Inter-domain routing for CommunicationNetworks Using Hierarchical Hopfield Neural Networks Hitalo O. Silva, Carmelo J. A. Bastos-Filho Polytechnic School of Pernambuco, University of Pernambuco Recife, Brazil Email: carmelofi
[email protected] Abstract—The establishment of routes between domains is one of the biggest challenger in the area of communication networks. This fact is because routing in networks with thousands of domains and millions nodes must be quick and acurate. To maintain Quality of Service (QoS). In this paper, we propose the inter-domain routing model and algorithm for communication network using Hopfield Neural Network (HNN). The proposed model restructure the neurons of HNN in a hierarchical manner. This new structure causes drastic decrease in the number of neurons and synapses of the neural network and in the time of establishment of routes. Index Terms—Routing Algorithms, Communication Net- works, Hopfield Neural Networks, Artificial Neural Networks. I. INTRODUCTION The dissemination of personal computers and smart- phones has generated an increasement on the demand for high data transmission networks. This occurs due to appli- cations that handle various types of media, such as social networks. In order to support these networks, new data transmission technologies, efficient routing protocols and robust equipments have being proposed. Service Providers (SPs) have incorporated some technologies in order to provide high quality services, minimize operational costs and generate a competitive advantage in the market. These adaptations in the network technology may require major changes in the network infrastructure of service providers. On the other hand, reliability is essential to achieve high throughput with a reasonable quality of service (QoS). Reliability is obtained by means of robustness. Robustness can be achieved by combining some different aspects, such as: redundant equipments and links; and adaptive routing protocols. The routing protocols must present some desired characteristics, such as fast response and scalability with the number of network nodes. In general, these characteristics lead to a better network performance. The establishment of routes between domains is one of the biggest challenges in the area of communication networks. Several algorithms and protocols have been proposed in the literature and standards. Among these algorithms, the most widely used is the Border Gateway Protocol (BGP). BGP is applied to establish routes in the Internet domain. Routes are defined based on the quantity of hops between Autonomous Systems (ASs) according to a pre-defined network policy. Despite the BGP qualities, there are some well known problems regarding the use of BGP algorithm. Among them, we can cite: the routing tables are large; different routes can be obtained for the same source-destination pair generating instability; sub-optimal routes can arise during the network operation; and the time needed to achieve convergence can be extremely high when the number of nodes increases. Computational Intelligence [1] techniques have being used for routing in communication networks. Some exam- ples are: Ant Colony Optimization (ACO) [2]–[4], Genetic Algorithms (GA) [5], [6] and Hopfield Neural Networks (HNN) [7]–[9]. HNN is a type of recurrent neural net- work used for routing in communication networks [10]– [13]. HNNs have been deployed for routing because of its efficiency and adaptability. Adaptability is a highly desirable feature for routing, since communication networks can change during the operation, leading to a dynamic environment. Besides, HNNs do not need a training process. However, the response time of HNN to provide a route between a pair of edge nodes is still higher than other widely applied routing algorithms, such as the Dijkstra’s algorithm [14]. Although some interesting approaches based on HNNs were proposed in the literature, they were not developed to be applied in networks with multiple domains and with a large number of nodes. Our hypothesis is that HNN models that present these two characteristics can be used in wide networks, such as the Internet. Therefore, we propose in this paper a hierarchical HNN model to mitigate these current limitations of the already proposed HNNs. The remainder of this paper is organized as follows: Section II presents the background concepts regarding HNN; Section III presents our proposal; Section IV shows the simulation setup and some preliminary results; and Section V presents the conclusions. II. STATE-OF-ART Hofield Neural Networks (HNN) is a kind of recurrent artificial neural network [17]. The processing elements are the neurons which are full-connected. This means that the output V i of each neuron is connected to the inputs of all other neurons via synaptic weights T ij , producing the change of the input neuron U i , and thus changing the network state. If the system is stable in Liapunov sense, the network reaches some minim of system energy. The convergence is detected when successive iterations lead to smaller and smaller output changes. Furthermore, each neuron can be externally excited by an input bias U i [18]. The HNN block diagram of is depicted in Figure 1. Fig. 1. Hopfield Neural Network Structure (Extracted of [18]). To solve routing problems in communication networks the HNN is organized in a (n×n) matrix, where n is the number of nodes in the network topology. Each element in the matrix is represented by a neuron. Each neuron is associated to each link between two adjacent nodes in the communication network; this neuron is described by xi (link from node x to node i). All diagonal elements in the matrix are zero, because one node can not be connected to itself. Therefore the computational network requires n(n − 1) neurons. The output of each neuron, described by V xi , depends on its input U xi via the sigmoidal function as in [19]. The parameter λ xi determines the shape of the sigmoidal function and significantly influences the computation time to convergence and the correctness of the algorithm. In our simulations we used the same λ xi for all neurons in the HNN. Hence, we will refer to the parameters λ xi as λ in the remaining of the paper. A communication network topology can be described by the undirected graph G = (N, A), where N is the number of nodes (the vertices on the topology), and A is a set of links (arcs or edge). ρ xi is the matrix that defines whether the arc xi exists in the topology of the communication network used in the simulation. If the arc xi does not exist then ρ xi = 1, otherwise ρ xi = 0. The cost matrix C xi represents a nonnegative cost associated to every link in the network. This cost will be zero for nonexisting links and for elements C ii , because one node can not be connected to itself. The goal of the HNN is to find the path that minimizes the total cost between two nodes (source-destination nodes). Therefore, the HNN should indicate the best path as an ordered sequence of nodes connecting source to destination. In this paper, we used the same energy function give in [20], whose minimization process drives the neural network into its lowest energy state, this stable state corresponds to the routing solution. The energy function of HNN for routing is described as following: E = µ 1 2 n x=1 (x,i)=(d,s) n i=1 i=x C xi V xi + µ 2 2 n x=1 (x,i)=(d,s) n i=1 i=x ρ xi V xi + µ 3 2 n x=1 n i=1 i=x V xi − n i=1 i=x V ix 2 + µ 2 2 n x=1 n i=1 i=x V xi (1 −V xi ) + µ 5 2 (1 −V ds ), (1) where C xi is the cost matrix, ρ xi is the topology matrix and µ 1 , µ 2 , µ 3 , µ 4 and µ 5 are constants. µ 1 minimizes the total cost of a path by taking into account the cost of existing links; µ 2 prevents nonexistent links from being included in the chosen path; µ 3 is zero for every node in the valid path; µ 4 forces the HNN to converge to a stable state and µ 5 is introduced to ensure that the source and the destination nodes belong to the solution. The input of each neuron U xi depends on the sum of all the outputs of the neurons of the HNN multiplied by a factor represented by the synaptic weights matrix T xi,yj , the external bias I xi and also by the influences of the previous U xi . The T xi,yj element represents the synaptic weights connecting the output of the neuron yj to the sum point in the input of the neuron xi. The bias and the synaptic weights of the HNN can be set as: I xi = µ 1 2 C xi (1 −δ xd δ is ) − µ 2 2 ρ xi (1 −δ xd δ is ) − µ 4 2 + µ 5 2 δ xd δ is (2) T xiyj = µ 4 δ xy δ iy −µ 3 δ xy −µ 3 δ ij −µ 3 δ yx −µ 3 δ iy . (3) When the HNN reaches some minimum in terms of the system energy, i e , the output variations are lower than a threshold value [20], this means that the simulation has converged. Thus, an adjustment is made in each output. If an output has a value greater than ”0.5” it is adjusted to ”1”, otherwise it is adjusted to ”0”. If the final value of V xi = 1, then the arc from node x to node i is in the shortest path, otherwise V xi = 0. To resolve the system, a simple difference equation based on discrete time proposed in [18] is used. The equation used to calculate the next input value of neurons is shown below. U xi [n + 1] = U xi [n] +AU xi [n − 1] +B n y=1 n j=1 j=y T xi,yj V yj +CI xi [n], (4) where U xi [n + 1] is the next input of the neuron xi, calculated based on its own input in previous instants U xi [n] and U xi [n − 1], on the output values of all the neurons of the network V yj [n] and on the external bias I xi [n]. A, B and C are constants that regulate the weight of the previous inputs. the intrinsic parallel behavior of neural networks, en- abled by the asynchronous functioning of neurons, allows one to implement HNNs on parallel processing platforms. Oliveira et al. [15] developed a HNN model for routing in communication networks based on Field Programmable Gate Array (FPGA) platform. The results of implementation demonstrated that the approach using HNN can establish routes faster than others routing algorithms, but the use of logical components increases non-linearly as the number of nodes increases linearly. The Internet is a conglomerate of Autonomous Systems (AS), which are controlled by different administrative au- thorities, interconnected by routing policies. ASs are com- posed by routers. The routers perform Interior Gateway Protocols (IGPs), for example: Open Shortest Path First (OSPF) and Intermediate System-to-Intermediate System (IS-IS), within their domains and are interconnected via Exterior Gateway Protocol (EGP). The current EGP used in the Internet is the Border Gateway Protocol Version 4 (BGPv4), defined in Request For Comment (RFC) 1771 [21], [22]. EGPs were introduced because IGPs do not support network with thousands of nodes and hundreds of thousands of routes. Even because, IGPs were not developed for this purpose. The establishment route between ASs is one of the biggest challenges in the area of communication net- work. The routes are defined based on the quantity of hops between Autonomous Systems (ASs) and network policies. Some problems are known about the BGP, for example: the routing table of routers are extensive; different routes to the same source-destination pair, sub-optimal routes, instability, other [23], [24]. III. HIERARCHICAL HOPFIELD NEURAL NETWORKS To enable inter-domain routing using HNN was necessary to redesign how the routing is performed. For this, the structure of HNN had to be recast. The new format is hierarchically organized and composed for independents HNNs virtually connected. This organization mimic the communication network structure, i.e., mimic the hierarchy of domains/networks, nodes, and links. A network is com- posed for nodes and links. However it may be constituted of other domains, forming a hierarchy. The Internet, for example, is a network of networks. The Figure 2 shows how is constituted a multilayer network. The ”Layer 0” represents the network as a whole. The ”Layer 1” is formed by domains that constitute the network. The inter-domain routing is performed in this layer. Lastly, in ”Layer 2” are the nodes and their connections. Is where the intra-domain routing happens. This reformulation will allow reducing the number of neurons, the convergence time and the establishment of inter-domain routes. For each network from hierarchy of networks there will be a HNN. The inter-domain routing will be performed by an HNN at a higher hierarchical level. This HNN will be formed by the nodes that represent the networks in lower hierarchical level and by the links that interconnect them. The process of establishment of routes occurs the Fig. 2. Formatting a communication network in hierarchical layers. upper layer to the lower layer. The route will be formed by the result of the execution of all HNNs involved in the path. For example, for establish a best route from ”Node 1 Domain 1” to ”Node 4 Domain 4” is necessary to cross all domains (Domains 1-2-3-4). The routing process starts with the inter-domain routing in HNN of Layer 1. In this routing, the source is o domain id (from 1 to 4) because the aim is find the best route between these domains. After that, the routing process is performed in others HNNs involved (HNNs representing the domains 1, 2, 3 and 4). In the routing of the HNN of the ”Domain 1” will have as source the node ”Node 1 Domain 1” and as destination the node ”Node 2 Domain 1”, which is the edge node. And in the routing of the HNN of the ”Domain 2”, will have as source the node ”Node 1 Domain 2” and as destination the node ”Node 3 Domain 2”, which is the edge node. After all routings, the individual routes are pooled and then the entire route between origin and destination is established. When there are over a link connecting two groups in one layer. To decide which link will be used was created Routing Policies. Routing Policies define routing preferences. The pseudocode of algorithm proposed in this paper is presented in Algorithm 1. IV. SIMULATION AND RESULTS With the aim to compare the performance and the struture of neurons and connections of the approaches HNN State- of-Art and HNN proposed in this paper, a experiment was conducted. It consisted of establishing 15 different routes, using the two approaches HNN, in a network topology composed of 16 nodes and 25 links (vide Figure 3). Each route was established 30 times. The structure of HNN State- of-Art is composed for 240 neurons (16 × 15) and 57, 600 connections (240 × 240). While the structure of the new approach is composed for 5 HNNs(one for each domain), 60 neurons {5(4 × 3) : 5 domains with 4 nodes each} and 720 connections {5(12 × 12): 5 HNNs with 12 neurons each}. The algorithms were implemented using the Java [25] programing language, in the Eclipse [26] development environment. In Table VI you can see the number of iterations and o time spent to establish each of the 15 routes using both Algorithm 1 Pseudocode of the routing algorithm using Hierarchical Hopfield Neural Networks. procedure GENERATEBESTPATHBYPAIR(pair) if pair.From.Level = pair.To.Level then currentPair = getMutualParentByPair(pair); CALCULATEROUTE(pair,currentPair); end if end procedure procedure CALCULATEROUTE(pair, currentPair) HNN hnn = getHNNByNode(pair.From.Parent); HNN.CALCULATEROUTEHNN(currentPair); if pair.Level = layer.Level then currentPair = getMutualParentByLayer(layer); pairs = generatePairsByNeurons; (currentPair); for all pairs do CALCULATEROUTE(pair,currentPair); end for end if end procedure procedure CALCULATEROUTEHNN( pair , layer ) Calculate T xi,yj ; Calculate I xi ; Insert noise in U xi ; do Calculate U xi ; Update neurons (U xi and V xi ); while ∆V xi < threshold Calculate V xi (binarization of the V xi ); end procedure Fig. 3. Network topology used to evaluate the performance of HNN State of the Art. approaches of HNN. The HNN Stat-of-Art uses only one neural network to perform routing. Thus, two columns are used to demonstrate the results obtained. On the other hand, the new approach uses 5 HNN to perform routing (vide Figure 4 to identify each HNN). Therefore, to demonstrate the results of the new proposal are used 12 columns: 2 columns for each HNN, one for the number of iterations and the other for the time spent; and 2 totaling the number of iterations and time spent. Obtaining the total time required to establish the route is not performed only through the sum of the times of HNNs. To do this, must be add all execution times to overhead from the algorithm. During the establish of routes using the new approach are not always all HNNs are used. This happens when both nodes belong to same domain (Intra-domain). For example, to establish the routes 00-01,00-02 and 00-03 only HNN 02 is needed, because all Fig. 4. Network topology formatted used to evaluate the performance of the new approach to HNN. TABLE I THIS TABLE SHOWS THE QUANTITY OF ITERATIONS NEEDED FOR THE PROPOSED APPROACHE TO CONVERGE. Pair HNN 01 HNN 02 HNN 03 HNN 04 HNN 05 Iterations 0-1 0 591.3 (1.62) 0 0 0 591.3 0-2 0 599.2 (6.28) 0 0 0 599.2 0-3 0 674.2 (6.28) 0 0 0 674.2 0-4 591.3 (1.62) 591.7 (3.77) 565.2 (0.48) 0 0 1748.2 0-5 591 (0.0) 591 (0.0) 591.3 (1.8) 0 0 1773.3 0-6 591 (0.0) 591 (0.0) 599.2 (6.28) 0 0 1781.2 0-7 591 (0.0) 591 (0.18) 674.2 (6.28) 0 0 1856.2 0-8 599.2 (6.28) 591 (0.0) 598.4 (2.33) 591.3 (1.62) 0 2379.9 0-9 598.0 (0.18) 591 (0.0) 598.1 (0.25) 565.2 (0.42) 0 2352.3 0-10 598.1 (0.25) 591 (0.0) 598 (0.0) 593.4 (2.51) 0 2380.5 0-11 598.0 (0.18) 591 (0.0) 598.0 (0.18) 618.2 (6.46) 0 2405.3 0-12 674.2 (6.1) 591 (0.0) 598.0 (0.18) 617 (0.0) 565.4 (0.55) 3045.6 0-13 673.1 (0.34) 591 (0.0) 598.0 (0.18) 616.9 (0.25) 591.3 (1.80) 3070.4 0-14 673.1 (0.25) 591 (0.0) 598 (0.0) 616.9 (0.18) 599.2 (6.28) 3078.3 0-15 673.1 (0.25) 591 (0.0) 598.0 (0.18) 616.9 (0.25) 674.3 (6.27) 3153.3 nodes are contained in the same domain. When a HNN is not used in routing, this is represented in the table by a ”-”. While the route to establish 00-15 we must use the HNNs 5, since the best route through all domains. Upon analyzing the results of the experiment, we found that when using the HNN proposed in this work there was: • An increase in the number of iterations required to converge. The increase in the number of iteration is due to the fact of the new proposal to use more HNNs to perform routing. However, the iterations are faster; • A reduction of 75% in the number of neurons; • A decrease of 98.75% in the number of connections; • Decrease in time spent to establish routes on average 97.34%; • And the emergence of sub-optimal routes. The sub- optimal routes happens when there are more than one link connecting two domains and link to the route selected is the most cost. In summary, the preliminary results indicate that perfor- mance of the HNN proposed in this paper is superior to the other approaches in both time and in the structure of neurons and connections needed to establish the routing. V. CONCLUSIONS In this work were presented the main events that moti- vated the major changes in communication networks and the major challenges that are to come. One of these is to adequate the routing protocols to large demands for TABLE II THIS TABLE SHOWS THE TIME NEEDED FOR THE PROPOSED APPROACH TO CONVERGE. Pair HNN 01 (µs) HNN 02 (µs) HNN 03 (µs) HNN 04 (µs) HNN 05 (µs) Total Time (µs) 0-1 0 1574,14 (354,78) 0 0 0 1587.84 0-2 0 1329,9 (78,51) 0 0 0 1337.00 0-3 0 1600,7 (420,38) 0 0 0 1611.75 0-4 1143,34 (74,4) 1152,69 (69,52) 1106,92 (78,08) 0 0 3478.50 0-5 1149,18 (94,95) 1133,08 (78,13) 1175,77 (102,33) 0 0 3523.92 0-6 1143,48 (86,99) 1171,22 (129,18) 1178,34 (97,95) 0 0 3560.08 0-7 1129,94 (82,77) 1122,78 (66,66) 1327,85 (292,68) 0 0 3649.32 0-8 1169,69 (200,59 1139,30 (128,75) 1131,53 (84,09) 1114,75 (76,10) 0 4631.75 0-9 1121,34 (57,10) 1124,34 (89,90) 1155,46 (182,13) 1065,70 (73,44) 0 4560.05 0-10 1200,70 (301,25) 1232,67 (355,49) 1151,41 (107,35) 1156,82 (144,04) 0 4827.10 0-11 1179,14 (218,56) 1122,74 (106,44) 1185,16 (263,81) 1765,49 (2776,83) 0 5453.73 0-12 1278,11 (85,94) 1115,19 (73,88) 1134,60 (73,88) 1173,06 (76,31) 1071,05 (82,76) 5880.18 0-13 1271,63 (96,75) 1105,61 (89,61) 1126,60 (99,47) 1148,21 (72,43) 1129,59 (244,58) 5886.51 0-14 1191,92 (19,60) 1075,00 (69,56) 1110,31 (92,35) 1097,00 (30,79) 1057,64 (25,46) 5625.48 0-15 1204,48 (42,41) 1180,55 (382,84) 1158,12 (311,45) 1170,50 (242,46) 1341,32 (377,96) 6157.61 TABLE III PARAMETERS USED TO SIMULATE. Name Value A 0.001 B 0.001 C 0.001 µ1 950 µ2 2500 µ3 1500 µ4 475 µ5 2500 ∆Vxi 10 −5 noise −0.0002 ≤ Uxi ≤ +0.0002 γ 1 transmission speed, reliability and the great increase in the number of nodes and routes. Then the routing protocol Border Gateway Protocol (BGP), which is applied in the establishment of routes in the Internet. BGP has some problems, for example: large routing tables; oscillation to establish routes, sub-optimal routes and instability. Sub- sequently, were presented the Hopfield Neural Networks (HNN) and how they were adapted for routing of commu- nication networks. Adaptations in HNN make it possible to establish routes adaptively and efficiently using neural networks. However, the performance of this new approach is not comparable to the routing algorithms used, such as Dijkstra, and does not scale linearly whereas the number of nodes increases linearly. The aims of this study were to produce, validate and test a new approach to inter-domain routing for communication networks based on HNN. A new approach to structure the TABLE IV COMPARISON BETWEEN THE RESULTS OF THE TWO ROUTING HNN APPROACHES. Pair HNN Estate-of-Art HNN Proposed Iterations Time (µs) Iterations Time (µs) 0-1 798,30 (1,64) 222748,38 (5917,22) 591,3 (1,6) 1587,84 (371,00) 0-2 830,93 (0,25) 235480,04 (13646,81) 599,2 (6,3) 1337,00 (82,116) 0-3 942,87 (1,43) 248704,09 (7990,36) 674,2 (6,3) 1611,75 (430,32) 0-4 700,73 (4,02) 201948,41 (5579,55) 1748,2 (3,8) 3478,50 (214,75) 0-5 687,80 (4,96) 221269,01 (18580,11) 1773,3 (1,8) 3523,92 (214,75) 0-6 779,57 (37,67) 222271,81 (11393,11) 1781,2 (6,3) 3560,08 (242,11) 0-7 817,27 (6,94) 222253,86 (17273,85) 1856,2 (6,3) 3649,32 (388,38) 0-8 767,33 (3,37) 219040,04 (4615,61) 2379,9 (6,3) 4631,75 (318,82) 0-9 779,93 (1,23) 225457,72 (5794,21) 2352,3 (0,4) 4560,05 (328,28) 0-10 724,93 (0,25) 194914,09 (4852,82) 2380,5 (2,5) 4827,10 (491,86) 0-11 788,50 (2,74) 224520,99 (4267,37) 2405,3 (6,5) 5453,73 (315,10) 0-12 839,60 (3,48) 223339,14 (4070,86) 3045,6 (6,1) 5880,14 (309,60) 0-13 688,70 (3,83) 196348,34 (4024,28) 3070,4 (1,8) 5886,51 (409,14) 0-14 703,27 (6,94) 202100,00 (3743,57) 3078,3 (6,3) 5625,48 (145,32) 0-15 761,77 (0,77) 215702,04 (13386,69) 3153,3 (6,3) 6157,61 (694,52) TABLE V COMPARISON BETWEEN THE RESULTS OF THE TWO ROUTING HNN APPROACHES TO STABLISH 240 DIFFERENT ROUTES 30 TIMES. HNN Estate-of-Art HNN Proposed Iterations Time (µs) Iterations Time (µs) 195852,13 (11,85) 55466177,26 (2030915,34) 1895,53 (777,85) 867264,51 (22082,91) T A B L E V I C O M P A R I S O N B E T W E E N T H E R E S U L T S O F T H E T W O R O U T I N G H N N A P P R O A C H E S . O r d e r S o u r c e D e s t i n a t i o n H N N E s t a t e - o f - A r t H N N P r o p o s e d H N N H N N 0 1 H N N 0 2 H N N 0 3 H N N 0 4 H N N 0 5 T o t a l I t e r a t i o n s T i m e ( m s ) I t e r a t i o n s T i m e ( m s ) I t e r a t i o n s T i m e ( m s ) I t e r a t i o n s T i m e ( m s ) I t e r a t i o n s T i m e ( m s ) I t e r a t i o n s T i m e ( m s ) I t e r a t i o n s T i m e ( m s ) 0 1 0 0 0 1 6 8 8 . 2 5 2 7 7 . 4 2 6 0 0 4 - - - - - - - - 6 0 0 7 0 2 0 0 0 2 7 0 6 . 1 6 2 3 4 . 7 1 6 3 3 3 - - - - - - - - 6 3 3 6 0 3 0 0 0 3 7 7 3 . 0 6 2 0 5 . 1 9 7 0 8 4 - - - - - - - - 7 0 8 6 0 4 0 0 0 4 7 1 0 . 7 0 1 9 0 . 7 4 6 0 0 4 5 6 4 2 6 1 2 1 - - - - 1 7 7 6 7 0 5 0 0 0 5 7 2 4 . 5 8 2 3 0 . 7 4 5 9 1 2 6 0 0 2 5 9 1 1 - - - - 1 7 8 2 5 0 6 0 0 0 6 7 3 3 . 4 5 2 1 7 . 9 4 5 9 1 1 6 3 3 1 5 9 1 2 - - - - 1 8 1 5 5 0 7 0 0 0 7 7 7 3 . 8 4 2 0 5 . 7 7 5 9 1 1 7 0 8 1 5 9 1 1 - - - - 1 8 9 0 3 0 8 0 0 0 8 8 1 8 . 3 5 2 2 5 . 0 6 6 3 3 1 6 0 0 1 6 1 1 2 5 9 1 1 - - 2 4 3 5 6 0 9 0 0 0 9 7 8 6 . 1 3 2 3 0 . 8 7 5 9 8 1 5 6 6 1 5 9 8 2 5 9 1 2 - - 2 3 5 3 7 1 0 0 0 1 0 8 3 9 . 5 8 2 3 4 . 1 9 5 9 8 2 6 0 8 1 5 9 8 1 5 9 1 2 - - 2 3 9 5 7 1 1 0 0 1 1 8 3 9 . 1 3 2 3 8 . 8 7 5 9 8 1 6 5 3 1 5 9 8 1 5 9 1 1 - - 2 4 4 0 5 1 2 0 0 1 2 8 4 3 . 3 2 2 2 3 . 6 4 7 0 7 1 5 6 4 2 6 1 7 1 5 9 8 1 5 9 1 1 3 0 7 7 6 1 3 0 0 1 3 8 8 1 . 2 9 2 3 3 . 4 8 6 7 3 2 6 0 1 1 6 1 7 1 5 9 8 1 5 9 1 1 3 0 8 0 6 1 4 0 0 1 4 8 8 5 . 1 9 2 2 5 . 9 0 6 7 3 2 6 3 3 2 6 1 7 1 5 9 9 1 5 9 1 2 3 1 1 3 8 1 5 0 0 1 5 9 2 0 . 2 6 2 2 5 . 3 5 6 7 3 1 7 0 8 1 6 1 7 2 5 9 8 1 5 9 1 1 3 1 8 7 6 HNN hierarchically. 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