlnear cost functon whose mnmzaton provdes a contnuous spectrum of solutons. In ths paper, the cost functon of a path P s defned as:
Neural Network Based Algorthm for Mult-Constraned Shortest Path Problem 781 f ( P) λ K k dk = k= 1 Ck P s feasble (6) otherwse Where C1, C2,, C are the K QoS constrants, K λk 1 s the sgnfcant coeffcent of the k th QoS parameter. 4.2 Vectoral ACNN Based Algorthm There are two mprovements n vectoral ACNN based algorthm as to the orgnal mult-label algorthm. One s to lmt the total labels of each node. In the orgnal mult-label algorthm, a path can be dscarded only when ts constrants are all worse than others or don t satsfy the QoS constrants. So a large number of paths would be reserved on each node, whch results n an exponental complexty of the algorthm. However, t s proved that the exponental complexty can be reduced to a polynomal one f the total labels of each node are lmted to be no more than M (constant). In our algorthm, a maxmal M thresholds on each neuron s lmted. Another s to select the paths for reservng. Because of the randomcty of the QoS parameters on each lnk, no cost functon can provde an exactly scalar evaluaton for a mult-parameter path. A path P(,) s = ( s ) wth good ftness from the source node s to a mddle node may become feasble when outspread to the destnaton node d and vce versa. So the optmal path mght not be found f we merely reserve paths accordng to ther ftness. The proportonal selecton scheme s a smple and effectve way to solve ths knd of problem, whch s well-documented n the heurstc algorthms, such as genetc algorthms [12]. In the proposed algorthm, a proportonal selecton scheme s used to prevent the optmal soluton from beng dscarded. The proposed algorthm can be descrbed as followng: Step 1: Intalzaton. Let y (0) = φ, θ (0) = φ, u (0) = φ, V. Step 2: Network startng. Let y s (1) = {( s, 0,, 0)}, then fre the source neuron s and run the network. Step 3: Internal actvty calculatng. Calculate the nternal actvty accordng to the followng equaton, u () t = {(( p,), d+ w ) ( p, d) y ( t 1), w 0, j V}. j j j Noted that u () t s a path set whch collects the arrvng autowaves at tme t. Step 4: Thresholds updatng. (1) Combne the two path set u () t and θ ( t 1) nto a new set A () t. (2) Remove from A () t the paths whose dstance components d1, d2,, d are all K worse than others. And remove the unfeasble paths from A () t. (3) Evaluate the paths n the new path set A () t accordng to the Eq.(6).
782 J. Dong, J. Zhang, and Z. Chen (4) Update the threshold θ () t wth M paths (f possble) chosen from A () t wth the probabltes n nverse proporton to ther ftness. Step 5: Autowaves generatng. The neurons generate M autowaves accordng to the paths n ther threshold (or less than M f there are not enough paths n ther threshold), and propagate those autowaves to ther neghborhoods,.e., y() t = { P= ( p, d ) P θ ()} t. Step 6: Network stop condton. Let t = t+ 1. If the optmal path from the source neuron s to the destnaton neuron d s obtaned or t > Maxtme, stop the network, otherwse, go to step 3. The algorthm mentoned-above s called vectoral ACNN. One can have an ntutonstc nterpretaton of the vectoral ACNN as follows: Once produced from the source neuron (see Step 2), the autowaves reproduce themselves and propagate from one neuron to another along wth the lnks (see Step 3). When gatherng n a neuron, the autowaves compete wth each other (see Step 4). The M wnners occupy the neuron s threshold watng for the next competton (see Step 4), and ther duplcates wll propagate to the next neurons for another competton (see Step 5). Ths process s recurrng agan and agan untl the approxmate optmal path s found (see Step 6). The computaton of ths algorthm focuses on the threshold updatng step (.e., Step 4). Assumng the average number of adjacent nodes to be m, and the node number of the network to be n, the algorthm needs to calculate and compare n ( m+ 1) M paths at Step 4 n each algorthmc loop. So the computatonal complexty of the algorthm 2 s On ( ( m+ 1) M) to fnd a feasble path, whch s a polynomal complexty. 5 Computer Smulaton Fg. 3 shows a network wth 10 nodes. Let node 3 be the source, and node 10 be the destnaton. The problem s to fnd a shortest (cheapest) path from node 3 to node 10, whose cost does not exceed 80 and whose constrant does not exceed 60,.e., the QoS request s R=(source=3,destnaton=10,cost 80,constrant 60). Fg. 3. A 10-node dgraph. The two numbers on lnks are the cost and the constrant respectvely, both of whch are addtve.
Neural Network Based Algorthm for Mult-Constraned Shortest Path Problem 783 The vectoral ACNN based algorthm s used to solve the problem. Where the maxmal number of paths reserved on each neuron s K=2, and the sgnfcant coeffcents of the two weghts are r 1 = 2 and r 2 = 1 respectvely (see Eq.(6)). Tables 1 and 2 show the result path sets of the neuron s thresholds at frst fve teratons. Where, a path s denoted as [(s, node 1, node 2, ), (cost, constrant)],.e., the number n frst bracket are the nodes sequence of the path, and the two numbers n second bracket are the cost and the constrant of the path respectvely. Table 1. Path sets on the threshold of dfferent neurons (1~5) at frst fve teratons (t) t Node 1 Node 2 Node 3 Node 4 Node 5 1 φ φ [(3),(0,0)] φ φ 2 φ [(3,2),(14,26)] [(3),(0,0)] [(3,4),(12,8)] φ 3 [(3,4,1),(24,30)] [(3,2,1),(20,56)] [(3,2),(14,26)] [(3),(0,0)] [(3,4),(12,8)] [(3,2,5),(19,39)] [(3,4,5),(20,38)] 4 [(3,4,1),(24,30)] [(3,2,1),(20,56)] [(3,2),(14,26)] [(3),(0,0)] [(3,4),(12,8)] [(3,2,5),(19,39)] [(3,4,5),(20,38)] 5 [(3,4,1),(24,30)] [(3,2,1),(20,56)] [(3,2),(14,26)] [(3),(0,0)] [(3,4),(12,8)] [(3,2,5),(19,39)] [(3,4,5),(20,38)] Table 2. Path sets on the threshold of dfferent neurons (6~10) at frst fve teratons (t) t Node 6 Node 7 Node 8 Node 9 Node 10 1 φ φ φ φ φ 2 [(3,6),(19,8)] φ φ φ φ 3 [(3,6),(19,8)] [(3,6,7),(36,28)] φ φ φ 4 [(3,6),(19,8)] [(3,6,7),(36,28)] [(3,4,1,7),(34,45)] [(3,4,5,8),(45,41)] [(3,2,5,8),(44,42)] [(3,6,7,9),(51,36)] φ 5 [(3,6),(19,8)] [(3,6,7),(36,28)] [(3,4,1,7),(34,45)] [(3,4,5,8),(45,41)] [(3,2,5,8),(44,42)] [(3,6,7,9),(51,36)] [(3,2,5,8,10),(76,56)] [(3,4,5,8,10),(77,55)]
784 J. Dong, J. Zhang, and Z. Chen Tables 1 and 2 show that the optmal soluton (the underlne path on the threshold of neuron 10 at ffth teraton) was found after fve teratons. Ths result path s [(3,2,5,8,10),(76,56)], see the thck path on Fg. 3, whch s better than the result path found by Lu s algorthm [7],.e., the path [(3,4,5,8,10),(77,55)]. Furthermore, comparatve smulaton experments of Jaffe s algorthm [6], Lu s algorthm [7] and our algorthm (the vectoral ACNN based algorthm) are done n dfferent scale networks wth dfferent constrant number. Table 3 shows the teraton number of one constrant problem n the networks wth 10, 100 and 200 nodes. Whle Table 4 shows the results of two constrant problem n dfferent scale networks. The cost and addtve constrants on the lnks of those networks are all produced randomly by computer, whch subject to Gaussan dstrbuton rangng from 1 to 100 wth mean 50 and stand devaton 20. The adjacent nodes number s also random rangng from 1 to 5 wth mean 3. M s set to be 2, 10 and 20 for 10-node networks, 100-node networks and 200-node networks respectvely. For the sake of smplcty, the sgnfcant coeffcents of the constrants λ k are all set to be 1. The parameters of Lu s algorthm are set to be the same to document [7]. Tables 3 and 4 show that the teraton number s dramatcally reduced usng the vectoral ACNN based algorthm. Wth the ncreasng of networks scale, the teraton number ncreases slower than that of the Lu s algorthm. The results are all better than that of Lu s algorthm. Table 3. Iteraton number of one constrants problem n 3 dfferent scale networks method 10 nodes 100 nodes 200 nodes Jaffe s algorthm 9 167 532 Lu s algorthm 5 89 204 Our algorthm 5 76 159 Table 4. Iteraton number of two constrants problem n 3-dfferent scale networks method 10 nodes 100 nodes 200 nodes Jaffe s algorthm 39 716 6113 Lu s algorthm 25 328 1527 Our algorthm 21 262 967 6 Conclusons Mult-constrant qualty-of-servce (QoS) routng, whch s n fact an MCSP problem, wll become ncreasngly mportant as the Internet evolves to support real-tme servces. In ths paper, a vectoral ACNN based algorthm s proposed for MCSP problem, whch can fnd the approxmate optmal path n polynomal tme. Performance of the proposed algorthm s mproved greatly comparng to the orgnal mult-label routng algorthm (.e., Jaffe s algorthm). Frstly, the proposed algorthm
Neural Network Based Algorthm for Mult-Constraned Shortest Path Problem 785 updates all neurons threshold synchronously,.e., t s a parallel algorthm. Secondly, only the maxmal M paths are reserved on the threshold of each neuron, whch reduces greatly the memory space and the computaton of the algorthm needed. Thrdly, although the M-paths lmted scheme may dscard the optmal paths sometmes because of the randomcty of the constrants and the rgdness of the cost functon, the proportonal selecton scheme would afford the optmal autowaves (.e., paths) enough opportunty to revve or to survve the search process. Comparng experments are gven n the paper, and the results show the effcency of the proposed algorthm. In concluson, the proposed algorthm has the characterstcs of parallelsm, effcency and lower computatonal complexty. Acknowledgment Ths work was supported by the Natonal Natural Scence Foundaton of Chna (Grant No. 60574039) and the 863 Project of Natonal Mnstry of Scence and Technology (Grant No. 2006AA03A175). References 1. Korkmaz, T., Krunz, M.: Mult-constraned optmal path selecton. The 20th Annual Jont Conference of the IEEE Computer and Communcatons Socetes 2 (2001) 834-843. 2. Xu, D., Chen, Y., Xong, Y., Qao, C.: On the Complexty of and Algorthms for Fndng the Shortest Path Wth a Dsjont Counterpart, IEEE/ACM Trans. Networkng.14 (2006) 147-158 3. Wang, Z., Croweroft, J.: Qualty-of-servce routng for supportng multmeda applcatons. IEEE J. Select. Area. Commun.14 (1996) 1219-1234 4. Ja, Z., Varaya,P.: Heurstc Methods for Delay Constraned Least Cost Routng Usng k- Shortest-Path, IEEE Trans. AC.17 (2006) 707-712 5. Dumtrescu, I., Boland, N.: Improved Preprocessng, Labelng and Scalng Algorthms for the Weght-Constraned Shortest Path Problem, Networks 42 (2003) 135-153 6. Jaffe, J.M.: Algorthm for fndng paths wth multple constrants, Networks 14 (1984) 95-116 7. Lu, J., Nu, Z., Zheng, J.: An mproved routng algorthm subject to multple constrants for ATM networks. ACTA ELECTRONICA SINICA 27 (1999) 4-8 (In Chnese) 8. Dong, J., Wang, W., Zhang, J.: Accumulatve competton neural network for shortest path tree computaton. Internatonal Conference on Machne Learnng and Cybernetcs,Vol.III, X an Chna (2003) 1157-1161 9. Dong, J., Zhang, J.: Accumulatng Competton Neural Networks based Multple Constraned Routng Algorthm. Control and Decson 19 (2004) 751-755 10. Wang, Z.: On the complexty of qualty of servce routng. Informaton Processng Letters 69 (1999) 111-114 11. Korkmaz, T., Krunz, M., Tragoudas, S.: An Effcent Algorthm for Fndng a Path Subject to Two Addtve Constrants. Proceedngs of the ACM SIGMENTRICS 1 (2000) 318-327 12. Gelenbe, E., Lu, P., Lane, J.: Genetc algorthms for route dscovery. IEEE Trans. on SMC--Part B 99 (2006) 1247 1254